An archive of previous A&C seminars. Use the search box to filter by speaker, institution, title, or year.

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2026

August 12th, 2026: Computing with Full Memory in 2026

A&C Seminar [video]

Speaker: Ian Mertz, Charles University

Abstract: Catalytic computing, the study of using full memory as a resource in space-bounded computation, has seen a resurgence of interest in the past few years, with new techniques, such as the compress-or-random paradigm, as well as applications, most notably the breakthrough by Williams on time versus space. More recently, there has been an emerging algorithmic direction within the field as well, where the goal is to solve basic primitives, such as graph connectivity, in a time-space efficient manner by adding the power of catalytic memory. Furthermore, catalytic space as a resource has now been studied in various settings beyond the usual machine model, such as streaming and communication complexity.

We will survey such new directions in catalytic computing, with a focus on a few elementary algorithms which illustrate and exemplify these trends.

Bio: Ian Mertz is a postdoctoral researcher at the Computer Science Institute (IUUK) at Charles University in Prague. His work revolves around catalytic computing, a branch of space-bounded algorithms dealing with the use of full memory as a computational resource, as well as the study of how the complexity of problems compose over many instances. He received a B.Sc./B.A. from Rutgers University in 2016, an M.Sc. from University of Toronto in 2018, and a Ph.D. in computer science from University of Toronto in 2022, and has since held positions at University of Warwick and Charles University

August 10th, 2026: Geometric Distances for Curves and Graphs: From Matching to Simplification

A&C Seminar [video]

Speaker: Carola Wenk, Tulane University

Abstract: Discrete geometric shapes lie at the core of many applications and have enjoyed thorough theoretical treatment in computational geometry. This talk will focus on continuous one-dimensional shapes, in particular curves and embedded graphs, and on Fréchet-type distances for comparing them. We will discuss how to compute these distances efficiently, how to match shapes under transformations such as translations and rigid motions, and how global simplification can produce lower-complexity shapes while guaranteeing a bound on their distance to the original.

August 5th, 2026: List Decoding Expander-Based Codes

A&C Seminar [video]

Speaker: Madhur Tulsiani, TTIC

Abstract: We will talk about a new framework based on graph regularity lemmas, for list decoding and list recovery of codes based on spectral expanders. These constructions often proceed by showing that the distance of local constant-sized codes can be lifted to a distance lower bound for the global code. The regularity framework allows us to similarly lift list size bounds for local base codes, to list size bounds for the global codes.

These methods yield novel combinatorial bounds and algorithmic results for list decoding and list recovery up to optimal radius, for Tanner codes of Sipser and Spielman, and for the distance amplification scheme of Alon, Edmonds, and Luby. Moreover, using existing algorithms for computing weak-regularity decompositions of sparse graphs, these tasks can be performed in (randomized) near-linear time.

Based on joint work with Shashank Srivastava.

July 16th, 2026: Paintability of Bipartite Graphs

A&C Seminar

Speaker: Annie Zeng, UIUC

Abstract: The list coloring problem on graphs is where each vertex has a list of colors, and the graph must be properly colored with each vertex only using colors from its list. The paintability game is the on-line version of list-coloring, where for each color an adversary reveals which vertices contain that color in their list, and one needs to make an on-the-spot decision about which vertices to use the color on. Due to the challenging nature of this on-line problem, many graph paintability bounds are significantly weaker than the corresponding off-line bounds for graph coloring. In 2023, Bernshtyen and Lee introduced a graph parameter known as weak degeneracy, based off of standard graph degeneracy, which serves as an upper bound for the list sizes needed in graph paintability and the related parameter of DP-Paintability. In this talk, we will demonstrate how a modified version of weak degeneracy can be utilized on a bipartite graph to obtain an upper bound for paintability and DP-Paintability. This talk is based on joint work with Peter Bradshaw

July 15th, 2026: Container Lemmas and the Query Complexity of Graph Property Testing

A&C Seminar [video]

Speaker: Cameron Seth, University of Waterloo

Abstract: Recent results have shown that the graph container method is a useful tool for analyzing property testing algorithms for a number of classic properties including k-colorability, rho-clique and satisfiability. In this talk we will demonstrate that there is a deeper connection between the container method and graph property testing. For hereditary properties, we relate one-sided testability to the existence of a type of container lemma, and we obtain quantitative bounds in both directions. As applications, we obtain closure results for partition properties and new testing results for graph and digraph properties such as the dichromatic number.

June 24th, 2026: An Improved Construction of Variety-Evasive Subspace Families

A&C Seminar [video]

Speaker: Robert Andrews, University of Waterloo

Abstract: Variety-evasive subspace families, introduced by Guo in 2020, are a primitive in the theory of “linear-algebraic pseudorandomness” that simultaneously generalizes both hitting sets and lossless rank condensers. Roughly speaking, these are collections of vector subspaces such that for any algebraic variety V, some subspace in the family is in general position with respect to V.

It is easy to construct small variety-evasive subspace families: with high probability, a random collection of subspaces will be variety-evasive. In this talk, I will describe a new explicit and elementary construction of small variety-evasive subspace families. This construction improves on the size of the evasive subspace families constructed by Guo and, in some regimes, comes within a polynomial factor of Guo’s lower bound on the size of such subspace families.

Based on joint work with Abhibhav Garg.

June 17th, 2026: A Strong Linear Programming Relaxation for Weighted Tree Augmentation

A&C Seminar

Speaker: Nathan Klein, Boston University

Abstract: In the weighted tree augmentation problem (WTAP) we are given a tree T and a set of weighted edges called links. The goal is to select a minimum weight set of links to add to the tree so that it becomes 2-edge-connected. This is a classic NP-Hard network design problem: phrased another way, it asks to increase the edge fault tolerance of a tree from 0 to 1.

Until the recent exciting work of Traub and Zenklusen, which culminated in a local-search based 1.5+epsilon approximation in 2023, it was unknown whether it was possible to beat a factor of 2 for WTAP. In this talk, I will discuss a 1.49 approximation algorithm obtained by rounding a strong LP relaxation. This goes below the integrality gap of 1.5 of the cut LP, a standard relaxation for WTAP, and is the first LP-based algorithm to get a factor below 2. The LP we round is similar to the standard one strengthened with constantly many rounds of Sherali-Adams, however we add additional constraints to help control its behavior.

Based on join work with Vincent Cohen-Addad, Marina Drygala, and Ola Svensson.

June 16th, 2026: Alan Turing and me

Joint A&C Seminar and Distinguished Lecture Series [video]

Speaker: Gilles Brassard, Université de Montréal

Abstract: Alan Turing is one of the greatest mathematicians of the 20th century, as well as the founding father of computer science. I am therefore very proud to have received the ACM award that bears his name. However, rather than talk about the work that earned me this accolade, I’m going to tell you a surprising story: in Turing’s foundational 1936 paper, in which he introduced what is known today as the Turing machine, he gave an inappropriate definition for computable real numbers. Indeed, according to his definition, multiplication by three and addition would not be computable! But alas, even with the contemporary standard definition of computable real numbers, testing for equality to zero remains undecidable. All of this is related to the fascinating intuitionistic mathematics of Brouwer, about which I shall not presume any prior knowledge.

This is joint work with Sophie Berthelette and Xavier Coiteux-Roy (https://doi.org/10.1016/j.tcs.2024.114573)

June 10th, 2026: Follow-the-Perturbed-Leader with Between-Action Dependence

A&C Seminar

Speaker: Jeffrey Negrea, University of Waterloo

Abstract: Follow-the-Perturbed-Leader (FTPL) algorithms offer a computationally efficient approach to online learning. An FTPL-based learner plays the best response to a randomly perturbed history. However, existing analyses of FTPL rely on strong “between-action” independence assumptions for the perturbation distribution, limiting their applicability when independence is not appropriate. We introduce a framework for analyzing Gaussian FTPL in the context of full-information online learning when the perturbation distribution exhibits between-action dependence. We apply our framework to two regimes: (i) infinite action spaces under bounded Lipschitz rewards, using Gaussian process perturbations, and (ii) linear polyhedral games. We demonstrate how to tightly account for dependence between actions in FTPL regret bounds, and we present an ansatz for the selection of the perturbation distribution.

May 27th, 2026: Robust Statistical Estimators with Bounded Empirical Sensitivity

A&C Seminar [video]

Speaker: Argyris Mouzakis, University of Waterloo

Abstract: We introduce a new measure of robustness for statistical estimators, which we call \emph{empirical sensitivity}. An estimator $\hat \theta$ has bounded empirical sensitivity if, with high probability over a dataset $X = (X_1, \dots, X_n) \sim \mathcal{D}^{\otimes n}$, for any dataset $Y$ obtained by modifying at most $\eta n$ points in $X$, we have that $\hat \theta(Y)$ is close to $\hat \theta(X)$. We study bounds on this quantity for the prototypical problem of Gaussian mean estimation. We prove new lower bounds, showing that for any estimator $\hat \mu$ which achieves an optimal $\ell_2$-error bound of $O(\sqrt{d/n})$, the empirical sensitivity is at least $\Omega(\eta + \sqrt{\eta d/n})$. The two terms arise due to obstructions on the mean and variance (via an Efron-Stein argument) of such an estimator. We show that this bound is tight up to logarithmic factors, by employing recent results for robust empirical mean estimation.

May 20th, 2026: Computing GCD of univariate polynomials in constant parallel time (over any characteristic)

A&C Seminar [video]

Speaker: Shanthanu Rai, TIFR

Abstract: The Euclidean algorithm for computing the greatest common divisor (GCD) is arguably the oldest non-trivial algorithm in print. Originally described for integers, it also applies to univariate polynomials. While efficient, the Euclidean algorithm is inherently sequential. Can it be implemented efficiently in parallel for polynomials? In their seminal work, Andrews and Wigderson answered this affirmatively: they showed that the GCD of two univariate polynomials can be computed in constant parallel time in the arithmetic PRAM model. Equivalently, in algebraic-complexity terms, they proved that the GCD can be computed in (piecewise) algebraic circuits of constant depth and polynomial size. However, their results hold only over fields of characteristic zero and fields of large (polynomially bounded) characteristic.

In this work, we extend their result to all sufficiently large fields, regardless of characteristic. The key ingredient is an analogue of the fundamental theorem of symmetric polynomials for constant-depth algebraic circuits, strengthening a result of Bläser and Jindal for unbounded-depth circuits. Our approach crucially uses the factor closure property of constant-depth circuits, established in our earlier work.

Based on joint work with Somnath Bhattacharjee, Mrinal Kumar, Varun Ramanathan, Ramprasad Saptharishi and Shubhangi Saraf.

May 13th, 2026: Adversarial Robustness on Insertion-Deletion Streams

A&C Seminar

Speaker: Elena Gribelyuk, Princeton

Abstract: We study adversarially robust algorithms for insertion-deletion (turnstile) streams, where future updates may depend on past algorithm outputs. While robust algorithms exist for insertion-only streams with only a polylogarithmic overhead in memory over non-robust algorithms, it was widely conjectured that turnstile streams of length polynomial in the universe size n require space linear in n. We refute this conjecture, showing that robustness can be achieved using space which is significantly sublinear in n. Our framework combines multiple linear sketches in a novel estimator-corrector-learner framework, yielding the first insertion-deletion algorithms that approximate: (1) the second moment F_2 up to a (1+eps)-factor in polylogarithmic space, (2) any symmetric function F with an O(1)-approximate triangle inequality up to a 2^{O(C)} factor in O(n^{1/C}) * S(n) bits of space, where S is the space required to approximate F non-robustly; this includes a broad class of functions such as the L_1-norm, the support size F_0, and non-normed losses such as the M-estimators, and (3) L_2 heavy hitters. For the F_2 moment, our algorithm is optimal up to poly(log n/eps) factors. Given the recent results of Gribelyuk et al. (STOC, 2025), this shows an exponential separation between linear sketches and non-linear sketches for achieving adversarial robustness in turnstile streams.

Based on joint work with Honghao Lin, David Woodruff, Huacheng Yu, and Samson Zhou.

May 6th, 2026: A Truly Subcubic Combinatorial Algorithm for Induced 4-Cycle Detection

A&C Seminar

Speaker: Shyan Akmal, MPI

Abstract: One of the most basic questions about finding patterns in data is the Induced Subgraph Detection problem. In this problem, we are given a small pattern graph and a large host graph, and are tasked with determining if the host contains the pattern as an induced subgraph. A dream goal of fine-grained complexity is to fully classify those patterns for which Induced Subgraph Detection is easy to solve, and those for which it becomes difficult. In this talk, we present some recent, surprising progress on this classification question, centered around a particularly suspicious pattern known as the 4-cycle.

Bio: Shyan Akmal is a Klaus Tschira Boost fellow and postdoctoral researcher at the Max Planck Institute for Informatics, working on problems in graph algorithms and algebraic methods in computer science. Previously, he was a Siebel scholar at MIT, where he completed his PhD under the supervision of Virginia Vassilevska Williams and Ryan Williams.

April 29th, 2026: Spectral Clustering in Birthday Paradox Time

A&C Seminar

Speaker: Weronika Wrzos-Kaminska, EPFL

Abstract: Given a vertex in a (k,φ,ϵ)-clusterable graph, i.e. a graph whose vertex set can be partitioned into a disjoint union of φ-expanders of size ≈n/k with outer conductance bounded by ϵ, can one quickly tell which cluster it belongs to?

This question goes back to the expansion testing problem of Goldreich and Ron'11. In the case k=2, a sample of ≈n^{1/2 +O(ϵ/φ^2)} logarithmic length walks from a given vertex suffice to determine its cluster, by the birthday paradox. The study of the general case k>2 was initiated by Czumaj, Peng and Sohler [STOC'15], and the works of Chiplunkar et al. [FOCS'18], Gluch et al. [SODA'21] showed that ≈poly(k)⋅n^{1/2+O(ϵ/φ^2)} random walk samples suffice for general k. This matches the k=2 result up to polynomial factors in k, but creates a conceptual inconsistency: if the birthday paradox is the guiding phenomenon, then the query complexity should decrease, as opposed to increase, with the number of clusters k!

We design a novel representation of vertices in a (k,φ,ϵ)-clusterable graph using a mixture of logarithmic length walks. This representation uses the optimal ≈(n/k)^{1/2+O(ϵ/φ^2)} walks per vertex, and allows for a fast nearest neighbor search. This gives a clustering oracle with query time ≈(n/k)^{1/2+O(ϵ/φ^2)} and space complexity k⋅(n/k)^{1/2+O(ϵ/φ^2)}, matching the birthday paradox bound.

Based on joint work with Michael Kapralov and Ekaterina Kochetkova.

April 22nd, 2026: Improved Lower Bounds for Privacy under Continual Release

A&C Seminar [video]

Speaker: Bardiya Aryanfard, ISTA

Abstract: In the continual observation model of differential privacy, problems are generally considered easy if they admit an additive error polylogarithmic in the stream length T and the universe size n. Conversely, problems that require additive error polynomial in n and T are considered difficult. Recently, Raskhodikova and Steiner (PODS ‘25) proved polynomial lower bounds on the additive error of many graph problems under fully dynamic edge differential privacy. This raises a natural question: are these problems difficult even in the insertions-only model, or does their hardness arise strictly from the fully dynamic setting? We show that for many problems, the former is true. We prove polynomial lower bounds for a variety of these problems (e.g., maximum matching) in the insertions-only setting. We then extend our techniques to the problem of estimating all symmetric norms simultaneously (SNE), providing the first polynomial lower bound for this problem.

Based on joint work with Monika Henzinger, David Saulpic, and A. R. Sricharan (https://arxiv.org/abs/2512.15981, to appear in PODS ‘26).

April 15th, 2026: Adaptive Robustness of Hypergrid Johnson-Lindenstrauss

A&C Seminar [video]

Speaker: Andrej Bogdanov, University of Ottawa

Abstract: In 1984 W. B. Johnson and J. Lindenstrauss showed that a random projection of an arbitrary point set S into low-dimensional space is approximately distance-preserving, as long as S is of size at most exponential in the target dimension. The resulting embedding has found many uses in data science.

If S is larger than exponential, however, its points can contract arbitrarily under the projection.

We give evidence that when S is the n-dimensional hypergrid of integral points with bounded infinity-norm, the task of finding a contracting pair of points exhibits a computational-statistical gap. In a certain parameter range, contracting pairs are abundant, but finding such a pair is computationally infeasible.

As a cryptographic application, we show that the rounded Johnson-Lindenstrauss embedding is a robust property-preserving hash function. Such hash functions h compress data while preserving distances between inputs up to some distortion factor, with the guarantee that even knowing h, no computationally bounded adversary can find a pair of points that violates the distortion bound.

The talk is based on joint work with Alon Rosen, Neekon Vafa, and Vinod Vaikutanathan.

April 1st, 2026: On the complexity of constrained reconfiguration and motion planning

A&C Seminar [video]

Speaker: Remy El Sabeh, University of Waterloo

Abstract: Abstarct:Coordinating the motion of multiple agents in constrained environments is a fundamental challenge in robotics, motion planning, and scheduling. A motivating example involves n robotic arms, each represented as a line segment. The objective is to rotate each arm to its vertical orientation, one at a time (clockwise or counterclockwise), without collisions nor rotating any arm more than once. This scenario is an example of the more general k-Compatible Ordering problem, where n agents, each capable of k state-changing actions, must transition to specific target states under constraints encoded as a set of k pairs of directed graphs.We show that k-Compatible Ordering is 𝖭𝖯-complete, even when is planar, degenerate, or acyclic. On the positive side, we provide polynomial-time algorithms for cases such as when k=1 or has bounded treewidth. We also introduce generalized variants supporting multiple state-changing actions per agent, broadening the applicability of our framework. These results extend to a wide range of scheduling, reconfiguration, and motion planning applications in constrained environments.

March 25th, 2026: New Perspectives on Trace Reconstruction

A&C Seminar [video]

Speaker: Paul Valiant, Purdue

Abstract: Trace reconstruction asks: if you repeatedly see randomly deleted versions of an unknown binary string, how many samples do you need to recover the original? Despite a lot of work, the best lower-bounds are polynomial and the best upper-bounds are exponential. Many variants of the model have been introduced in hopes of motivating or revealing new approaches to narrow this gap. Here we discuss perspectives inspired by the *circular* trace reconstruction model introduced by Narayanan and Ren (ITCS 2021), in which traces undergo a random cyclic shift in addition to random deletions. We introduce several new characterizations of the “low order statistics” of a trace, the most surprising being a Fourier-based analysis that shows that two sparse strings x,y must *always* differ in some statistic of order at most 6, leading to an n^6 algorithm for the sparse circular setting.

March 18th, 2026: Collaboration Opportunities with the National Institute of Informatics, Tokyo

A&C Seminar

Speaker: Emanuel Planas, National Institute of Informatics, Tokyo

Abstract: This seminar introduces the National Institute of Informatics (NII) and opportunities for international cooperation with NII.

NII is the ROIS Inter-University Research Institute for Advanced Research on Informatics and Global Science Data Infrastructure support, situated in the heart of Tokyo, Japan. NII is ranked top 2 for publications in Computer Science in Japan. Research activity is divided in four Departments: 1.Principles of Informatics: Mathematical Informatics, Intelligent Informatics, Quantum Information, Algorithmics, Machine Learning Theory, Robotics 2.Information Systems Architecture Science: Network Architecture, Computer Architecture, Information Network, Software Architecture, Software Engineering 3. Digital Content and Media Sciences: Natural Language Processing, Human and Knowledge Media, Pattern Media, Image processing, Fondations of Content Management Information and Society: Information Use, Privacy, Human and Knowledge Media, Science Information, Information Public Policy.

The Global Liaison Office (GLO) at NII has established partnerships with more than 115 MOU partners. GLO supports international exchanges via the “MOU Grant Program” under which a NII researcher can invite a colleague from overseas for a stay, or send some member of his team for a research visit (up to 1.5 months, around 40 researches a year supported). GLO also offers the “NII International Internship Program” with which we welcome and fund MOU partners Master’s and Ph.D. students for a research stay in one of our teams (3 to 6 months, around 140 students per year).

March 11th, 2026: Hilbert’s Nullstellensatz is in the Counting Hierarchy

A&C Seminar [video]

Speaker: Abhibhav Garg, University of Waterloo

Abstract: We show that Hilbert’s Nullstellensatz, the problem of deciding if a system of multivariate polynomial equations has a solution in the algebraic closure of the underlying field, lies in the counting hierarchy. More generally, we show that the number of solutions to a system of equations can be computed in polynomial time with oracle access to the counting hierarchy. Our results hold in particular for polynomials with coefficients in either the rational numbers or a finite field. Previously, the best-known bounds on the complexities of these problems were PSPACE and FPSPACE, respectively. Our main technical contribution is the construction of a uniform family of constant-depth arithmetic circuits that compute the multivariate resultant.

March 4th, 2026: Query-Efficient Locally Private Hypothesis Selection via the Scheffe Graph

A&C Seminar [video]

Speaker: Matthew Regehr, University of Waterloo

Abstract: We propose an algorithm with improved query-complexity for the problem of hypothesis selection under local differential privacy constraints. Given a set of k probability distributions Q, we describe an algorithm that satisfies local differential privacy, performs ~O(k^3/2) non-adaptive queries to individuals who each have samples from a probability distribution p, and outputs a probability distribution from the set Q which is nearly the closest to p. Previous algorithms required either Ω(k^2) queries or many rounds of interactive queries.

Technically, we introduce a new object we dub the Scheffé graph, which captures structure of the differences between distributions in Q, and may be of more broad interest for hypothesis selection tasks.

https://arxiv.org/abs/2509.16180

February 25th, 2026: Derandomizing Matrix Concentration Inequalities from Free Probability

A&C Seminar [video]

Speaker: Robert Wang, University of Waterloo

Abstract: Recently, sharp matrix concentration inequalities were developed using the theory of free probability. These results lead to a powerful a new framework for analyzing random matrices, with many applications in algorithm analysis, statistical inference, and graph theory. In this work, we design polynomial time deterministic algorithms to construct outcomes that satisfy the guarantees of these inequalities. As direct consequences, we obtain polynomial time deterministic algorithms for the matrix Spencer problem and for constructing near-Ramanujan graphs. Our proofs show that the concepts and techniques in free probability are useful not only for mathematical analyses but also for efficient computations.

February 18th, 2026: Better Bounds for Semi-Streaming Single Source Shortest Paths

A&C Seminar [video]

Speaker: Janani Sundaresan, University of Waterloo

Abstract: In the semi-streaming model, an algorithm must process any $n$-vertex graph by making one or few passes over a stream of its edges using $O(n \text{poly }\log(n))$ words of space, and at the end of the last pass, output a solution to the problem of interest. We focus on approximating the shortest path distance between two known vertices $s$ and $t$, and make progress on this question from both directions.

We give a simple randomized $(1+\epsilon)$-approximation algorithm for any $\epsilon > 0$ which runs in $O( (n/\epsilon) \cdot \log^3(n))$ space and $O((1/\epsilon) \cdot \log^2(n))$ passes. Prior to our work, the best known algorithms used $O((1/\epsilon) \cdot \log^c(n))$ passes for some large unspecified constant $c$.

We also give a lower bound that any algorithm estimating the distance between $s, t$ up to any constant approximation factor requires $\Omega(\log n/\log \log n)$ passes over the edges in semi-streaming. Previously, only constant-pass lower bounds were known and only for small approximation ratios below two. In this talk, however, we only focus on the upper bound result.

Joint work with Sepehr Assadi and Gary Hoppenworth (https://arxiv.org/abs/2507.17841)

February 11th, 2026: Pointer chasing in fewer rounds is even harder

A&C Seminar [video]

Speaker: Parth Mittal, University of Waterloo

Abstract: In the pointer chasing problem in communication complexity, two players are given functions f_A and f_B from [n] to [n] respectively, and wish to compute the last pointer in the sequence v_0 = 1, v_1 = f_A(v_0), v_2 = f_B(v_1), … , v_k obtained by applying f_A and f_B alternately. It is easy to see that we can solve this task in O(k log n) bits of communication if we are allowed k rounds of communication. On the other hand, a series of works ended with Yehudayoff (Combin. Probab. Comput. 2020) showing that any randomized protocol using even 1 fewer round must spend Ω(n / k) communication on this problem. In this talk, I will prove that the (k / 1000)-round randomized communication complexity of the function where the players are required to compute all the pointers in the sequence (rather than just the final one) is Ω(n). The proof uses the gadgetless lifting framework of Mao, Yang, and Zhang (ITCS 25).

February 4th, 2026: Optimal Pure Differentially Private Sparse Histograms in Deterministic Linear Time

A&C Seminar

Speaker: Hao Wu, University of Waterloo

Abstract: We present an algorithm that releases a pure differentially private (under the replacement neighboring relation) sparse histogram for $n$ participants over a domain of size $d \gg n$. Our method achieves the optimal $\ell_\infty$-estimation error and runs in strictly $O(n)$ time in the Word-RAM model, improving upon the previous best deterministic-time bound of $\Tilde{O}(n^2)$ and resolving the open problem of breaking this quadratic barrier (Balcer and Vadhan (2019)).

Arxiv Link: https://arxiv.org/abs/2507.17017v2

January 28th, 2026: Data Attribution in High Dimensions and without Strong Convexity

A&C Seminar

Speaker: Ittai Rubinstein, MIT

Abstract: Data attribution estimates the effect of removing a set of samples from a model’s training set without retraining the model from scratch and are used for interpretability, credit assignment, privacy and more. However, key approaches to data attribution significantly underestimate removal effects in the high-dimensional regime (#params >= Omega(#samples)), and existing theoretical analyses require strong convexity assumptions that rarely hold in practice, even for simple linear probes. In this talk, we will present a correction to the leading approaches to data attribution that improve accuracy in the high-dimensional regime and the first theoretical analysis of these data attribution methods without strong convexity

January 21st, 2026: The Direct-Sum Problem in Communication and Information Complexity

A&C Seminar

Speaker: Pachara Sawettamalya, Princeton

Abstract: Communication complexity, introduced by Yao in the 1970s, studies the amount of communication required to compute a function when the input is distributed among multiple players. Since its inception, the area has been extensively developed and has revealed deep connections with many branches of mathematics such as combinatorics, linear algebra, convex geometry, information theory, etc. It has also become a fundamental tool for proving lower bounds in streaming algorithms, distributed computing, dynamic data structures, and many more related models.

In this talk, I hope to give a fairly-light survey on the evolution of the direct-sum problem in communication complexity, which asks whether computing n independent instances of a function f requires n times the resources needed to compute a single instance of f. As time permits, I will discuss topics such as an early combinatorial approaches, a more-recent information-theoretic approach, and its applications to lower bounds for dynamic data structures. I will also highlight several open questions in this area, including problems that may have been “overlooked” in past work.

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2025

December 10th, 2025: Computing Graph Cuts Privately

A&C Seminar [video]

Speaker: Mina Dalirrooyfard, Morgan Stanley

Abstract: With the increasing availability of publicly shared statistics derived from private datasets, safeguarding users’ personal information has become crucial. Differential privacy (DP) has emerged as a widely adopted framework for quantifying the level of individual privacy an algorithm preserves.Over the past decade, numerous fundamental algorithms have been studied within the context of DP. This talk will focus on recent advances in the differentially private computation of graph cuts, including differentially private min-st-cut, Gomory-Hu tree and computing differentially private synthetic graphs that maintain all cuts. We will also touch on multiple interesting open problems in this area of research.

December 3rd, 2025: Coloring Graphs with Few Colors in the Streaming Model

A&C Seminar

Speaker: Helia Yazdanyar, University of Waterloo

Abstract: Graph streaming algorithms are well understood when we allow many colors, but far less is known about how to estimate the chromatic number. In this talk, I will present recent progress on the question: how much space is required to distinguish whether an input graph is q-colorable or requires significantly more than q colors? Our work develops upper and lower bounds across three natural streaming models: adversarial, random-order, and dynamic. In this talk, we will focus primarily on the adversarial setting, where we prove space lower bounds for distinguishing q- vs 2^{\Omega(q)}-colorable graphs. A key component of the proof is a new graph construction we introduce — cluster packing graphs — which enables us to characterize space requirements in this regime. I will also briefly discuss our results for random-order streams, where we obtain upper and lower bounds for distinguishing q- vs q^t-colorable graphs.

https://arxiv.org/abs/2510.18177

November 26th, 2025: Space complexity for approximate all-pairs max flows

A&C Seminar [video]

Speaker: Thatchaphol Saranurak, UMichigan

Abstract: I will show that any data structure for sqrt{n}-additive approximate max flow requires \Omega(n^2) space, generalizing the trivial lower bound of n^2 in the exact case.

This is based on a joint work with Seth Pettie and Longhui Yin https://arxiv.org/abs/2201.00408

November 19th, 2025: On the Gradient Complexity of Private Optimization with Private Oracles

A&C Seminar [video]

Speaker: Michael Menart, University of Toronto

Abstract: Convex optimization over finite sum losses is an important primitive in machine learning, and differential privacy is an increasingly important standard for trustworthy learning. As a result, a robust literature on differentially private convex optimization has emerged. The optimal utility rate for this problem has been known for over a decade [BST14], but a formalruntime characterization has remained elusive, with prior work focusing largely on improving runtime upper bounds. In this talk, I will present work on some of the first runtime lower bounds for a class of differentially private optimizers. Specifically, we introduce the notion of private oracle methods, which are the predominant approach used in practice. This algorithm class includes, for example, the ubiquitous DP-SGD algorithm. For such methods, we provide tight runtime lower bounds in high dimensional regimes. Our results show that private oracle methods necessarily suffer a dimension dependent penalty in their runtime as compared to their non-private counterparts. Our results also characterize the impact of minibatch size in differentially private optimization, and show that small minibatches can lead to even larger runtime lower bounds. Our techniques are information theoretic in nature, and naturally extend to provide lower bounds for (non-private) gradient quantization schemes as well. This talk is based on joint work with Aleksandar Nikolov.

November 14th, 2025: The power of in-place space-bounded computation

A&C Seminar

Speaker: Edward Pyne, MIT

Abstract: When complexity theorists (like me!) talk about computing functions in small space, we picture a read-only input, a small worktape, and a write-only output. When practitioners think about computing functions in small space, they think of an input x that must be transformed into an output f(x), using few additional bits. Because this model is arguably less clean, it previously escaped the attention of complexity theorists.

We show that complexity can play a useful role in understanding “in-place space-bounded computation”. For instance, assuming cryptography exists, it is impossible to transform (x,y) to x*y in-place. We give a variety of algorithms and lower bounds for the model, and use these to prove new results in catalytic computing. I will also discuss open problems and directions.

Joint work with James Cook, Surendra Ghentiyala, Ian Mertz, and Nathan Sheffield. ECCC-TR25-150

November 12th, 2025: Recursive lattice reduction—A simple framework for finding short lattice vectors

A&C Seminar

Speaker: Noah Stephen-Davidowitz, Cornell

Abstract: We’ll present a new framework called recursive lattice reduction for finding short non-zero vectors in a lattice. This gives new algorithms for solving the computational problem whose hardness underlies the security of lattice-based cryptography. These new algorithms are much simpler than prior work, and they provably match the state of the art. The analysis of the algorithms is also quite simple, and in particular, the analysis provides a much clearer explanation of why the algorithms perform as they do (i.e., the amount of time needed for these algorithms to find vectors of a given length, which is the key quantity that governs the security of lattice-based cryptography in practice).

The framework is based entirely on the following idea: in order to find a short non-zero vector in an n-dimensional lattice, one should first find a dense \ell-dimensional sublattice for some \ell < n and then recursively solve the \ell-dimensional problem of finding short non-zero vectors in this sublattice. After doing this repeatedly, we are eventually left with the problem of finding short non-zero vectors in a k-dimensional sublattice for k small enough that we can simply find an optimal solution. One obtains a family of algorithms depending on how k and \ell are chosen.

Based on joint work with Divesh Aggarwal, Thomas Espitau, and Spencer Peters, which appeared in SOSA 2025. https://arxiv.org/abs/2311.15064 .

October 29th, 2025: The Sample Complexity of Membership Inference and Privacy Auditing

A&C Seminar [video]

Speaker: Jonathan Ullman, Northeastern University

Abstract: A membership-inference attack gets the output of a learning algorithm, and a target individual, and tries to determine whether this individual is a member of the training data or an independent sample from the same distribution. A successful membership-inference attack typically requires the attacker to have some knowledge about the distribution that the training data was sampled from, and this knowledge is often captured through a set of independent reference samples from that distribution.

In this work we study how much information the attacker needs for membership inference by investigating the sample complexity-the minimum number of reference samples required-for a successful attack. We study this question in the fundamental setting of Gaussian mean estimation where the learning algorithm is given n samples from a Gaussian distribution in d dimensions, and tries to estimate the mean up to some specified error. Our result shows that for membership inference in this setting, many samples.

Our result is the first to show that the attacker sometimes needs many more samples than the algorithm used to train the model. This result has significant implications for practice, as all attacks used in practice have a restricted form that uses O(n) samples and cannot benefit from more samoples. Thus, these attacks may be underestimating the possibility of membership inference, and better attacks may be possible when information about the distribution is easy to obtain.

Joint work with Mahdi Haghifam and Adam Smith.

October 22nd, 2025: Recent developments in negative length shortest paths

A&C Seminar [video]

Speaker: Kent Quanrud, Purdue University

Abstract: The textbook algorithm for real-weighted single-sourceshortest paths takes O(mn) time on a graph with m edges and n vertices. Thebreakthrough algorithm by Fineman [Fin24] takes Õ(mn8/9) randomizedtime. The running time was subsequently improved to Õ(mn4/5)randomized time [HJQ1]. Building on the work of [Fin24] and [HJQ1], we developed a single-source shortest pathsalgorithm running in Õ(mn3/4+m 4/5n) randomized runningtime [HJQ2]. Equivalently, is translates to Õ(mn3/4) time for m ≥ n5/4,and Õ(m4/5n) for m ≤ n5/4. Our main new technique replaces Fineman’shop-reducing auxiliary graph from with a bootstrapping process whereconstant-hop reducers for small subgraphs of the input graph are iterativelyamplified and expanded until the desired polynomial-hop reduction is achievedover the entire graph. [Fin24]Jeremy T. Fineman. Single-Source Shortest Paths with Negative Real Weights in Õ(mn8/9) Time. Proceedings ofthe 56th Annual ACM Symposium on Theory of Computing, STOC 2024, June 24-28,2024, pp. 3–14. [HJQ1]Yufan Huang, Peter Jin, Kent Quanrud. Faster single-source shortest paths withnegative real weights via proper hop distance. SODA 2025: 5239-5244. [HJQ2] Yufan Huang, Peter Jin, Kent Quanrud. Fasternegative length shortest paths by bootstrapping hop reducers. CoRRabs/2506.00428 (2025).

October 15th, 2025: Sublinear-Time Lower Bounds for Approximating Matching Size using Non-Adaptive Queries

A&C Seminar [video]

Speaker: Vihan Shah, University of Birmingham

Abstract: We study the problem of estimating the size of the maximum matching in the sublinear-time setting. This problem has been extensively studied, with several known upper and lower bounds. A notable result by Behnezhad (FOCS 2021) established a 2-approximation in O~(n) time.

However, all known upper and lower bounds are in the adaptive query model, where each query can depend on previous answers. In contrast, non-adaptive query models—where the distribution over all queries must be fixed in advance—are widely studied in property testing, often revealing fundamental gaps between adaptive and non-adaptive complexities. This raises the natural question: is adaptivity also necessary for approximating the maximum matching size in sublinear time? This motivates the goal of achieving a constant or even a polylogarithmic approximation using O~(n) non-adaptive adjacency list queries, similar to what was done by Behnezhad using adaptive queries.

We show that this is not possible by proving that any randomized non-adaptive algorithm achieving an n^{1/3 - gamma}-approximation, for any constant gamma > 0, with probability at least 2/3, must make Omega(n^{1 + eps}) adjacency list queries, for some constant eps > 0 depending on gamma. This result highlights the necessity of adaptivity in achieving strong approximations. However, non-trivial upper bounds are still achievable: we present a simple randomized algorithm that achieves an n^{1/2}-approximation in O(n \log n) queries.

October 8th, 2025: Composing Low-Space Algorithms

A&C Seminar

Speaker: Roei Tell, University of Toronto

Abstract: Given two algorithms $A_1$ and $A_2$ each using low space, what is the complexity of computing the composed function $A_2( A_1(x) )$? The two known approaches for composition are expensive either in time or in space, and interpolating them yields a quadratic time-space curve. This work asks whether quadratic time-space is optimal.

We prove an unconditional lower bound for composing linear-time algorithms, and then show that proving the same lower bound in various other natural settings (e.g., for algorithms running in large polynomial time) would imply a breakthrough on a major open problem in complexity theory, namely the BPL = L problem. The main contribution in the work is connecting three questions: the complexity of composition, time-space tradeoffs for deterministic algorithms, and BPL = L.

Based on an upcoming work with Ted Pyne, and the lower bound proof uses an idea of Ryan Williams

October 1st, 2025: Distributed Triangle Detection is Hard in Few Rounds

A&C Seminar

Speaker: Janani Sundaresan, University of Waterloo

Abstract: In the CONGEST model, $n$ vertices with information only about their neighborhoods are allowed to communicate to each other over the edges of the input graph. Communication happens in synchronous rounds with a bandwidth of $O(log n)$. We prove that detecting a triangle in this model requires $Omega(log log n)$ rounds. Prior to our work, the only lower bound was that at least two rounds are necessary.

It is known that standard communication complexity arguments that have been used to get lower bounds in the CONGEST model in the past are incapable of giving any meaningful multi-round lower bounds for this problem. Our main contribution is a new information-theoretic technique that combines classical round elimination arguments of communication complexity with the point-to-point communication aspects of distributed networks and can be of independent interest.

Joint work with Sepehr Assadi (https://arxiv.org/abs/2504.01802)

September 24th, 2025: Surveying Graph Parameters for Algorithmic Meta-Theorems for Solution Discovery

A&C Seminar

Speaker: Stephanie Maaz, University of Waterloo

Abstract: Solution discovery asks whether a given (infeasible) starting configuration to a problem can be transformed into a feasible solution using a limited number b of transformation steps. This paper investigates meta-theorems for solution discovery for graph problems definable in monadic second-order logic (MSO$_1$) and first-order logic (FO) where the transformation step is to slide a token to an adjacent vertex, focusing on parameterized complexity and structural graph parameters that do not involve the transformation budget b. We present both positive and negative results. On the algorithmic side, we prove that MSO$_1$-Discovery is fixed-parameter tractable when parameterized by neighborhood diversity. On the hardness side, we establish that FO-Discovery is W[1]-hard when parameterized by twin cover number, modulator to stars, or modulator to paths numbers. Additionally, we prove that MSO$_1$-Discovery is W[1]-hard when parameterized by bandwidth. These results complement the straightforward observation that solution discovery for the studied problems is fixed-parameter tractable when the budget b is included in the parameter (in particular, parameterized by cliquewidth+b, where cliquewidth is at most each of the studied parameters), and provide a near-complete (fixed-parameter tractability) meta-theorems investigation for solution discovery problems for MSO$_1$- and FO-definable graph problems and structural parameters larger than cliquewidth.

September 17th, 2025: An O(log log n)-approximate budget-feasible mechanism for subadditive valuations

A&C Seminar [video]

Speaker: Rian Neogi, University of Waterloo

Abstract: In budget-feasible mechanism design, there is a set of items U. A buyer wishes to purchase a set of items from the sellers of maximum value, where the value of a subset S of items is provided by a valuation function v. Each element e is held by a distinct seller, who incurs a private cost c_e for supplying her item. The buyer also has a budget of B on the total payments made to the sellers. The private costs c_e are known only to the sellers, and not to the buyer. Each seller e reports a cost r_e to the mechanism, which may or may not be equal to her true cost c_e. As a result, a seller may choose to misreport her cost if she sees that she is better off when doing so (for example, the mechanism might be giving her a higher payment under the misreported cost). Our goal is to design a mechanism such that (1) sellers are incentivized to report their true cost (i.e. sellers are no better off by misreporting their costs) (2) the total payments made to sellers is at most B, and, (3) the set of purchased items has a high value, i.e. it’s a good approximation to OPT = max{v(S) : \sum_{e in S} c_e <= B}.

We consider valuation functions v that are subadditive, meaning that for any two sets A,B of items, we have the inequality v(A)+v(B) >= v(A U B). These functions capture the concept of complement-free valuations: roughly speaking, this means that the combined value of a pair of goods is at most the sum of their individual values. In this talk, I will present a recent result which shows that there is a polynomial-time budget-feasible mechanism for subadditive valuations achieving an approximation ratio of O(log log n). This improves on the previous best bound of O(log n / log log n).

This is joint work with Chaitanya Swamy and Kanstantsin Pashkovich.

September 10th, 2025: Low depth arithmetic circuits for multivariate resultants

A&C Seminar

Speaker: Abhibhav Garg, University of Waterloo

Abstract: Checking the satisfiability of a system of polynomial equations is a basic task in computational science. The complexity of this task varies significantly depending on the parameter regime of interest: satisfiability of linear equations can be checked efficiently, while satisfiability of nonlinear equations is easily seen to be NP-hard. Less well-known is the fact that satisfiability of polynomial equations in a constant number of variables can be checked efficiently, by reducing to linear algebra. Recent work shows that checking satisfiability of polynomials in one variable is strictly easier than linear algebra: satisfiability of univariates can be checked in constant depth, while most linear algebraic tasks cannot be performed in constant depth. In this talk, we continue this line of work in the multivariate setting, showing that satisfiability of polynomial equations in a constant number of variables is also strictly easier than linear algebra. Specifically, in this parameter regime, the multivariate resultant, which is a nonlinear generalization of the determinant, can be computed by arithmetic circuits of polynomial size and constant depth, a fact which is false for the determinant itself.

August 7th, 2025: Towards efficient algorithms on compressed graph databases

A&C Seminar

Speaker: Prof. Stefan Boettcher, Paderborn University

Abstract: The speed of algorithms on massive graphs depends on the size of the given data. Grammar-based compression is a technique to compress the size of a graph while still allowing to read or to modify the graph with a little time overhead. When data access methods to compressed data are chosen carefully, the speed-up gained by data size reduction significantly predominates the time overhead needed to partially uncompress compressed data. The talk gives an overview of the key ideas behind grammar-based compression for large graphs and shows how to apply graph compression to graph databases. Furthermore, it introduces recompression as a fast technique to keep compressed graphs small when they are frequently modified.

August 5th, 2025: Extremal Combinatorial Objects in Hardness of Approximation in P

A&C Seminar

Speaker: Karthik C. S., Rutgers University

Abstract: In the last decade, the area of hardness of approximation for problems solvable in polynomial time has emerged. One of the popular frameworks for proving such results is called threshold graph composition and relies on the existence and efficient construction of non-trivial extremal combinatorial objects.

In this talk, we shall focus on defining and constructing these extremal objects while briefly outlining their applications to hardness of approximation in P.

July 30th, 2025: Computing moment polytopes of tensors, and applications in TCS and quantum information

A&C Seminar [video]

Speaker: Maxim van den Berg, RU Bochum

Abstract: Moment polytopes of tensors are convex polytopes that appear in a variety of areas in TCS and physics. In algebraic complexity theory they have been proposed as a way to obtain lower bounds for the complexity of matrix multiplication. In quantum information theory they are also known as entanglement polytopes, and they naturally characterize the reachable marginals in multipartite systems under entanglement monotone transformations. The two perspectives together led to the construction of the so-called quantum functionals, relevant in Strassen’s duality theory to asymptotic tensor rank. Moreover, moment polytopes appear more generally as the central dual objects in non-commutative convex optimization, which has many further applications.

Despite the importance, little is known about these objects. In particular, examples were only known in very small dimensions. We developed an algorithm which can compute examples an order of magnitude larger than what was possible before (and our algorithm applies to general moment polytopes, not just for tensors). Inspired by the examples, we prove several new theoretical results. For instance, we show matrix multiplication tensors have non-maximal moment polytopes. In the context of quantum information, this shows a limitation of expressiveness of matrix product states.

In this talk I will give an accessible introduction to moment polytopes of tensors, explain our algorithm and discuss our results. The talk is based on our paper from STOC 2025 of (almost) the same title.

July 23rd, 2025: Almost Tight Additive Guarantees for k-Edge Connectivity

A&C Seminar [video]

Speaker: Chaitanya Swamy, University of Waterloo

Abstract: We consider the k-edge connected spanning subgraph} (kECSS) problem, where we are given an undirected graph with nonnegative edge costs, and we seek a minimum-cost subgraph that is k-edge connected, i.e., there are k edge-disjoint paths between every pair of nodes.

For even k, we present a polytime algorithm that computes a (k-2)-edge connected subgraph of cost at most the optimal value of the natural LP-relaxation for kECSS; for odd k, this leads to a (k-3)-edge connected subgraph of cost at most the LP optimum. Since kECSS is APX-hard for all k\geq 2, the tightest connectivity guarantee one can hope for without exceeding the optimal value is (k-1)-edge connectivity, so our results are nearly tight. They also significantly improve upon the recent work of Hershkowitz et al., both in terms of solution quality and the simplicity of algorithm and its analysis. Our result is obtained using iterative rounding, with additional insights involving uncrossing tight sets for kECSS. A slight variant of our approach yields a (k-1)-edge connected subgraph of cost at most 1.5 times the LP optimum; with unit edge costs, the cost guarantee improves to (1+4/3k) times the LP optimum, which improves upon the state-of-the-art approximation for unit edge costs, but with a unit loss in edge connectivity. The kECSS-result also yields results for the k-edge connected spanning multigraph (kECSM) problem, where multiple copies of an edge can be selected. Our techniques extend to the degree-bounded versions of kECSS and kECSM, wherein we also impose degree lower- and upper- bounds on the nodes.

The talk will be self contained. Joint work with Nikhil Kumar, C&O.

July 16th, 2025: Closure results for polynomial factorization and some applications

A&C Seminar

Speaker: Shubhangi Saraf, University of Toronto

Abstract: I will talk about a recent result showing that algebraic formulas and constant-depth circuits are closed under taking factors. In other words, the complexity of factors of polynomials computable by algebraic formulas or constant depth algebraic circuits is not much more than the complexity of the original polynomial itself.

This result turns out to be an elementary consequence of a fundamental and surprising result of Furstenberg from the 1960s, which gives a non-iterative description of the power series roots of a bivariate polynomial. Combined with standard structural ideas in algebraic complexity, we observe that this theorem yields the desired closure results.

We will see applications of this result to deterministic algorithms for factoring, hardness/randomness tradeoffs, as well as GCD computation of polynomials.

This talk is based on joint works with Somnath Bhattacharjee, Mrinal Kumar, Shanthanu Rai, Varun Ramanathan and Ramprasad Saptharishi.

July 2nd, 2025: Synthesis and Arithmetic of Quantum Circuits

A&C Seminar

Speaker: Amolak Ratan Kalra, University of Waterloo

Abstract: Efficient decomposition of a unitary operator U using words from a universal gate set G is a fundamental problem in quantum computing. The process by which this is achieved is called circuit synthesis. This problem arises naturally in the context of quantum circuit compilation. In this talk, I will introduce this problem and explain how one can use tools from number theory to solve it. I will then explain some of our more recent results that build on this connection.

June 16th, 2025: An Improved Fully Dynamic Algorithm for Counting 4-Cycles in General Graphs

A&C Seminar [video]

Speaker: Vihan Shah, University of Waterloo

Abstract: We study subgraph counting over fully dynamic graphs, which undergo edge insertions and deletions. Maintaining the number of triangles in fully dynamic graphs is very well studied and has an upper bound of O(m^{1/2}) for the update time [KNN+20]. There is also a conditional lower bound of approximately Omega(m^{1/2}) for the update time [HKNS15] under the OMv conjecture implying that Theta(m^{1/2}) is the ``right answer’’ for the update time of counting triangles. More recently, [HHH22] studied the problem of maintaining the number of 4-cycles in fully dynamic graphs and designed an algorithm with O(m^{2/3}) update time which is a natural generalization of the approach for counting triangles. Thus, it seems natural that O(m^{2/3}) might be the correct answer for the complexity of the update time for counting 4-cycles.

In this work, we present an improved algorithm for maintaining the number of 4-cycles in fully dynamic graphs. Our algorithm achieves a worst-case update time of O(m^{2/3-eps}) for some constant eps>0. Our approach crucially uses fast matrix multiplication and leverages recent developments therein to get an improved runtime. Using the current best value of the matrix multiplication exponent omega=2.371339 we get eps=0.009811 and if we assume the best possible exponent i.e. omega=2 then we get eps=1/24. The lower bound for the update time is Omega(m^{1/2}), so there is still a big gap between the best-known upper and lower bounds. The key message of our paper is demonstrating that O(m^{2/3}) is not the correct answer for the complexity of the update time.

June 11th, 2025: Linear Hashing Is Optimal

A&C Seminar [video]

Speaker: Vinayak Kumar, UT Austin

Abstract: When n balls are independently and uniformly tossed into n bins, the expected max-load—the number of balls in the heaviest bin—is O(logn/loglogn). This classical result plays a central role in the analysis of hashing with chaining and load balancing. However, implementing a truly random hash function is often impractical due to its high computational and storage costs.In this talk, I will present a recent result showing that hashing n balls into n bins via a random matrix over F2 achieves the same expected max-load of O(logn/loglogn). This simple and efficient hash family matches the performance of a fully random function and resolves an open question posed by Alon, Dietzfelbinger, Miltersen, Petrank, and Tardos.

Based on joint work with Michael Jaber and David Zuckerman.

June 4th, 2025: Polynomial-Time PIT from (Almost) Necessary Assumptions

A&C Seminar

Speaker: Deepanshu Kush, University of Toronto

Abstract: The celebrated result of Kabanets and Impagliazzo (Computational Complexity, 2004) showed that PIT algorithms imply circuit lower bounds, and vice versa. Since then, it has been a major challenge to understand the precise connections between PIT and lower bounds. A main goal has been to understand which lower bounds suffice to obtain efficient PIT algorithms, and how close are they to lower bounds that are necessary for the conclusion.

We construct polynomial-time PIT algorithms from lower bounds that are, up to relatively minor remaining gaps, necessary for the existence of such algorithms. That is, we prove that these lower bounds are, up to the mentioned minor gaps, both sufficient and necessary for polynomial-time PIT.

The key to these improvements is studying PIT versus lower bounds in the uniform setting, in which we focus on proving lower bounds for uniform arithmetic circuits and their variants (and on deducing algorithms from such lower bounds). Indeed, by working in this setting we obtain results that are significantly tighter than previously known results concerning polynomial-time PIT vs lower bounds and are in fact also tighter than known hardness-vs-randomness connections in the Boolean setting.

This talk is based on joint work with Robert Andrews and Roei Tell.

May 28th, 2025: Optimal Differentially Private Sampling of Unbounded Gaussians

A&C Seminar [video]

Speaker: Argyris Mouzakis, University of Waterloo

Abstract: We provide the first $\widetilde{\mathcal{O}}(d)$-sample algorithm for sampling from unbounded Gaussian distributions under the constraint of (ε, δ)-differential privacy. This is a quadratic improvement over previous results for the same problem, settling an open question of Ghazi, Hu, Kumar, and Manurangsi.

This is joint work with Valentio Iverson and Gautam Kamath. The paper will appear in COLT 2025, and is available on arxiv https://arxiv.org/abs/2503.01766

May 21st, 2025: Time Lower Bounds for Property Testing

A&C Seminar

Speaker: Renato Ferreira, University of Waterloo

Abstract: We initiate a systematic study of the computational complexity of property testing, focusing on the relationship between query and time complexity. While traditional work in property testing has emphasized query complexity—often via information-theoretic techniques—relatively little is known about the computational hardness of property testers. Our goal is to chart the landscape of time-query trade-offs and develop tools for proving time complexity lower bounds.

Our first contribution is a pair of time-query hierarchy theorems for property testing. For all suitable nondecreasing functions $q(n)$ and $t(n)$ with $t(n) = \Omega(\text{polylog}(n))$, we construct properties with query complexity $\widetilde{\Theta}(q(n))$ and time complexity at least $\Omega(t(n))$. Our weak hierarchy holds unconditionally, while the strong version—assuming the Strong Exponential Time Hypothesis—guarantees time complexity between $\widetilde{\Omega}(t(n))$ and $\widetilde{O}(t(n)^{1+\gamma})$ for arbitrarily small $\gamma > 0$.

We then turn to halfspaces in $\mathbb{R}^d$, a fundamental class in property testing and learning theory. For the distribution-free distance approximation problem, known algorithms achieve query complexity $O(d/\epsilon^2)$, but run in time $\widetilde{O}(1/\epsilon^d)$. We provide a fine-grained justification for this gap: assuming the (integer) $k$-SUM conjecture, any algorithm must have running time ${\Omega}(1/\epsilon^{d/2})$.

Joint work with Diptaksho Palit and Sofya Raskhodnikova.

May 14th, 2025: Can SGD Select Good Fishermen?

A&C Seminar

Speaker: Felix Zhou, Yale University

Abstract: Self-selection has a long history in Econometrics and applications in various theoretical and applied fields, including treatment effect estimation, imitation learning, learning from strategically reported data, and learning from markets at disequilibrium. In the classical setting of self-selection, the goal is to learn k models, simultaneously from observations (x(i),y(i)) where y(i) is the output of one of k underlying models on input x(i). Note that the observed model depends on the outputs themselves and is determined by some known selection criterion.

We revisit the problem of estimating k linear regressors with maximum self-selection bias in d dimensions, as introduced by Cherapanamjeri, Daskalakis, Ilyas, and Zampetakis [CDIZ23, STOC'23]. Our main result is a poly(d,k,1/ε)+k^O(k) time algorithm for this problem, which yields an improvement in the running time of the algorithms of [CDIZ23] and Gaitonde & Mossel [GM24]. This is achieved by providing the first local convergence algorithm for self-selection, thus resolving the main open question of [CDIZ23].

Based on joint work with Alkis Kalavasis and Anay Mehrotra.

May 7th, 2025: Maximum Matching in $O(\log \log n) $ Passes in Dynamic Streams

A&C Seminar

Speaker: Janani Sundaresan, University of Waterloo

Abstract: In the dynamic streaming model, an $n$-vertex input graph is defined through a sequence of edge insertions and deletions in a stream. The algorithms are allowed to process this stream in multiple passes while using O(n \poly\log (n)) space. This model has been introduced in the seminal work of Ahn, Guha, and McGregor (AGM) in 2012 and has been studied extensively since.

In this talk, we focus on approximating maximum matching in the dynamic stream model. An $O(1)$-approximation algorithm in $O(\log n)$ passes was already introduced by [AGM12], but improving the number of passes has remained elusive. We give a randomized sketching-based algorithm that achieves an $O(1)$-approximation in only $O(\log \log n)$-passes and $O(n \poly \log n)$ space, which exponentially improves the state-of-art. Using standard techniques, the approximation ratio of this algorithm can be improved to (1+eps)-approximation for any constant eps > 0 with asymptotically the same space and number of passes.

Furthermore, we prove the first multi-pass lower bound for this problem, showing that $\Omega(\log \log n)$ passes are also necessary for any algorithm which finds an $O(1)$-approximation to maximum matching in $O(n \poly \log n)$ space. Our upper and lower bounds collectively settle the pass complexity of this fundamental problem in the dynamic streaming model. The talk, however, will primarily focus on the algorithmic part of our results.

This is joint work with Sepehr Assadi, Soheil Behnezhad, Christian Konrad and Kheeran K. Naidu. Available online at https://arxiv.org/abs/2407.21005.

May 6th, 2025: Gradient Optimization Methods: The Benefits of a Large Step-size

A&C Seminar

Speaker: Peter Bartlett, UC Berkeley

Abstract: Deep learning, the technology underlying the recent progress in AI, has revealed some major surprises from the perspective of theory. Optimization in deep learning relies on simple gradient descent algorithms that are traditionally viewed as a time discretization of gradient flow. However, in practice, large step sizes — large enough to cause oscillation of the loss — exhibit performance advantages.This talk will review recent results on gradient descent with logistic loss with a step size large enough that the optimization trajectory is at the “edge of stability.” We show the benefits of this initial oscillatory phase for linear functions and for multi-layer networks, and identify an asymptotic implicit bias that gradient descent imposes for a large family of deep networks. Based on joint work with Yuhang Cai, Michael Lindsey, Song Mei, Matus Telgarsky, Jingfeng Wu, Bin Yu and Kangjie Zhou.

April 21st, 2025: Efficient Algorithms for RDV Graphs

Thesis Presentation

Speaker: Prashant Gokhale

Abstract: In this thesis, we study the maximum matching and minimum dominating set problem in RDV graphs, i.e., graphs that are vertex-intersection graphs of downward paths in a rooted tree. A straightforward implementation of these algorithms would require O(n+m) time. We improve their efficiency by transforming the question about the neighborhood of v into a type of range query amid a set of horizontal and vertical line segments. Our algorithms run in O(nlogn) time, presuming a O(n)-sized intersection representation of the graph is given. In addition, our techniques can also be used to obtain faster algorithms for maximum independent set and perfect k-clique packing in RDV graphs.

April 9th, 2025: Almost tight exponential lower bounds for binary linear 3-LCCs

A&C Seminar

Speaker: Rafael Oliveira, University of Waterloo

Abstract: In this talk we will see the recent elegant and near optimal lower bound for 3-LCCs proved by Alrabiah and Guruswami. The talk will be based on the following paper: https://arxiv.org/abs/2404.05864

April 7th, 2025: Adam and friends via duality

Joint A&C Seminar and Probability Seminar

Speaker: Matus Telgarsky, NYU

Abstract: This talk shows how various optimization methods can be derived and analyzed via a duality framework. Superficially this seems nice due to faster convergence rates than non-duality proofs, but the real value is that unifying in the dual highlights a different set of similarities than in the primal. The work will primarily be in unpublishable settings (linear separability), but will use slightly less unpublishable settings (deep networks from ten years ago) for motivation. The talk will also feature a brief general-audience discussion of some issues facing the ml theory and general academic communities. Joint work with Ziwei Ji, Danny Son, Zihan Wang.

April 2nd, 2025: Foundations of Multi-objective Machine Learning

A&C Seminar [video]

Speaker: Nika Haghtalab, Berkeley

Abstract: In this talk, I will introduce the framework of multi-objective learning as a modern generalization of statistical learning.This framework that acts as a unifying paradigm for addressing needs such as robustness, collaboration, and fairness aims to optimize a set of complex and unstructured objectives from only a small amount of sampled data. I will also discuss how the multi-objective learning paradigm relates to the classical and modern considerations in machine learning broadly, such as generalization, introducing technical tools with versatile provable guarantees, and empirical evidence for its performance on existing benchmarks.

March 21st, 2025: Streaming and Communication Complexity of Load-Balancing via Matching Contractors

A&C Seminar [video]

Speaker: Robert Wang, University of Waterloo

Abstract: In the load-balancing problem, we have an n-vertex bipartite graph G=(L,R,E) between a set of clients and servers. The goal is to find an assignment of all clients to the servers, while minimizing the maximum load on each server, where load of a server is the number of clients assigned to it. We study load-balancing in the one-way communication model: the edges of the input graph are partitioned between Alice and Bob, and Alice needs to send a message to Bob for him to output the solution. We show that settling the one-way communication complexity of load-balancing is equivalent to a natural sparsification problem for load-balancing. We then prove a dual interpretation of this sparsifier, showing that the minimum density of a sparsifier is effectively the same as the maximum density one can achieve for an extremal graph family that is new to this paper, called Matching-Contractors; these graphs are intimately connected to the well-known Ruzsa-Szemeredi graphs and generalize them in certain aspects. Our chain of equivalences thus shows that the one-way communication complexity of load-balancing can be reduced to a purely graph theoretic question: what is the maximum density of a Matching-Contractor on n vertices? Finally, we present a novel combinatorial construction of some-what dense Matching-Contractors, which implies a strong one-way communication lower bound for load-balancing: any one-way protocol (even randomized) with O~(n) communication cannot achieve a better than n^{1/4−o(1)}-approximation. Previously, no non-trivial lower bounds were known for protocols with even O(nlogn) bits of communication. Our result also implies the first non-trivial lower bounds for semi-streaming load-balancing in the edge-arrival model, ruling out n^{1/4−o(1)}-approximation in a single-pass.

March 19th, 2025: Vizing’s Theorem in Near-Linear Time

A&C Seminar [video]

Speaker: Martin Costa, Warwick

Abstract: Vizing’s theorem states that any n-vertex m-edge graph of maximum degree Δ can be edge colored using at most Δ + 1 different colors [Vizing, 1964]. Vizing’s original proof is algorithmic and shows that such an edge coloring can be found in O(mn) time. This was subsequently improved to Õ(mn^(1/2)) time, independently by [Arjomandi, 1982] and by [Gabow et al., 1985]. Very recently, this runtime bound was further improved to Õ(n^2) by [Assadi et al., 2024] and Õ(mn^(1/4)) by [Bhattacharya et al., 2024]. In this talk, I will present a randomized algorithm that computes a (Δ + 1)-edge coloring in near-linear time—in fact, only O(m log Δ) time—with high probability, giving a near-optimal algorithm for this fundamental problem.

March 17th, 2025: Polycubes with small perimeter defect

A&C Seminar

Speaker: Gill Barequet, Technion

Abstract: Lattice animals are connected sets of cells on a lattice. For example, \emph{polyominoes} are connected sets of cells on the planar square lattice, where connectivity is through edges (sides of the squares). One of the main problems in this area is finding a formula for the number of animals (or for the number of members of some family of animals) on some lattice. In this talk, I will explore methods for setting formulae for the number of animals on the $d$-dimensional cubical lattice, whose perimeter deviates by a fixed constant from the maximum possible perimeter. Joint work with Andrei Asinowski (Alpen-Adria-Universität Klagenfurt) and Yufei Zheng (UMass Amherst).

March 14th, 2025: Testing Support Size More Efficiently Than Learning Histograms|

A&C Seminar

Speaker: Renato Ferreira, University of Waterloo

Abstract: Consider two problems about an unknown probability distribution $p$:1. How many samples from $p$ are required to test if $p$ is supported on $n$ elements or not? Specifically, given samples from $p$, determine whether it is supported on at most $n$ elements, or it is “$\epsilon$-far” (in total variation distance) from being supported on $n$ elements.2. Given $m$ samples from $p$, what is the largest lower bound on its support size that we can produce?The best known upper bound for problem (1) uses a general algorithm for learning the histogram of the distribution $p$, which requires $\Theta(\tfrac{n}{\epsilon^2 \log n})$ samples. We show that testing can be done more efficiently than learning the histogram, using only $O(\tfrac{n}{\epsilon \log n} \log(1/\epsilon))$ samples, nearly matching the best known lower bound of $\Omega(\tfrac{n}{\epsilon \log n})$. This algorithm also provides a better solution to problem (2), producing larger lower bounds on support size than what follows from previous work. The algorithm builds upon the Chebyshev polynomial technique of Wu and Yang (Annals of Statistics 2019), and the proof relies on an analysis of Chebyshev polynomial approximations outside the range where they are designed to be good approximations. Joint work with Nathan Harms (EPFL), to appear at STOC 2025.

March 12th, 2025: A Tolerant Independent Set Tester

A&C Seminar

Speaker: Cameron Seth, University of Waterloo

Abstract: In this talk we study a simple algorithm for tolerant testing the property of having a rho n independent set in the dense graph model. In particular, we give an algorithm that inspects a random subgraph on O(rho^3/eps^2) vertices and, with high probability, distinguishes between graphs that have an induced subgraph of size rho n with less than (eps/polylog(eps)) n^2 edges from graphs for which every induced subgraph of size rho n has at least eps n^2 edges. The notion of tolerant testing generalizes the standard notion of testing, however, our result matches the sample complexity bounds of the standard testing problem (Blais and Seth 2023, Feige, Langberg and Schechman 2004), and, surprisingly, shows that tolerant testing the independent set property is no harder than the standard testing problem.

Our main technique is the graph container method, which is a powerful tool for many problems related to counting independent sets, and that has recently seen applications in graph property testing (Blais and Seth 2023, 2024). In this talk we will give a brief introduction to the container method and discuss our main technical contribution: a new graph container lemma that applies to sparse subgraphs instead of independent sets.

March 7th, 2025: Space Complexity of Minimum Cut Problems in Single-Pass Streams

A&C Seminar

Speaker: Vihan Shah

Abstract: We consider the problem of finding a minimum cut of a weighted graph presented as a single-pass stream. Since finding the exact minimum cut essentially requires you to store the entire stream, we study the (1+eps)-approximate minimum cut problem. An upper bound for streaming (1+eps)-approximate minimum cut easily follows from streaming (1+eps) cut sparsification which is a very well-studied problem. It has an upper bound using O(n polylog n / eps^2) space [AG09] and a lower bound of Omega(n/eps^2) bits [CKST19]. This immediately provides an upper bound of O(n polylog n / eps^2) bits for streaming (1+eps)-approximate minimum cut, but the lower bound does not carry over. The best lower bound for (1+eps)-approximate minimum cut is Omega(n log 1/eps) bits [AG09]. This raises the fundamental question, what is the streaming space complexity of (1+eps)-approximate minimum cut? In our paper we resolve this question up to polylogarithmic factors. Additionally, we explore the random-order streaming model, a relaxation of the adversarial streaming model, and demonstrate that significantly better bounds can be achieved in this setting.

March 5th, 2025: Treewidth and Linear Algebra

A&C Seminar

Speaker: Luke Schaeffer, University of Waterloo

Abstract: We look at the complexity of solving sparse linear systems as a function of the treewidth of the instance. That is, the sparse matrix is associated with a sparse graph, and solutions can be found faster when that graph has low treewidth. We give a parameterized algorithm in system size and treewidth achieving the conjectured optimal performance. No prior knowledge of treewidth is necessary. This is joint work with Daniel Grier.

February 26th, 2025: Interactive proofs for primality testing of special classes of ideals

A&C Seminar [video]

Speaker: Abhibhav Garg, University of Waterloo

Abstract: Given a set of polynomials as an algebraic circuit, we study the problem of testing if the ideal generated by these polynomials is prime. This is a generalisation of the problem of testing if the polynomials have a common solution. It is also a generalisation of the problem of testing if a polynomial is absolutely irreducible. Assuming GRH, we show that for certain classes of ideals, namely radical ideals and complete intersection ideals, the problem of testing primality lies in the third level of the polynomial hierarchy. This is almost optimal, since the problems are NP-hard. Previously, the best known bound for these problems was PSPACE. Our method is a vast generalisation of the method used by Koiran to show that satisfiability of polynomials is in PH. We study how algebraic and geometric properties of ideals such as irreducibility and dimension behave under mod p reduction of coefficients. We give new effective versions of classical results from algebraic geometry and commutative algebra that may have independent applications. This is joint work with Rafael Oliveira and Nitin Saxena.

February 19th, 2025: Estimating Euclidean distance to linearity

A&C Seminar [video]

Speaker: Andrej Bogdanov, University of Ottawa

Abstract: Learning a linear approximation of an unknown function is the basis of many algorithms. Is it possible to calculate the quality of the approximation before the effort is spent in learning it? We study this question for real-valued functions f over the Boolean cube with respect to query complexity. We show that, given oracle access to f, an ϵ-additive approximation of the squared Euclidean distance to its closest linear function can be computed with O(log^3(1/ϵ) · 1/ϵ^2) queries (assuming f has bounded 4-norm). This query complexity is independent of the dimension and optimal up to the polylogarithmic factor. In contrast, learning a linear approximation requires query complexity that is linear in the dimension. We also obtain tight bounds for the sample complexity of estimating the distance to linearity.

The talk is based on joint work with Lorenzo Taschin.

February 12th, 2025: An exponential lower bound for 2-query Relaxed Locally Decodable codes

A&C Seminar

Speaker: Elena Grigorescu, University of Waterloo

Abstract: Locally Decodable Codes (LDCs) are error-correcting codes with super-fast decoding algorithms. They are important mathematical objects in many areas of theoretical computer science, yet the best constructions so far have codeword length that is super-polynomial in the message length, for codes with constant query complexity and constant alphabet size.

In a very surprising result, Ben-Sasson, Goldreich, Harsha, Sudan, and Vadhan (SICOMP 2006) showed how to construct a relaxed version of LDCs (RLDCs) with constant query complexity and almost linear codeword length over the binary alphabet and used them to obtain significantly improved constructions of Probabilistically Checkable Proofs.

In this talk I will present an exponential lower bound on the length of RLDCs making 2 queries, over the binary alphabet. This answers a question explicitly raised by Gur and Lachish (SICOMP 2021) and is the first exponential lower bound for RLDCs. Combined with the results of Ben-Sasson et al., our result exhibits a “phase-transition”-type behavior on the codeword length for some constant query complexity.

Joint work with Alex Block, Jeremiah Blocki, Kuan Cheng, Xin Li, Yu Zheng, and Minshen Zhu (Appeared in CCC 2023).

February 5th, 2025: Fun with Diophantine approximation

A&C Seminar

Speaker: Ingrid Vukusic, University of Waterloo

Abstract: Let’s play around with some numbers and make them cancel each other out. For example, 2*2*9/(6*6) = 1. We call the triple of numbers (2,9,6) multiplicatively dependent. Now here is a little Diophantine problem: Can you find integers 1<a<b<c such that (a,b,c) and (a+1,b+1,c+1) are both multiplicatively dependent? This problem per se is not a question of Diophantine approximation (“How well can real numbers be approximated by rational ones?”). However, in my talk, I will explain how methods and results from the field of Diophantine approximation are used to solve problems such as the above. I will give an introduction to Diophantine approximation, and, in particular talk about lower bounds for linear forms in logarithms and continued fractions. I will also make connections to metric number theory, which asks about properties of “almost all real numbers”.

January 15th, 2025: Online Algorithms for Spectral Hypergraph Sparsification

A&C Seminar [video] [slides]

Speaker: Kam Chuen (Alex) Tung

Abstract: We provide the first online algorithm for spectral hypergraph sparsification. In the online setting, hyperedges with positive weights are arriving in a stream, and upon the arrival of each hyperedge, we must irrevocably decide whether or not to include it in the sparsifier. Our algorithm produces an (eps, delta)-spectral sparsifier with multiplicative error eps and additive error delta that has O(eps^{-2} n log n log r log (1 + eps*W / (delta*n))) hyperedges with high probability, where n is the number of nodes, r is the rank of the hypergraph, and W is the sum of edge weights. The space complexity of our algorithm is O(n^2), while previous algorithms required space complexity Omega(m), where m is the number of hyperedges. This provides an exponential improvement in the space complexity since m can be exponential in n.

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2024

December 18th, 2024: Solution Discovery for Source Problems in P

A&C Seminar

Speaker: Stephanie Maaz

Abstract: In the recently introduced class of solution discovery problems [Fellows et al., ECAI 2023], we are given an initial infeasible solution for an instance under a source problem Π and tasked with transforming it into a feasible solution (under Π) via a bounded number b of modification steps. While the paper of Fellows et al, have mainly focused on fundamental NP-hard graph source problems such as Dominating Set and Coloring, my joint work with Mario Grobler, Nicole Megow, Amer E. Mouawad, Vijayaragunathan Ramamoorthi, Daniel Schmand and Sebastian Siebertz explores solution discovery problems of graph vertex (or edge) subset source problems in P, namely Spanning Tree, Shortest Path, Matching, and Vertex/Edge Cut. When the modification step consists of changing the current infeasible solution by removing a vertex (resp. edge) from the infeasible set and replacing it with an adjacent vertex (resp. edge), we show that almost all considered problems become NP-complete. This motivates the study of these problems under parameterized complexity assumptions. Under these assumptions, our work considered the size of the (infeasible or feasible) solutions and the bound on the number of modification steps (b) as parameters. Solution discovery problems are particularly compelling due to their utility in dynamic real-life environments, where systems must adapt to changing circumstances. Through the talk, we will discuss the results of my joint work, familiarize ourselves with related works and see how solution discovery algorithms are different than other algorithms considered in the literature before.

December 13th, 2024: A Fine-grained Approach to Algorithms and Complexity

Joint A&C Seminar and Distinguished Lecture Series [video]

Speaker: Virginia Vassilevska Williams, MIT

Abstract: A central goal of algorithmic research is to determine how fast computational problems can be solved in the worst case. Unfortunately, for many central problems, the best known running times are essentially those of their classical algorithms from the 1950s and 1960s. For years, the main tool for explaining computational difficulty have been NP-hardness reductions, basing hardness on P ≠ NP. However, if one cares about exact running time (as opposed to merely polynomial vs non-polynomial), NP-hardness is not applicable, especially if the problem is already solvable in polynomial time. In recent years, a new theory has been developed, based on “fine-grained reductions” that focus on exact running times. In this talk I will give an overview of this area, and will highlight some new developments.

December 11th, 2024: Combinatorial List Decodability of Random Reed–Solomon Codes and Beyond

A&C Seminar [video]

Speaker: Zeyu Guo, Ohio State

Abstract: A code C \subseteq A^n over an alphabet A is said to be (r, L)-combinatorially list decodable if any Hamming ball of radius r n in A^n contains at most L codewords from C. In this talk, I will discuss a recent line of research that has demonstrated the optimal combinatorial list decodability of random Reed–Solomon codes. Additionally, I will briefly touch on related progress concerning rank-metric codes and folded Reed–Solomon codes.

December 4th, 2024: The Quest for the One True Definition of Fault Tolerant Spanners

A&C Seminar [video]

Speaker: Greg Bodwin, UMich

Abstract: A spanner is a graph sparsifier that approximately preserves shortest path distances. Spanners are often applied in contexts where nodes and edges can temporarily fail. We would like our spanners to still have good approximation properties, even after these failures. How should we formalize this? We will survey 10 or so papers, spanning the last 20 or so years. We will discuss what they reveal about fault tolerance, how it should (or shouldn’t) be defined, and where the area might go next.

November 29th, 2024: Sparse graph counting and Kelley-Meka bounds for binary systems

A&C Seminar [video]

Speaker: Hamed Hatami, McGill

Abstract: Abstract: Graph counting lemma of Chung, Graham, and Wilson (Combinatorica 1988) is a fundamental result in combinatorics, which states that if a large graph is pseudorandom, then the number of copies of any small graph $H$ in $G$ is close to what is expected from a random graph of the same density. However, this result is only nontrivial when $G$ is a dense graph. In this work, we obtain a counting lemma that works in the sparse setting, and it is well-suited for the density increment arguments in additive combinatorics.In a recent remarkable breakthrough, Kelley and Meka (FOCS 2023) obtained a strong upper bound on the density of sets of integers without nontrivial three-term arithmetic progressions. We combine our counting lemma with other ideas to establish Kelley–Meka type bounds for all linear patterns defined by translation-invariant systems of binary linear forms, i.e., each form depends on exactly two variables. In particular, we obtain strong bounds for the Turan problem in Abelian Cayley graphs, i.e. an upper bound on the maximum edge density of an Abelian Cayley graph with a clique of a given size. To prove our results, we employ some of the recent technology developed by Kelley and Meka and also the follow-up work by Kelley, Lovett, and Meka (STOC 2024). This talk is based on a joint work with Esty Kelman, Yuval Filmus, and Kaave Hosseini.

November 20th, 2024: Lower Bounds on Entanglement and Quantum Gates in Non-local Quantum Computation

A&C Seminar [video]

Speaker: Vahid Asadi, University of Waterloo

Abstract: A non-local quantum computation (NLQC) replaces an interaction between two quantum systems with a single simultaneous round of communication and shared entanglement. In this talk, we will review two classes of NLQC, f-routing and f-BB84, which are of relevance to classical information-theoretic cryptography and quantum position verification, and we will show the first non-trivial lower bounds on entanglement in both settings but are restricted to lower bounding protocols with perfect correctness. Within this setting, we give a lower bound on the Schmidt rank of any entangled state that completes these tasks for a given function f(x,y) in terms of the rank of a matrix g(x,y) whose entries are zero when f(x,y)=0, and strictly positive otherwise. This also leads to a lower bound on the Schmidt rank in terms of the non-deterministic quantum communication complexity of f(x,y). We also prove the number of quantum gates plus single qubit measurements needed to implement a function f is lower bounded linearly by the communication complexity of f in the simultaneous message-passing model with shared entanglement. Because of a relationship between f-routing and the conditional disclosure of secrets (CDS) primitive studied in information-theoretic cryptography, we obtain a new technique for lower bounding the randomness complexity of CDS.

November 13th, 2024: Maximum Coverage in Turnstile Streams with Applications to Fingerprinting Measures

A&C Seminar [video]

Speaker: Hoai-an Nguyen, Carnegie Mellon University

Abstract: In the maximum coverage problem we aim to choose at most k subsets such that the number of distinct items covered by the union of the subsets is maximized. The input can be formalized as a n x d matrix A where there are n items in the universe and d subsets. A_ij is nonzero if item i is in subset j and is 0 otherwise. In this paper, to our knowledge, we create the first linear sketch to solve maximum coverage, potentially leading to large runtime improvements and has direct applications to distributed and streaming settings. We specifically focus on the direct application to turnstile streams which allow deletions. Here, the updates are of the form (i, j, ±1) which performs A_ij = A_ij ± 1. Previous work mainly considers the more restrictive set-arrival model where each update reveals an entire column of A or the insertion-only model which does not allow deletions. We design an algorithm with an O(d/ε^3) space bound, which is nearly optimal for constant k.

We then turn to fingerprinting for risk measurement. The input is again a n × d matrix A where there are n users and d features, and the aim is to determine which k columns together pose the greatest re-identification risk. Our maximum coverage sketch directly enables a solution of targeted fingerprinting for risk measurement. Furthermore, we give a result of independent interest: a sketch related to the complement of Fk for k ⩾ 2. We use this sketch to solve general fingerprinting for risk management. Empirical evaluation confirms the practicality of our fingerprinting algorithms and shows a speedup of up to 210x over prior work. We also demonstrate the use of our general fingerprinting algorithm as a dimensionality reduction technique facilitating enhanced feature selection efficiency.

Based on joint work with Alina Ene, Alessandro Epasto, Vahab Mirrokni, Huy Nguyen, David P Woodruff, and Peilin Zhong

November 13th, 2024: Kernelization Complexity of some Solution Discovery Problems

A&C Seminar

Speaker: Stephanie Maaz

Abstract: The goal of this talk is to introduce the solution discovery framework and discuss a joint work with Mario Grobler, Amer E. Mouawad, Naomi Nishimura, Vijayaragunathan Ramamoorthi, and Sebastian Siebertz on the kernelization complexity of some solution discovery problems. Each of those solution discovery problems relates to one vertex (edge) subset source problem Π on graphs. It also takes as input an initial configuration of tokens on the vertices (edges) of an input graph G together with a budget b, and asks whether we can transform the given configuration into a feasible solution of Π on G with at most b modification steps. Here, each modification step consists of sliding a token to an adjacent vertex (edge). Our work focused on foundational source problems such as Vertex Cover, Independent Set, Shortest Path, and Vertex Cut and considered diverse parameters for kernelization, including the number of tokens k, the budget b, as well as structural parameters such as pathwidth. The solution discovery framework itself, which is the space where the previously described solution discovery problems can be defined, effectively models several interesting real-world applications. However, despite its potential, this framework has seen limited research and was only recently formally introduced by Fellows et al. [ECAI 2023]. Through the talk, we will familiarize ourselves with some of the key developments under this framework.

November 6th, 2024: Combinatorial aspects of matrix multiplication

A&C Seminar [slides]

Speaker: Kevin Pratt, NYU

Abstract: One of the major open problems in computer science to determine the exponent of matrix multiplication — the smallest number \omega such that two n-by-n matrices can be multiplied using n^{\omega +o(1)} arithmetic operations. While it is popularly conjectured that \omega = 2, we seem to be far from a proof (or disproof) of this. In this talk I will discuss this problem in the context of the group-theoretic approach of Cohn and Umans. After giving background on this approach, I will highlight some open problems in additive combinatorics which could potentially yield \omega = 2, and discuss some recent progress towards these problems.

October 30th, 2024: What is New in Join-Aggregate Query Processing?

A&C Seminar [video]

Speaker: Xiao Hu, University of Waterloo

Abstract: Join-aggregate queries defined over commutative semirings subsume a wide variety of common algorithmic problems, such as graph pattern matching, graph colorability, matrix multiplication, and constraint satisfaction problems. Developing efficient algorithms for computing join-aggregate queries in the conventional RAM model has been a holy grail in database theory. One of the most celebrated results in this area is the Yannakakis algorithm dating back to 1981. Despite its prominence as a textbook solution, no improvements in its complexity have been made over the past 40 years. In this talk, I will present the first algorithm that improves upon Yannakakis for computing acyclic join-aggregate queries. Moreover, this algorithm is proven to be output-optimal among all combinatorial algorithms. One application is an output-optimal algorithm for chain matrix multiplication over sparse matrices. Beyond combinatorial algorithms, I will also show how fast matrix multiplication can further speed up the processing of conjunctive queries, a critical subclass of join-aggregate queries. Finally, I will highlight a few interesting open problems in this area.

October 25th, 2024: Entrywise Approximate Laplacian Solving

A&C Seminar

Speaker: Jingbang Chen

Abstract: We will be discussing a joint work with Mehrdad Ghadiri, Hoai-An Nguyen, Richard Peng, Junzhao Yang. We study the escape probability problem in random walks over graphs. Given vertices, s,t, and p, the problem asks for the probability that a random walk starting at s will hit t before hitting p. Such probabilities can be exponentially small even for unweighted undirected graphs with polynomial mixing time. Therefore current approaches, which are mostly based on fixed-point arithmetic, require n bits of precision in the worst case. We present algorithms and analyses for weighted directed graphs under floating-point arithmetic and improve the previous best running times in terms of the number of bit operations. We believe our techniques and analysis could have a broader impact on the computation of random walks on graphs both in theory and in practice.

October 23rd, 2024: Separations in query complexity for total search problems

A&C Seminar [video]

Speaker: Srijita Kundu, University of Waterloo

Abstract: We study the query complexity analogue of the class TFNP of total search problems. We give a way to convert partial functions to total search problems under certain settings; we also give a way to convert search problems back into partial functions. Our results include exponential separations between quantum query complexity and approximate degree (and in fact non-negative approximate degree), between approximate degree and the positive-weights adversary bound, for TFNP problems. Using our conversion between TFNP problems and partial functions, we give a partial function separation between quantum query complexity and approximate degree, thus reproving a result of Ambainis and Belovs (2023). Finally, lifting some of our results, we prove new separations in communication complexity.

Based on joint work with Shalev Ben-David.

October 9th, 2024: Private Mean Estimation with Person-Level Differential Privacy

A&C Seminar

Speaker: Argyris Mouzakis, University of Waterloo

Abstract: We study person-level differentially private (DP) mean estimation in the case where each person holds multiple samples. DP here requires the usual notion of distributional stability when all of a person’s datapoints can be modified. Informally, if n people each have m samples from an unknown d-dimensional distribution with bounded k-th moments, we show that

n = \tilde{\Theta}(d/(α^2 m) + d/(α \sqrt{m} ε) +d/{α^{k / (k − 1)} m ε} +d/ε)

people are necessary and sufficient to estimate the mean up to distance α in ℓ2-norm under ε-differential privacy (and its common relaxations). In the multivariate setting, we give computationally efficient algorithms under approximate-DP and computationally inefficient algorithms under pure DP, and our nearly matching lower bounds hold for the most permissive case of approximate DP. Our computationally efficient estimators are based on the standard clip-and-noise framework, but the analysis for our setting requires both new algorithmic techniques and new analyses. In particular, our new bounds on the tails of sums of independent, vector-valued, bounded-moments random variables may be of interest. The paper is joint work with Sushant Agarwal, Gautam Kamath, Mahbod Majid, Rose Silver, and Jon Ullman, and is set to appear in SODA 2025. It is also available on arxiv.

October 2nd, 2024: Nearly Optimal List Labeling

A&C Seminar [video]

Speaker: Hanna Komlós, NYU

Abstract: The list-labeling problem captures the basic task of storing a dynamically changing set of up to N elements in sorted order in an array of size M=cN where c is a constant. The goal is to support insertions and deletions while moving around elements within the array as little as possible. Until recently, the best known upper bound stood at O(log^2(N)) amortized cost. This bound, which was first established in 1981, was finally improved two years ago, when a randomized O(log^{1.5}(N)) expected-cost algorithm was discovered. The best randomized lower bound for this problem remains Ω(log(N)), and closing this gap is considered to be a major open problem in data structures. We present the See-Saw Algorithm, a randomized list-labeling solution that achieves a nearly optimal bound of O(log(N) polyloglog(N)) amortized expected cost. This bound is achieved despite at least three lower bounds showing that this type of result is impossible for large classes of solutions. This work will appear at FOCS 2024.

September 25th, 2024: Algebraic Pseudorandomness in VNC^0

A&C Seminar [video]

Speaker: Robert Andrews, University of Waterloo

Abstract: Polynomial identity testing (PIT) is a central problem in theoretical computer science with numerous algorithmic applications, including applications to problems with no obvious algebraic character. Known polynomial-time algorithms for PIT crucially rely on the use of randomness. Designing a deterministic polynomial-time algorithm for PIT is a major goal of algebraic complexity theory.

Often, algorithms for PIT are obtained by constructing pseudorandom objects known as hitting set generators, which are analogous to the pseudorandom generators seen in boolean derandomization problems like P versus BPP. Recently, there has been interest in constructing such hitting set generators in a “cryptographic” regime of parameters, where the complexity of the generator is very small compared to the class of algebraic circuits it fools.

In this talk, I will describe a new construction of a generator where each output can be computed by a formula of constant size and the generator itself fools algebraic circuits of polynomial size and constant depth. Under strong (but reasonable) hardness assumptions, this generator also fools polynomial-size algebraic branching programs. No background in algebra or pseudorandomness is necessary!

September 18th, 2024: Spectral Sparsification by Deterministic Discrepancy Walk

A&C Seminar

Speaker: Lap Chi Lau, University of Waterloo

Abstract: Spectral sparsification and discrepancy minimization are two well-studied areas that are closely related. Building on recent connections between these two areas, we generalize the “deterministic discrepancy walk” framework by Pesenti and Vladu [SODA~23] for vector discrepancy to matrix discrepancy, and use it to give a simpler proof of the matrix partial coloring theorem of Reis and Rothvoss [SODA~20]. Moreover, we show that this matrix discrepancy framework provides a unified approach for various spectral sparsification problems, from stronger notions including unit-circle approximation and singular-value approximation to weaker notions including graphical spectral sketching and effective resistance sparsification. In all of these applications, our framework produces improved results with a simpler and deterministic analysis. Joint work with Robert Wang and Hong Zhou.

September 11th, 2024: New Reductions and Upper Bounds for Distance Preservers

A&C Seminar [video]

Speaker: Gary Hoppenworth, UMich

Abstract: Given a graph G, a distance preserver is a sparse subgraph H of G that exactly preserves shortest path distances between a small set of demand pairs P \subseteq V(G) \times V(G) in G. The study of distance preservers is primarily focused on the following question: How many edges are needed to construct a distance preserver of an n-vertex graph G with respect to a set P of p=|P| demand pairs? Distance preservers were first introduced by Coppersmith and Elkin in [SODA'06] and have connections to extremal graph theory and incidence geometry.

In the first part of this talk, we will review what is known about distance preservers. Some highlights include: 1) We will introduce a property of shortest paths known as ‘consistency’, and we will use consistency to bound the size of distance preservers by the number of edges in a graph avoiding a certain forbidden subgraph. 2) We will discuss a conjecture on distance preservers that generalizes the classical Szemeredi-Trotter Theorem of discrete geometry to finite metrics.

In the second part of this talk, we will present two new results related to distance preservers. First, we will present an extremal reduction from distance preservers in directed graphs to distance preservers in undirected graphs. This reduction has the property that any improvement in undirected distance preserver upper bounds in the regime of p \leq n would imply improved directed distance preserver upper bounds as well. Along the way, we prove the following algorithmic result: All-Pairs Shortest Paths (APSP) in directed acyclic graphs can be reduced to APSP in undirected graphs in O(m) time. Second, we will present new upper bounds for distance preservers in unweighted directed graphs. Upper bounds for distance preservers in unweighted undirected graphs were previously studied by Bodwin and Vassilevska Williams in [SODA'20].

The second part of this talk is based on joint work with Yinzhan Xu and Zixuan Xu.

August 20th, 2024: Constant-Cost Communication

A&C Seminar [video]

Speaker: Mika Goos, EPFL

Abstract: Some of the most extreme examples of the power of randomness in computing come from communication complexity, where shared randomness can allow two parties to solve non-trivial problems with communication cost independent of the input size. The textbook example is the Equality problem, where two parties hold n-bit strings x and y, respectively, and they wish to decide whether x = y. While n bits of deterministic communication are necessary, there is a randomised protocol that communicates only 2 bits. Can we characterise all communication problems that admit such hyperefficient protocols? We discuss recent progress and open problems.

Based on joint work with Yuting Fang, Nathaniel Harms, and Pooya Hatami.

August 14th, 2024: Scalable Algorithm for Finding Balanced Subgraphs with Tolerance in Signed Networks

A&C Seminar

Speaker: Jingbang Chen

Abstract: We will be discussing a joint work with Qiuyang Mang, Hangrui Zhou, Richard Peng, Yu Gao, Chenhao Ma, which is accepted by KDD2024. Signed networks, characterized by edges labeled as either positive or negative, offer nuanced insights into interaction dynamics beyond the capabilities of unsigned graphs. Central to this is the task of identifying the maximum balanced subgraph, crucial for applications like polarized community detection in social networks and portfolio analysis in finance. Traditional models, however, are limited by an assumption of perfect partitioning, which fails to mirror the complexities of real-world data. Addressing this gap, we introduce an innovative generalized balanced subgraph model that incorporates tolerance for irregularities. Our proposed region-based heuristic algorithm, tailored for this NP-hard problem, strikes a balance between low time complexity and high-quality outcomes. Comparative experiments validate its superior performance against leading solutions, delivering enhanced effectiveness (notably larger subgraph sizes) and efficiency (achieving up to 100x speedup) in both traditional and generalized contexts.

July 16th, 2024: Finding maximum matchings in RDV graphs efficiently

A&C Seminar

Speaker: Prashant Gokhale

Abstract: Abstract: We study the maximum matching problem in RDV graphs, i.e., graphs that are vertex-intersection graphs of downward paths in a rooted tree. We show that this problem can be reduced to a problem of testing (repeatedly) whether a vertical segment intersects one of a dynamically changing set of horizontal segments,which in turn reduces to an orthogonal ray shooting query. Using a suitable data structure, we can therefore find a maximum matching in O(n log n) time (presuming a linear-sized representation of the graph is given), i.e., without even looking at all edges.

July 3rd, 2024: Online Omniprediction

A&C Seminar [video]

Speaker: Sumegha Garg, Rutgers

Abstract: A recent line of work has shown a surprising connection between multicalibration, a multi-group fairness notion, and omniprediction, a learning paradigm that provides simultaneous loss minimization guarantees for a large family of loss functions [GKR+22, GHK+23 , GKR23 , GHHK+23]. Prior work studies omniprediction in the batch setting. Our work initiates the study of omniprediction in the online adversarial setting. In the talk, we will briefly see the definitions and motivations for these theoretic notions, and then survey the new results for online omniprediction. Our contributions are two-fold:

1. We develop a new online multicalibration algorithm that is well defined for infinite benchmark classes F (e.g. the set of all linear functions), and is oracle efficient — i.e. for any class F, the algorithm has the form of an efficient reduction to a no-regret learning algorithm for F. This implies an oracle efficient online omnipredictor — an online prediction algorithm that can be used to simultaneously obtain no regret guarantees to all Lipschitz convex loss functions.

2. We show upper and lower bounds on the extent to which our regret rates can be improved.

Joint work with Christopher Jung, Omer Reingold and Aaron Roth.

June 19th, 2024: The Border and its Demystification

Joint A&C Seminar and Distinguished Lecture Series [video]

Speaker: Nitin Saxena, IIT Kanpur

Abstract: Border (or approximative) complexity of polynomials plays an integral role in algebraic algorithms and the geometric complexity theory approach to P!=NP. This raises an important open question: can a border circuit be efficiently debordered (i.e., convert from approximative to exact)? Or, could the approximation involve exponential-precision which may not be efficiently simulable? It’s unclear how such circuits can be presented in practice if at all. We will discuss our new presentable-border definition and its application in circuit factorization.

Kumar (ToCT’20) proved the universal power of the border of top-fanin-2 depth-3 circuits. We recently solved some of the related open questions. If time permits, the talk will outline our result: border of bounded-top-fanin depth-3 circuits is relatively easy — it can be computed by a polynomial-size algebraic branching program (ABP).

Based on the works with C.S. Bhargav, Prateek Dwivedi and Pranjal Dutta; (CCC 2021) + (FOCS 2021, invited to SICOMP) + (FOCS 2022) + (STOC 2024).

June 12th, 2024: Deterministic Recoverable Consensus Numbers

A&C Seminar

Speaker: Sean Ovens, University of Waterloo

Abstract: Herlihy’s wait-free consensus hierarchy classifies the power of object types in asynchronous shared memory systems where processes can permanently crash (i.e. stop taking steps). In this hierarchy, a type has consensus number n if objects of that type can be used along with (read/write) registers to solve consensus among n processes that can permanently crash, but not among n+1 or more processes. In systems where processes can recover after crashing, the power of an object type to solve consensus may be different. Golab’s recoverable consensus hierarchy classifies the power of object types in such a system. In the recoverable consensus hierarchy, a type has recoverable consensus number n if objects of that type can be used along with registers to solve consensus among n processes that can recover after crashing, but not among n+1 or more processes. In this paper, we prove that the recoverable consensus hierarchy of deterministic, readable types is robust, i.e., if consensus can be solved among n processes that can recover after crashing using a collection of objects of deterministic, readable types, then one of these types has recoverable consensus number at least n. This is important for comparing the relative computational power of different deterministic, readable types, because it implies that one cannot combine various objects to obtain an algorithm that is better at solving recoverable consensus than any of the individual object types. Our result can be used to show that, for all n≥4, there exists a readable type with consensus number n and recoverable consensus number n−2. We also show that, for all n>n′≥1, there exists a non-readable type that has consensus number n and recoverable consensus number n′.

June 5th, 2024: Sandpile Prediction on Undirected Graphs

A&C Seminar

Speaker: Jingbang Chen

Abstract: We will be discussing a joint work on the sandpile prediction problem with Ruinian Chang, Ian Munro, Richard Peng, Qingyu Shi, Zeyu Zheng. The Abelian Sandpile model is a well-known model used in exploring self-organized criticality. Despite a large amount of work on other aspects of sandpiles, there have been limited results in efficiently computing the terminal state, known as the sandpile prediction problem. On graphs with special structures, we present algorithms that compute the terminal configurations for sandpile instances in O(nlogn) time on trees and O(n) time on paths, where n is the number of vertices. Our algorithms improve the previous best runtime of O(nlog^5n) on trees [Ramachandran-Schild SODA ‘17] and O(nlogn) on paths [Moore-Nilsson ‘99]. To do so, we move beyond the simulation of individual events by directly computing the number of firings for each vertex. The computation is accelerated using splittable binary search trees. In addition, we give algorithms in O(n) time on cliques and O(nlog^2n) time on pseudotrees. On general graphs, we propose a fast algorithm under the setting where the number of chips N could be arbitrarily large. We obtain a logN dependency, improving over the poly(N) dependency in purely simulation-based algorithms. Our algorithm also achieves faster performance on various types of graphs, including regular graphs, expander graphs, and hypercubes. We also provide a reduction that enables us to decompose the input sandpile into several smaller instances and solve them separately.

May 29th, 2024: Constant-Depth Arithmetic Circuits for Linear Algebra Problems

A&C Seminar [video]

Speaker: Robert Andrews, IAS

Abstract: How do you compute the greatest common divisor (GCD) of two univariate polynomials? The Euclidean algorithm provides a polynomial-time solution, and fast variants of the Euclidean algorithm can solve this problem in nearly-linear time. The GCD can also be expressed in a linear-algebraic form. Basic tasks in linear algebra like computing determinants and solving linear systems can be performed in O(log^2 n) parallel time, and this can be used to compute the GCD in O(log^2 n) parallel time. This algorithm does not take advantage of any structure present in the resulting linear systems, so in principle one could compute the GCD in parallel even faster.In this talk, I will describe a new algorithm that computes the GCD in O(log n) parallel time by using a combination of polynomial interpolation and Newton’s identities for symmetric polynomials. In fact, this algorithm can be implemented as an arithmetic circuit of constant depth. Similar ideas yield constant-depth circuits to compute the resultant, Bézout coefficients, and squarefree decomposition. Based on joint work with Avi Wigderson.

May 22nd, 2024: Directed Isoperimetry and Monotonicity Testing: A Dynamical Approach

A&C Seminar

Speaker: Renato Ferreira Pinto Jr

Abstract: This work explores the connection between classical isoperimetric inequalities, their directed analogues, and monotonicity testing. We study the setting of real-valued functions $f : [0,1]^d \to \mathbb{R}$ on the solid unit cube, where the goal is to test with respect to the $L^p$ distance. Our goals are twofold: to further understand the relationship between classical and directed isoperimetry, and to give a monotonicity tester with sublinear query complexity in this setting.

Our main results are 1) an $L^2$ monotonicity tester for $M$-Lipschitz functions with query complexity $\widetilde O(\sqrt{d} M^2 / \epsilon^2)$ and, behind this result, 2) the directed Poincaré inequality $\mathsf{dist}^{\mathsf{mono}}_2(f)^2 \le C \mathbb{E}[|\ abla^- f|^2]$, where the “directed gradient” operator $\ abla^-$ measures the local violations of monotonicity of $f$.

To prove the

April 30th, 2024: Privacy, Learning, and Inference

Joint A&C Seminar and Distinguished Lecture Series [video]

Speaker: Adam Smith, Boston University

Abstract: The results of learning and statistical inference reveal information about the data they use. This talk discusses the possibilities and limitations of fitting machine learning and statistical models while protecting the privacy of individual records.

I will begin by explaining what makes this problem difficult, using recent examples of training-data memorization and other breaches. I will present differential privacy, a rigorous definition of privacy in statistical databases that is now widely studied, and increasingly used to analyze and design deployed systems.

Time permitting, I will also present recent algorithmic results on a fundamental problem: differentially private mean estimation. We give an efficient and (nearly) sample-optimal algorithm for estimating the mean of “nicely” distributed data sets. When the data come from a Gaussian or sub-Gaussian distribution, the new algorithm matches the sample complexity of the best nonprivate algorithm.

April 5th, 2024: The Sublinear Lens and the Matching Problem

Joint A&C Seminar and Dean’s Distinguished Visiting Professor Lecture [video]

Speaker: Sanjeev Khanna, University of Pennsylvania

Abstract: The maximum matching problem occupies a central place in combinatorial optimization, and its study is intimately connected to major algorithmic advances. While much of the long and rich history of the matching problem revolves around classical algorithms, characterized by linear-time and linear-space as gold standards of efficiency, the past few decades have witnessed the emergence of an exciting new area of sublinear algorithms. The field of sublinear computation is driven by the challenges of computing over very large datasets, and focuses on the design of algorithms that use computational resources significantly smaller than the input size. Remarkably, much like in the classical setting, the matching problem has come to play a pivotal role in the study of sublinear algorithms.

This talk will be a (personally biased) journey through some surprising algorithmic results and unexpected connections discovered by exploring the matching problem through the lens of sublinear algorithms.

April 3rd, 2024: Towards Practical Distribution Testing

A&C Seminar [video]

Speaker: Yash Pote, NUS

Abstract: Inferring information about probability distributions with limited samples is a fundamental challenge in computer science. In this talk, we will focus on the problem of estimating the distance between pairs of distributions. Specifically, given distributions P and Q, and a parameter epsilon, we will estimate the total variation distance TV(P, Q) up to an additive tolerance of epsilon. In high dimensions, i.e., when the domain is {0,1}^n for a large n, black-box distance estimation is intractable; hence, richer “grey-box” models, such as conditional sampling, have emerged as promising alternatives. In this talk, I will present our work in developing grey-box estimators that are fast in theory and practice.

March 27th, 2024: Stochastic Minimum Vertex Cover with Few Queries: a 3/2-approximation

A&C Seminar [video]

Speaker: Mahsa Derakhshan, Northeastern University

Abstract: In this talk, we discuss the stochastic vertex cover problem. In this problem, G is an arbitrary known graph, and G* is an unknown random subgraph of G containing each of its edges independently with a known probability p. Edges of G* can only be verified using edge queries. The goal in this problem is to find a minimum vertex cover of G* using a small number of queries. In this talk, we present a 3/2-approximation algorithm using only O(n/p) non-adaptive queries. This is an improvement over the prior 2-approximation algorithm by Behnezhad et al., who also show that Ω(n/p) queries are necessary to achieve any constant approximation. We will also discuss how this result can be extended to instances where edge realizations are not fully independent. We complement this upper bound with a tight 3/2-approximation lower bound for stochastic graphs whose edge realizations demonstrate mild correlations.

February 22nd, 2024: Randomness without randomness: Structural explanations of the power of randomness in communication

A&C Seminar

Speaker: Nathan Harms, EPFL

Abstract: Understanding the power of randomness in communication is one of the main goals in communication complexity. Towards this goal, many recent works have studied *constant-cost* communication problems, which are the most extreme cases of the power of randomized communication. In this talk I will survey some of these works and discuss several results which present structural explanations for the power of certain constant-cost randomized protocols.

February 5th, 2024: New Techniques for Convex Optimization and Sparsification

A&C Seminar

Speaker: Arun Jambulapati, University of Michigan

Abstract: The computational challenges posed by recent massive datasets motivate the design of more efficient algorithms. My work takes problems motivated by modern machine learning and develops theoretical tools to answer two central questions: 1) how can we learn from data faster, and 2) how can we select representative data (to potentially speed up downstream processing). I will first present several results on faster convex optimization leveraging a new theoretical primitive called a “ball optimization oracle”. I will give near-optimal algorithms for minimizing convex functions assuming access to this oracle and show how this framework can be applied to yield faster algorithms for important problems in computer science such as regression, differentially private optimization, and robust optimization.

I will follow with results on problems motivated by dataset compression. Leveraging techniques from high-dimensional geometry, I give near-optimal constructions of sparsifiers for sums of norms and generalized linear models. This directly implies new sparsifiers for hypergraphs, sums of symmetric submodular functions, and gives faster algorithms for linear regression.

January 24th, 2024: Fast Algorithms for Directed Graph Partitioning Using Flows and Reweighted Eigenvalues

A&C Seminar

Speaker: Robert Wang, University of Waterloo

Abstract: We consider a new semidefinite programming relaxation for directed edge expansion, which is obtained by adding triangle inequalities to the reweighted eigenvalue formulation. Applying the matrix multiplicative weight update method to this relaxation, we derive almost linear-time algorithms to achieve O(\sqrt{logn})-approximation and Cheeger-type guarantee for directed edge expansion, as well as an improved cut-matching game for directed graphs. This provides a primal dual flow-based framework to obtain the best known algorithms for directed graph partitioning. The same approach also works for vertex expansion and for hypergraphs, providing a simple and unified approach to achieve the best known results for different expansion problems and different algorithmic techniques.

January 19th, 2024: Sorting and Selection in Rounds with Adversarial Comparisons

A&C Seminar

Speaker: Chris Trevisan, University of Waterloo

Abstract: We continue the study of selection and sorting of n numbers under the adversarial comparator model, where comparisons can be adversarially tampered with if the arguments are sufficiently close.We derive a randomized sorting algorithm that does O(nlog2n) comparisons and gives a correct answer with high probability, addressing an open problem of Ajtai, Feldman, Hassadim, and Nelson [AFHN15]. Our algorithm also implies a selection algorithm that does O(nlogn) comparisons and gives a correct answer with high probability. Both of these results are a log factor away from the naive lower bound. [AFHN15] shows an Ω(n1+ε) lower bound for both sorting and selection in the deterministic case, so our results also prove a discrepancy between what is possible with deterministic and randomized algorithms in this setting. We also consider both sorting and selection in rounds, exploring the tradeoff between accuracy, number of comparisons, and number of rounds. Using results from sorting networks, we give general algorithms for sorting in d rounds where the number of comparisons increases with d and the accuracy decreases with d. Using these algorithms, we derive selection algorithms in d+O(logd) rounds that use the same number of comparisons as the corresponding sorting algorithm, but have a constant accuracy. Notably, this gives selection algorithms in d rounds that use n1+o(1) comparisons and have constant accuracy for all d=ω(1), which still beats the deterministic lower bound of Ω(n1+ε). The paper is available in https://arxiv.org/abs/2310.09483 and has appeared in SODA 2024.

January 17th, 2024: A Distributed Palette Sparsification Theorem

A&C Seminar [video]

Speaker: Maxime Roland René Flin, Reykjavik University

Abstract: The celebrated palette sparsification result of [Assadi, Chen, and Khanna SODA'19] shows that to compute a Delta+1 coloring of the graph, where Delta denotes the maximum degree, it suffices if each node limits its color choice to O(log n) independently sampled colors. They showed that it is possible to color the resulting sparsified graph – with edges between neighbors that sampled a common color – and obtain a Delta+1 coloring for the original graph. However, to compute the actual coloring, that information must be gathered at a single location for centralized processing.

In this talk, I will explain how we can compute such a coloring distributively. The Distributed Palette Sparsification Theorem states that after palette sparsification with O(log^2 n) colors, there exists a O(log^2 Delta + log^3 log n)-round CONGEST algorithm on the sparsified graph that computes the coloring. We will see how this relates to the issue of distributively computing a perfect matching in a random graph.

In addition to combinatorial insight about palette sparsification, our theorem deepens the connections between sublinear and distributed coloring algorithms: it provides the first poly(log n)-round coloring algorithms in known constrained distributed models where vertices cannot afford to communicate on all their adjacent edges.

No prior knowledge about palette sparsification or distributed algorithms is required.

Based on joint work with Mohsen Ghaffari, Magnus M. Halldorsson, Fabian Kuhn, and Alexandre Nolin (https://arxiv.org/abs/2301.06457)

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2023

December 6th, 2023: Graph Morphing via Orthogonal Box Drawings

Thesis Presentation

Speaker: Jack Spalding-Jamieson

Abstract: A graph is a set of vertices, with some pairwise connections given by a set of edges. A graph drawing, such as a node-link diagram, visualizes a graph with geometric features. One of the most common forms of a graph drawings are straight-line point drawings, which represent each vertex with a point and each edge with a line segment connecting its relevant points, and poly-line point drawings, which more generally allow edges to be represented by poly-lines. Of particular interest to this work are planar straight-line drawings and planar poly-line drawings, in which no two vertices share a location, and no two edges cross (except at shared endpoints).We study the morphing problem for planar drawings: Given two planar drawings of the same graph, can we output a continuous transformation (a “morph”) from one to the other, such that each intermediate drawing is also a planar drawing? It is quite easy to test if a morph exists, but the test is non-constructive. We are interested in the problem of constructing morphs with simple representations. Specifically, we study sequences of linear morphs, which represent the overall morph with a sequence of drawings, so that each pair of adjacent drawings in the sequence can be linearly interpolated. Each drawing in the sequence is called an “explicit” intermediate drawing, since it given explicitly in the output.Previous work has shown that a pair of straight-line drawings of an n-vertex graph can be morphed using O(n) linear morphs, so that every explicit intermediate drawing is a straight-line drawing. We show that an additional constraint can be added, at the cost of a small tradeoff: We further restrict the explicit intermediate drawings to lie on an O(n)×O(n) grid, while allowing them to be poly-line drawings with O(1) bends per edge. Additionally, we give an algorithm that computes this sequence in O(n^2) time, which is known to be tight. Our methods involve morphing another class of drawings—orthogonal box drawings—which represent each vertex with an axis-aligned rectangle, and each edge with an orthogonal poly-line. Our methods for morphing orthogonal box drawings make use of methods known for morphing orthogonal point drawings, which are poly-line drawings that restrict each poly-line to use only axis-aligned line segments.

November 29th, 2023: New Codes on High Dimensional Expanders

A&C Seminar [video]

Speaker: Rachel Yun Zhang, MIT

Abstract: A code, which is a set of strings called codewords, is locally testable if one can test whether a given word is close to a codeword by reading only a few bits. Locally testable codes have been studied since the 1990s as key ingredients in the construction of probabilistically checkable proofs.

In this talk, I’ll describe a new construction of locally testable codes implemented on high dimensional expanders. It has been known for several years that high dimensional expanders give rise to locally testable codes assuming the existence of a compatible base code. Our main contribution is the construction and analysis of such a base code compatible with the coset complex of Kaufman and Oppenheim.

November 27th, 2023: Top-Down Lower Bounds for Depth-Four Circuits

A&C Seminar

Speaker: Mika Göös, EPFL

Abstract: We present a top-down lower-bound method for depth-4 boolean circuits. In particular, we give a new proof of the well-known result that the parity function requires depth-4 circuits of size exponential in n^{1/3}. Our proof is an application of robust sunflowers and block unpredictability.

November 22nd, 2023: Dynamic matching with (2-eps) approximation in polylog time

A&C Seminar

Speaker: David Wajc, Technion

Abstract: In this talk I will present dynamic algorithms with polylogarithmic update time for estimating the size of the maximum matching of a graph undergoing edge insertions and deletions with approximation ratio strictly better than 2. These algorithms answer in the affirmative (the value version of) a major open question repeatedly asked in the dynamic graph algorithms literature. I will give a high-level idea of the key ingredients needed for this result, and discuss subsequent developments. Based on a SODA 2023 best paper, joint with Sayan Bhattacharya, Peter Kiss and Thatchaphol Saranurak.

November 15th, 2023: Interior-point methods on manifolds: theory and applications

A&C Seminar [video]

Speaker: Harold Nieuwboer

Abstract: Interior-point methods offer a highly versatile framework for convex optimization that is effective in theory and practice. A key notion in their theory is that of a self-concordant barrier. We give a suitable generalization of self-concordance to Riemannian manifolds and show that it gives the same structural results and guarantees as in the Euclidean setting, in particular local quadratic convergence of Newton’s method. We analyze a path-following method for optimizing compatible objectives over a convex domain for which one has a self-concordant barrier, and obtain the standard complexity guarantees as in the Euclidean setting. We provide general constructions of barriers, and show that on the space of positive-definite matrices and other symmetric spaces, the squared distance to a point is self-concordant. To demonstrate the versatility of our framework, we give algorithms with state-of-the-art complexity guarantees for the general class of scaling and non-commutative optimization problems, which have been of much recent interest, and we provide the first algorithms for efficiently finding high-precision solutions for computing minimal enclosing balls and geometric medians in nonpositive curvature.

Joint work with Hiroshi Hirai and Michael Walter, based on https://arxiv.org/abs/2303.04771.

October 25th, 2023: Fiat-Shamir in the Plain Model from Derandomization

A&C Seminar

Speaker: Roei Tell, University of Toronto

Abstract: Fiat-Shamir in the Plain Model from Derandomization

Do efficient algorithms believe that NP = PSPACE? Under strong complexity-theoretic assumptions, we prove that every problem in PSPACE can be solved by an NP-type verifier running in polynomial time that looks correct to any efficient observer (i.e., it has computational soundness). For example, the PSPACE-complete problem TQBF can be decided by a polytime NP-type verifier V, interacting with an honest prover whose runtime is T(n) = 2^O(n), such that no adversary running in time poly(T) can find an input and a proof that mislead V (except with probability negligible in T over the adversary’s coins).

The proof uses recent techniques from non-black-box derandomization in order to compile the interactive proof system underlying IP = PSPACE into a deterministic (non-interactive) argument system. This is an instance from a broader set of results applying derandomization-based techniques and assumptions to instantiate (versions of) the Fiat-Shamir heuristic in cryptography, in the plain model and without (or with weak) cryptographic assumptions.

From an upcoming joint work with Lijie Chen and Ron Rothblum.

October 18th, 2023: Testing Graph Properties with the Container Method

A&C Seminar

Speaker: Cameron Seth, University of Waterloo

Abstract: We establish nearly optimal sample complexity bounds for testing the clique property in the dense graph model. Specifically, we show that it is possible to distinguish graphs on n vertices that have a clique of size rho n from graphs for which at least epsilon n^2 edges must be added to form a rho n clique by sampling and inspecting a random subgraph on only O(rho^3/epsilon^2) vertices.

The new bounds for testing the clique property are obtained via a new extension of the graph container method. This method has been an effective tool for tackling various problems in graph theory and combinatorics. Our results demonstrate that it is also a powerful tool for the analysis of property testing algorithms.

This is joint work with Eric Blais.

October 13th, 2023: Fast multivariate multipoint evaluation

A&C Seminar [slides]

Speaker: Mrinal Kumar, TIFR

Abstract: Multipoint evaluation is the computational task of evaluating a polynomial given as a list of coefficients at a given set of inputs. A straightforward algorithm for this problem is to just iteratively evaluate the polynomial at each of the inputs. The question of obtaining faster-than-naive (and ideally, close to linear time) algorithms for this problem is a natural and basic question in computational algebra. In addition to its own inherent interest, faster algorithms for multipoint evaluation are closely related to fast algorithms for other natural algebraic questions like polynomial factorization and modular composition. In this talk, I will briefly survey the state of art for this problem, and discuss some recent improvements and applications.

October 12th, 2023: Ray-volution: Unveiling the Potentials of GPU-Accelerated Ray Tracing in Non-Graphical Domains

A&C Seminar

Speaker: Rosina Kharal

Abstract: Over the past decade, the field of computational science has experienced significant advancements, largely fueled by the evolution of parallel computing. Utilizing the parallel processing capabilities of GPUs has facilitated previously inconceivable gains in computationally demanding tasks. Initially designed for graphics rendering, GPUs now form the foundation of vast domains such as data analytics and machine learning. Currently, we are navigating through another transformation led by GPUs in the realm of high-performance computing (HPC).Recently, companies like NVIDIA and AMD have launched GPU acceleration hardware to substantially enhance real-time ray tracing capabilities. These specialized components, known as Accelerated Ray Tracing Cores (RTX cores), are now available on consumer-grade GPUs, including the NVIDIA GeForce RTX series and the Titan RTX series. Initially aimed at accelerating ray tracing for graphics, these cores have begun attracting research interest for a variety of other applications. This seminar will shed light on the progression and potential of accelerated ray tracing technology, from its roots in graphics to its expanding role in non-graphical applications. We will explore how this technology can work alongside existing GPU multiprocessors to foster new developments in computational science. The session’s primary goal is to provide a deeper understanding of RTX technology, including an analysis of existing research where RTX is applied to non-graphical tasks, and a focus on current work in progress.

October 11th, 2023: Cheeger Inequalities for Directed Graphs and Hypergraphs using Reweighted Eigenvalues

A&C Seminar

Speaker: Kam Chuen (Alex) Tung

Abstract: We derive Cheeger inequalities for directed graphs and hypergraphs using the reweighted eigenvalue approach that was recently developed for vertex expansion in undirected graphs. The goal is to develop a new spectral theory for directed graphs and an alternative spectral theory for hypergraphs. The first main result is a Cheeger inequality relating the vertex expansion of a directed graph to the vertex-capacitated maximum reweighted second eigenvalue. This provides a combinatorial characterization of the fastest mixing time of a directed graph by vertex expansion, and builds a new connection between reweighted eigenvalues, vertex expansion, and fastest mixing time for directed graphs. The second main result is a stronger Cheeger inequality relating the edge conductance of a directed graph to the edge-capacitated maximum reweighted second eigenvalue. This provides a certificate for a directed graph to be an expander and a spectral algorithm to find a sparse cut in a directed graph, playing a similar role as Cheeger’s inequality in certifying graph expansion and in the spectral partitioning algorithm for undirected graphs. We also use this reweighted eigenvalue approach to derive the improved Cheeger inequality for directed graphs, and furthermore to derive several Cheeger inequalities for hypergraphs that match and improve the existing results. These are supporting results that this provides a unifying approach to lift the spectral theory for undirected graphs to more general settings.

(Joint work with Lap Chi Lau and Robert Wang. arXiv version: https://arxiv.org/abs/2211.09776)

October 4th, 2023: Notions of “rank” for boolean matrices

A&C Seminar

Speaker: Kaave Hosseini, University of Rochester

Abstract: In this talk I will discuss a few well-known complexity parameters for boolean matrices that are relaxations of rank (over the reals). These are approximate rank, sign-rank/dimension complexity, margin/discrepancy, gamma2 norm, and approximate-gamma2. The focus of this talk is to study the meta question: “what is the relationship between these parameters?”. Surprisingly, this meta question is not yet fully understood. I will try to answer some of these pairwise relations using different tools such Fourier analysis, topology, and also ideas from discrete geometry.

It turns out that study of this meta question connects many different branches of mathematics and equivalent stories are to be told in learning theory, communication complexity, convex geometry, theory of dimensionality reduction, etc. For example, in learning theory one related question is: if a binary concept class is learnable via large margin classifiers (such as SVM) does it imply that it is learnable via linear programming, and vice versa? In communication complexity, one question is whether one can turn a public-randomness protocol into a private-randomness protocol without a super constant overhead, if one allows the error probability to get arbitrarily close to 1/2.

September 27th, 2023: Hidden Permutations to the Rescue: Multi-Pass Semi-Streaming Lower Bounds for Approximate Matchings

A&C Seminar

Speaker: Sepehr Assadi, University of Waterloo

Abstract: A semi-streaming (graph) algorithm processes its input graph by making one or a few passes over its edges and using a space proportional to the number of vertices, hence, (potentially) quadratically smaller than the input size. Semi-streaming algorithms have been at the forefront of theoretical research on processing massive graphs in recent years. In this talk, we will consider the maximum (bipartite) matching problem in the semi-streaming model.

There is a large body of semi-streaming algorithms for the maximum matching problem, developed over the last two decades, that achieve a (1+eps)-approximation in progressively smaller number of passes as a function of eps and size of the graph. In sharp contrast to this progress however, there is scarcely any lower bounds known for this problem, beyond single-pass and very recently two-pass algorithms. For instance, it is not even ruled out whether one can achieve a (1+eps)-approximation in a fixed constant number of passes, say, even three passes.

In this talk, we present the first pass-approximation tradeoff for (small) constant-factor approximation of semi-streaming matchings: semi-streaming (1-ε)-approximation of bipartite matching (even its size) requires Omega(log(1/ε)) passes under a natural combinatorial hypothesis that moderately dense Ruzsa-Szemeredi graphs do exist. The key to our analysis is a novel way of analyzing communication complexity of hiding permutations directly, instead of relying on tools from Boolean function analysis of prior work in this context. This maybe of its own independent interest.

The talk will be self-contained and no prior background in streaming algorithms or communication complexity will be needed to enjoy this talk!

This is joint work with Janani Sundaresan (to appear in FOCS 2023).

September 25th, 2023: “A Tale of Two Subspaces” [Regev]

A&C Seminar

Speaker: Akshay Ramachandran, CWI

Abstract: Optimization problems with orthogonality constraints arise in many fields in science and engineering. For example, optimization over subspaces in physics and signal processing, and over rotations in computational geometry. The key step in solving these problems often boils down to understanding the relation between two subspaces. It turns out that this question has a surprisingly elegant answer given by the CS decomposition from numerical linear algebra. In this talk, we will discuss the CS decomposition in the context of the geodesic geometry of subspaces. This perspective further gives unifying framework for understanding the various numerical algorithms used to solve problems with orthogonality constraints. As our main illustration, we study the problem of computing eigenspaces of a matrix, giving a rigorous convergence analysis of the well-known power method and its subspace generalization. If time permits, we also present some new results on tractable algorithms for constrained optimization over rotation matrices. This is joint work with Kevin Shu and Alex Wang.

September 13th, 2023: How to Build a Distributed System with Real-time Constraints using Concurrent Functional Programming Tools

A&C Seminar

Speaker: Armando Santos, Well-Typed/Haskell Consultants

Abstract: Building complex distributed systems with real-time constraints is challenging; when they are intended to be run by end users in an adversarial environment with no central control it becomes especially challenging! These constraints make off-the-shelf solutions inappropriate, particularly when the only reliable assumption is that only half of the network’s nodes are ‘honest’, which makes creating a robust system extremely difficult. To accomplish this, we exploit the finest of what Strongly Statically Typed Purely Functional Programming has to offer. Type-safeness, referential transparency, compositionality, explicit control of effects, STM, and clever testing approaches are among the concepts and tools in our repertoire for managing complexity and ensuring the continuous operation of a distributed network.This presentation will highlight some of the important items the IOG networking team is working on and where we believe we are succeeding.

July 21st, 2023: Combinatorial and complexity theoretic aspects of Stabilities and Controllabilities of linear switched systems(discrete and continuous time)

Joint A&C Seminar and Tutte Colloquium

Speaker: Leonid Gurvits, The City College of New York

Abstract: I will talk about my “pre-hyperbolic” research, some of it done jointly with Alex Samorodnitsky and Alex Olshevsky.

For example:

1. One of the results I plan to present gives a criteria for the existence of n×n real Hadamard Matrix expressed in terms of best approximation of so called joint spectral radius by quadratic Lyapunov Functions.

2. Another one is about a complexity theoretic separation in analysis of Switched Absolute Stability for Open Classical(easy) and Quantum(hard) Linear Continuous Time Systems.

3. Time permitting, a separation between separable vs entangled states will be presented in terms regular vs nonregular languages.

The talk should be accessible to wide mathematical audience as all necessary notions will be introduced/explained on the fly.

July 19th, 2023: Succinct Intersection Graphs Revisited

A&C Seminar

Speaker: Kevin Wu

Abstract: We enhance several data structures for intersection graphs. The data structures of Chakraborty and Jo for bounded degree and bounded chromatic number interval graphs, Balakrishnan et al. We show that both bounded degree and bounded chromatic number interval graphs have a tight lower bound of $n\log \sigma$ (lower order terms omitted) bits where $\sigma$ is the maximum degree or chromatic number of the graph. This improves the lower bound of Chakraborty and Jo from $\frac{1}{6}n\log\sigma$. For bounded chromatic number interval graphs, we give a $n\log \sigma$ bit space succinct data structure with $O(\sigma\log n)$ query times. To match the time complexity of $O(\log\log\sigma)$ we use $2n\log\sigma$ bits rather than $(\sigma -1)n$ bits. For path graphs, we give a succinct $n\log n$ bit data structure with query times $O(\frac{\log n}{\log\log n})$ rather than $O(\log^2 n)$ of Balakrishnan et al. To achieve $O(1)$ query times, we give a data structure using $(2+\varepsilon)n\log n$ bits rather than $O(n\log^2 n)$ bits.

July 14th, 2023: Causal structure learning and online algorithms with ML advice

A&C Seminar

Speaker: Themis Gouleakis, NUS

Abstract: The enormous success of the field of machine learning in recent years and its ability to make accurate predictions using data has also influenced research in other areas. In this talk, we will explore such settings where “machine learned advice” can be exploited. We first introduce the problem of active causal structure learning with advice. In the typical well studied setting, the learning algorithm is given the essential graph for the observational distribution and is asked to recover the underlying causal directed acyclic graph (DAG) G∗ while minimizing the number of interventions made. In our setting, we are additionally given side information about G∗ as advice, e.g. a DAG G purported to be G∗ . We ask whether the learning algorithm can benefit from the advice when it is close to being correct, while still having worst-case guarantees even when the advice is arbitrarily bad. When the advice is a DAG G, we design an adaptive search algorithm to recover G∗ whose intervention cost is at most O(max{1, log ψ}) times the cost for verifying G∗ ; here, ψ is a distance measure between G and G∗ that is upper bounded by the number of variables n, and is exactly 0 when G = G∗ . Our approximation factor matches the state-of-the-art for the advice-less setting.

Another area of theoretical computer science that could benefit from that is online algorithms, where the goal is to exploit the ability of machine learning algorithms to make predictions of future input to the algorithm. The classical analysis of online algorithms, due to its worst-case nature, can be quite pessimistic when the input instance at hand is far from the worst-case. Often this is not an issue with machine learning approaches, which shine in exploiting patterns in past inputs in order to predict the future. However, such predictions, although usually accurate, can be arbitrarily poor. We will also discuss recent results in this context for the Online Traveling Salesperson Problem (OLTSP) with predictions.

July 12th, 2023: The basis number of 1-planar graphs

A&C Seminar

Speaker: Bobby Miraftab, Carleton University

Abstract: 1-planar graphs refer to graphs that can be drawn in the plane with at most one crossing per edge. MacLane’s planarity criterion provides a characterization of planar graphs based on their cycle spaces. More precisely, it says a graph is planar if and only if it contains a 2-basis (a basis in which each edge belongs to at most two elements of the basis). The basis number of a graph G is defined as the smallest integer k for which G has a k-basis B,( B spans the cycle space of G and satisfies the condition that each edge of G is contained in at most k members of B). We address the following question: does there exist a universal constant c such that every 1-planar graph has a c-basis? We give an affirmative answer for several classes of 1-planar graphs.

July 7th, 2023: Speculative Smart Contracts in the Algorand Blockchain

A&C Seminar

Speaker: Victor Luchangco

Abstract: The Algorand blockchain is a fast pure proof-of-stake blockchain with native (“layer 1”) support for atomic transactions, user-defined assets, and smart contracts. Layer-1 smart contracts are tightly constrained in both storage and execution to ensure that they do not unduly impact the overall performance of the blockchain. To accommodate applications with greater storage and computation requirements, we propose speculative smart contracts, which maintain storage and execute contract calls off-chain and issue “effects transactions” that summarize the effects of the calls on the blockchain itself.

In this talk, we describe how to implement speculative smart contracts, and the features of the Algorand blockchain that we exploit, and some extensions that are required, to do so. Speaker Bio: Victor Luchangco is a Principal Algorithms Researcher at Algorand, where he works on protocols and languages for blockchains. He is a co-author of The Art of Multiprocessor Programming, and is a recipient of the 2022 Dijkstra Prize in Distributed Computing. Before joining Algorand, he worked at Oracle Labs and Sun Labs on concurrent algorithms and data structures for shared-memory multiprocessors and the Fortress programming language.

He has authored over 50 papers and holds more than 50 patents. He received an Sc.D. in Computer Science from the Massachusetts Institute of Technology with a dissertation on models for weakly consistent memories.

July 5th, 2023: Optimal PAC Bounds without Uniform Convergence

A&C Seminar

Speaker: Ishaq Aden-Ali, UC Berkeley

Abstract: In statistical learning theory, determining the sample complexity of realizable binary classification for VC classes was a long-standing open problem. The results of Simon and Hanneke established sharp upper bounds in this setting. However, the reliance of their argument on the uniform convergence principle limits its applicability to more general learning settings such as multiclass classification. In this talk, we will discuss a simple technique that addresses this issue. We will present optimal high probability risk bounds through a framework that surpasses the limitations of uniform convergence arguments.

In addition to binary classification, we will see applications in three settings where uniform convergence is provably suboptimal. For multiclass classification, we prove an optimal risk bound scaling with the one-inclusion hypergraph density of the class, improving on the suboptimal analysis of Daniely and Shalev-Shwartz. In partial concept classification, we determine the optimal sample complexity bound, resolving a question posed by Alon, Hanneke, Holzman, and Moran. In the context of realizable bounded regression with absolute loss, we derive an optimal risk bound that relies on a modified version of the scale-sensitive dimension, refining a result of Bartlett and Long. Our rates surpass standard uniform convergence-based results due to the smaller complexity measure in our risk bound.

Joint work with Yeshwanth Cherapanamjeri, Abhishek Shetty, and Nikita Zhivotovskiy.

May 24th, 2023: Is Planted Coloring Easier than Planted Clique?

A&C Seminar

Speaker: Alex Wein, UC-Davis

Abstract: The task of finding a planted clique in the random graph G(n,1/2) is perhaps the canonical example of a statistical-computational gap: for some clique sizes, the task is statistically possible but believed to be computationally hard. Really, there are multiple well-studied tasks related to the planted clique model: detection, recovery, and refutation. While these are equally difficult in the case of planted clique, this need not be true in general. In the related planted coloring model, I will discuss the computational complexity of these three tasks and the interplay among them. Our computational hardness results are based on the low-degree polynomial model of computation.

By taking the complement of the graph, the planted coloring model is analogous to the planted clique model but with many planted cliques. Here our conclusion is that adding more cliques makes the detection problem easier but not the recovery problem.

Based on joint work with Pravesh Kothari, Santosh Vempala, and Jeff Xu, available at https://arxiv.org/abs/2303.00252

March 15th, 2023: Differentially Private Approximate Histograms using the Misra-Gries Sketch

A&C Seminar

Speaker: Christian Janos Lebeda, IT University of Copenhagen

Abstract: Estimating the histogram of a dataset is a fundamental task in differential privacy. Privacy is typically achieved by first computing the histogram and then independently adding noise to each entry. However, this approach assumes that we can compute the histogram exactly. This is not always possible such as in the data stream model where memory is very limited. In the non-private setting, we can approximate the histogram of a stream of elements with optimal worst-case guarantees using the Misra-Gries sketch. Chan, Li, Shi, and Xu [PETS 2012] first introduced an algorithm for releasing a Misra-Gries sketch under differential privacy, but the magnitude of the noise required to achieve privacy scales linearly in the size of the sketch.In this talk, I will present a new algorithm for differentially private Misra-Gries sketches. I first introduce relevant techniques from differential privacy and the data stream model. I then present a new observation about the structure of Misra-Gries sketches. We utilize this structure to achieve differential privacy by adding noise almost identical to the non-streaming setting. This is joint work with Jakub Tětek.

March 1st, 2023: The Long Arm of Theoretical Computer Science: A Case Study in Blockchains/Web3

Joint A&C Seminar and Distinguished Lecture Series [video]

Speaker: Tim Roughgarden, Columbia University; Head of Research, a16z crypto

Abstract: Blockchains that support a general contract layer (e.g., Ethereum) export the functionality of a general-purpose, ownerless, and open-access computer that can enforce property rights for digital data. How is such functionality implemented? Using a lot of extremely cool computer science ideas! And like everywhere else in computer science, theory plays an undeniable role in the understanding and advancement of this technology.

In this talk, I’ll highlight three examples (among many):

• Possibility and impossibility results for permissionless consensus (i.e., implementing an “ownerless” computer).
• Incentive-compatible transaction fee mechanism design (part of implementing an “open-access” computer).
• Succinct proofs of computation (for boosting the computer’s power by piggybacking on off-chain computation).

January 17th, 2023: Title not recorded

A&C Seminar

Speaker: Daniela Opocenska, UWaterloo exchange student

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2022

December 6th, 2022: On Matrix Multiplication and Polynomial Identity Testing

A&C Seminar [video]

Speaker: Robert Andrews, University of Illinois Urbana-Champaign

Abstract: Determining the complexity of matrix multiplication is a fundamental problem of theoretical computer science. It is popularly conjectured that ω, the matrix multiplication exponent, equals 2. If true, this conjecture would yield fast algorithms for a wide array of problems in linear algebra and beyond. If instead ω > 2, can we leverage the hardness of matrix multiplication to design algorithms for other problems?

In this talk, I will describe how lower bounds on ω can be used to make progress on derandomizing polynomial identity testing.

November 11th, 2022: Isoperimetric Inequalities for Real-Valued Functions with Applications to Monotonicity Testing

A&C Seminar [video]

Speaker: Iden Kalemaj

Abstract: We study sublinear algorithms for monotonicity of real-valued functions. An algorithm for testing monotonicity queries a function at a few locations and distinguishes between when the function is monotone and when it is far from monotone. An algorithm for approximating the distance to monotonicity returns the distance to monotonicity of the function with some additive and multiplicative error. We improve previous algorithms for these two tasks by showing that several isoperimetric inequalities about Boolean functions on the hypercube can be generalized to real-valued functions.

A celebrated line of work [Margulis ‘74, Talagrand ‘93, Chakrabarty and Seshadhri ‘13, Khot Minzer Safra ‘15] studied the size of the “boundary” between the points on which a Boolean function takes value 0 and the points on which it takes value 1. This boundary is defined in terms of the edges of the d-dimensional hypercube, where the edges can be directed or undirected. In particular, the inequality of Khot, Minzer, and Safra ‘15 implies all previous inequalities. It bounds the average square root of the number of decreasing edges incident on a vertex of the hypercube by the distance to monotonicity of the function. We generalize all the inequalities in this line of work to real-valued functions. Our main contribution is a Boolean decomposition technique that represents a real-valued function as a collection of Boolean functions that roughly capture the distance to monotonicity of the original function. In this talk, we prove the Boolean decomposition theorem and touch on the improved algorithms for monotonicity testing and distance approximation.

This is joint work with Hadley Black and Sofya Raskhodnikova.

October 28th, 2022: New Lower Bounds for Set Multilinear Formulas

A&C Seminar [video]

Speaker: Shubhangi Saraf

Abstract: The recent exciting breakthrough by Limaye, Srinivasan, and Tavenas showing superpolynomial lower bounds for constant-depth algebraic circuits has underscored the importance of studying the complexity of set multilinear formulas. In this talk, along with discussing the impact of this work and several others in the literature, I will discuss some new results in this area, namely strong (and “sharp”) lower bounds for set-multilinear circuits/formulas in the constant (or low) depth setting. This is based on joint work with Deepanshu Kush.

October 21st, 2022: The Brascamp-Lieb polytope: matroid matching and rank of matrix spaces

A&C Seminar [slides]

Speaker: Akshay Ramachandran

October 3rd, 2022: Title not recorded

A&C Seminar

Speaker: Argyris Mouzakis

September 21st, 2022: Title not recorded

A&C Seminar

Speaker: Daniel Gabric

August 3rd, 2022: Computability and Complexity in Analytic Combinatorics

A&C Seminar [video] [slides]

Speaker: Stephen Melczer

Abstract: Enumerative combinatorics studies discrete objects by capturing aspects of their behaviour (such as the number of objects of a given size) using sequences. In this talk we explore how to combine pure mathematical tools with computational methods to answer questions about the computability and complexity of asymptotic behaviour for sequences under different encodings that arise frequently in combinatorics. Applications discussed will include (time permitting) the analysis of classical algorithms, models predicting the shape of biomembranes, queuing theory, random walks, ratchet models for gene expression, transcendence of zeta values, restricted permutations, maximum likelihood degree in algebraic statistics, sampling algorithms for perfect matchings in bipartite graphs, and parallel synthesis for DNA storage.

March 23rd, 2022: Almost linear time algorithms for max-flow and more

A&C Seminar [video]

Speaker: Sushant Sachdeva, University of Toronto

Abstract: We give the first almost-linear time algorithm for computing exact maximum flows and minimum-cost flows on directed graphs. By well known reductions, this implies almost-linear time algorithms for several problems including bipartite matching, optimal transport, and undirected vertex connectivity.

Our algorithm is designed using a new Interior Point Method (IPM) that builds the flow as a sequence of almost-linear number of approximate undirected minimum-ratio cycles, each of which is computed and processed very efficiently using a new dynamic data structure.

Our framework extends to give an almost-linear time algorithm for computing flows that minimize general edge-separable convex functions to high accuracy. This gives the first almost-linear time algorithm for several problems including entropy-regularized optimal transport, matrix scaling, p-norm flows, and Isotonic regression.

Joint work with Li Chen, Rasmus Kyng, Yang Liu, Richard Peng, and Maximilian Probst Gutenberg.

March 14th, 2022: Parameterized Complexity of Reconfiguration of Atoms

A&C Seminar

Speaker: Stephanie Maaz

Abstract: Our work is motivatedby the challenges presented in preparing arrays of atoms for use in quantum simulation. The recently-developed process of loading atoms into traps results in approximatelyhalf of the traps being filled. To consolidate the atoms so that they form a dense and regular arrangement, such as all locations in a grid, atoms are rearranged usingmoving optical tweezers. Time is of the essence, as the longer that the process takes and the more that atoms are moved, the higher the chance that atoms will be lostin the process. Viewed as a problem on graphs, we wish to solve the problem of reconfiguring one arrangement of tokens (representing atoms) to another using as fewmoves as possible. Because the problem is NP-complete on general graphs as well as on grids, we focus on the parameterized complexity for various parameters, consideringboth undirected and directed graphs, and tokens with and without labels. For unlabelled tokens, the problem is fixed-parameter tractable when parameterized by the numberof tokens, the number of moves, or the number of moves plus the number of vertices without tokens in either the source or target configuration, but intractablewhen parameterized by the difference between the number of moves and the number of differences in the placement of tokens in the source and target configurations. Whenlabels are added to tokens, however, most of the tractability results are replaced by hardness results.

March 10th, 2022: Anaconda: A Non-Adaptive Conditional Sampling Algorithm for Distribution Testing

Student Seminar

Speaker: Argyris Mouzakis

Abstract: We investigate distribution testing with access to non-adaptive conditional samples. In the conditional sampling model, the algorithm is given the following access to a distribution: it submits a query set S to an oracle, which returns a sample from the distribution conditioned on being from S. In the non-adaptive setting, all query sets must be specified in advance of viewing the outcomes.

Our main result is the first polylogarithmic-query algorithm for equivalence testing, deciding whether two unknown distributions are equal to or far from each other. This is an exponential improvement over the previous best upper bound, and demonstrates that the complexity of the problem in this model is intermediate to the the complexity of the problem in the standard sampling model and the adaptive conditional sampling model. We also significantly improve the sample complexity for the easier problems of uniformity and identity testing. For the former, our algorithm requires only ̃O(log n) queries, matching the information-theoretic lower bound up to a O(log log n)-factor. Our algorithm works by reducing the problem from ℓ1-testing to ℓ∞-testing, which enjoys a much cheaper sample complexity. Necessitated by the limited power of the non-adaptive model, our algorithm is very simple to state. However, there are significant challenges in the analysis, due to the complex structure of how two arbitrary distributions may differ.

March 10th, 2022: Words that Almost Commute and Anti-Commute

A&C Seminar

Speaker: Daniel Gabric

Abstract: The \emph{Hamming distance} $\text{ham}(u,v)$ between two equal-length words $u$, $v$ is the number of positions where $u$ and $v$ differ. The words $u$ and $v$ are said to be \emph{conjugates} if there exist non-empty words $x,y$ such that $u=xy$ and $v=yx$. The smallest value $\text{ham}(xy,yx)$ can take on is $0$, when $x$ and $y$ commute. But, interestingly, the next smallest value $\text{ham}(xy,yx)$ can take on is $2$ and not $1$. We provide an efficient formula to count the number $h(n)$ of length-$n$ words $u=xy$ over a $k$-letter alphabet that have a conjugate $v=yx$ such that $\text{ham}(xy,yx)=2$. We also provide efficient formulae for other quantities closely related to $h(n)$. Additionally, we give some results on the asymptotic behaviour of $h(n)$. Finally, we show some preliminary results on conjugates $u=xy$ and $v=yx$ that differ in all positions (i.e., $\text{ham}(xy,yx)=|xy|$).

March 3rd, 2022: Computing Realistic Terrains from Imprecise Elevations

Thesis Presentation

Speaker: Graeme Stroud

Abstract: It is ideal for triangulated terrains to have characteristics or properties that are realistic. In the imprecise terrain model, each vertex of a triangulatedterrain has an imprecise elevation value only known to lie within some interval. Under some objective function, the goal is to compute a precise terrain by assigning a single elevation value to each point, so that the objective function is optimized. This thesis examines various objectives, such as minimizing the number of local extrema and minimizing the terrain’s surface area. We give algorithmsin some cases, hardness results in other cases. Specifically, we consider four objectives: (1) minimizing the number of local extrema; (2) optimizing coplanar features; (3) minimizing the surface area; (4) minimizing the maximum steepness.

Problem (1) is known to be NP-hard, but we give an algorithm for a special case. For problem (2) we give an NP-hardness proof for the general case anda positive result for a special case. Meanwhile, problems (3) and (4) can be approximated using Second Order Cone Programming. We also consider versions of these problems for terrains one dimension down, where the output is a polyline. Here we give very efficientalgorithms for all objective functions considered.

Finally, we go beyond terrains and briefly consider the Distant Representatives problem, where the goal is to choose precise points from segments to be as far from each other as possible.For this problem, we give a parameterized algorithm for vertical segments, prove NP-hardness for unit vertical segments, and show hardness of approximation for vertical and horizontal segments.

February 24th, 2022: A Tale of Two Reconstructions

A&C Seminar [video]

Speaker: Vishwas Bhargava

Abstract: Arithmetic circuits are a natural model for computing polynomials using basic arithmetic operations like addition and multiplication. The problem of learning arithmetic circuits, a.k.a. reconstruction, is an important and well-studied problem. Here we are given a polynomial (either explicitly as a list of coefficients or using black-box access) and the goal is to find the smallest (or approximately smallest) arithmetic circuit computing it.

In this talk, we will discuss two vastly different paradigms in learning arithmetic circuits, which are worst-case learning and average-case (non-degenerate) learning. We will elaborate on various techniques involved in these paradigms, their similarity, and their differences. Along the way, we will see a worst-case as well as a non-degenerate algorithm for decomposing tensors. Time permitting, we will also discuss some recent progress in non-degenerate learning of “generalized” depth 3 arithmetic circuits.

Bio: Vishwas is a final-year Ph.D. student at Rutgers, advised by Shubhangi Saraf. he is broadly interested in the theoretical aspects of computer science; specifically, problems in Computational Complexity theory, Pseudorandomness/derandomization, and problems having Algebraic or Number Theoretic flavor.

February 18th, 2022: Optimization Beyond Minimization

A&C Seminar [video]

Speaker: Manolis Gkaragkounis

Abstract: Motivated by recent advances in both theoretical and applied aspects of multiplayer games, spanning from e-sports to multi-agent generative adversarial networks, a surge of different studies have been focused on the core problem of understanding the behavior of game dynamics in general N-player games. From the seminal settings of two competitive players and Min-Max Optimization to the complete understanding of how the day-to-day behavior of the dynamics correlates to the game’s different notion of equilibria is much more limited, and only partial results are known for certain classes of games (such as zero-sum or congestion games). In this talk, we study from two different perspectives arguably the most well-studied class of no-regret dynamics, “Follow-the-regularized-leader” (FTRL) and Discretizations of Gradient Flow (GDA/OGDA/EG), and we establish a sweeping negative result showing that the notion of mixed Nash equilibrium is antithetical to no-regret learning. Specifically, we show that any Nash equilibrium which is not strict (in that every player has a unique best response) cannot be stable and attracting under the dynamics of FTGL. This result has significant implications for predicting the outcome of a learning process as it shows unequivocally that only strict (and hence, pure) Nash equilibria can emerge as stable limit points thereof. For a final happy end story, we present either structural examples of families where convergence is possible providing the last-iterate convergence rates or even new methods inspired from other areas like control theory & planning.

Bio:

Emmanouil (Manolis) V. Vlatakis Gkaragkounis is a final year PhD student in the Department of Computer Science at Columbia University, under the supervision of prof. Mihalis Yannakakis and Rocco Servedio. Currently, he is Simons-Google Research fellow at the University of California at Berkeley. Before joining Columbia University, he interned at “Athena” Research & Innovation Center in Athens, Greece. He received his integrated B.s & M.s in ECE Department of National Technical University of Athens, where he was advised by Dimitris Fotakis. Manolis’s primary interest is in the intersection of Theoretical Computer Science & Machine Learning, with a particular focus in Algorithmic Game Theory, Optimization, Computational Complexity and Beyond Worst-case Analysis of Algorithms .

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2021

November 3rd, 2021: The maximum number of unbordered conjugates among binary words

A&C Seminar

Speaker: Daniel Gabric, University of Waterloo

Abstract: A word is said to be \emph{bordered} if it contains a non-empty proper prefix that is also a suffix. Otherwise, it is said to be \emph{unbordered}. Two words $u$ and $v$ are said to be conjugates if they are cyclic shifts of each other. For example, the words {\tt eat} and {\tt ate} are conjugates. Using a decision procedure based on automatic sequences, we complete the classification, due to Harju and Nowotka, of binary words with the maximum number of unbordered conjugates. Furthermore, we prove that for every possible number, up to the maximum, there exists a word having that number of unbordered conjugates.

November 1st, 2021: SGD: The role of implicit regularization, batch size and multiple epochs

A&C Seminar [video]

Speaker: Ayush Sekhari, Cornell University

Abstract: Multi-epoch, small-batch, Stochastic Gradient Descent (SGD) has been the method of choice for learning with large over-parameterized models. A popular theory for explaining why SGD works well in practice is that the algorithm has an implicit regularization that biases its output towards a good solution. Perhaps the theoretically most well understood learning setting for SGD is that of Stochastic Convex Optimization (SCO), where it is well known that SGD learns at the minimax optimal rate. In this talk, we will consider the problem of SCO and discuss various surprising results on the role of implicit regularization, batch size, and multiple epochs for SGD. We will also discuss extensions of these results for the general learning setting, and deep learning. Paper:https://arxiv.org/pdf/2107.05074.pdf Brief bio: Ayush is a PhD student in the Computer Science department at Cornell University, advised by Professor Karthik Sridharan and Professor Robert D. Kleinberg. His research interests span across optimization, online learning, reinforcement learning and control, and the interplay between them. Before coming to Cornell, he spent a year at Google as a part of the Brain residency program. Before Google, he completed his undergraduate studies in computer science from IIT Kanpur in India where he was awarded the President’s gold medal.

April 21st, 2021: Title not recorded

A&C Seminar

Speaker: Lingyi Zhang, University of Waterloo

Abstract: We show the O(log n) time extract minimum function of efficient priority queues can be generalized to the extraction of the k smallest elements in O(k log(n/k)) time. We first show the heap-ordered tree selection of Kaplan et al. can be applied on the heap-ordered trees of the classic Fibonacci heap to support the extraction in O(k log(n/k)) amortized time.

We then show selection is possible in a priority queue with optimal worst-case guarantees by applying heap-ordered tree selection on Brodal queues, supporting the operation in O(k log(n/k)) worst-case time. Via a reduction from the multiple selection problem, Omega(k log(n/k)) time is necessary.

We then apply the result to the lazy search trees of Sandlund & Wild, creating a new interval data structure based on selectable heaps. This gives optimal O(B + n) lazy search tree performance, lowering insertion complexity into a gap Delta_i to O(log(n/|Delta_i|)) time. An O(1)-time merge operation is also made possible under certain conditions. If Brodal queues are used, all runtimes of the lazy search tree can be made worst-case. The presented data structure uses soft heaps of Chazelle, biased search trees, and efficient priority queues in a non-trivial way, approaching the theoretically-best data structure for ordered data.

Master’s thesis presentation. Supervisor: Professor J. Ian Munro.

April 21st, 2021: VC Dimension and Distribution-Free Sample-Based Testing

A&C Seminar [video]

Speaker: Nathan Harms, University of Waterloo

Abstract: We consider the problem of determining which classes of functions can be tested more efficiently than they can be learned, in the distribution-free sample-based model that corresponds to the standard PAC learning setting. Our main result shows that whileVC dimension by itself does not always provide tight bounds on the number of samples required to test a class of functions in this model, it can be combined with a closely-related variant that we call “lower VC” (or LVC) dimension to obtain strong lower bounds on this sample complexity.

We use this result to obtain strong and in many cases nearly optimal bounds on the sample complexity for testing unions of intervals, halfspaces, intersections of halfspaces, polynomial threshold functions, and decision trees. Conversely, we show thattwo natural classes of functions, juntas and monotone functions, can be tested with a number of samples that is polynomially smaller than the number of samples required for PAC learning.

Finally, we also use the connection between VC dimension and property testing to establish new lower bounds for testing radius clusterability and testing feasibility of linear constraint systems.

April 14th, 2021: An optimal separation of randomized and quantum query complexity

A&C Seminar [video] [slides]

Speaker: Pei Wu

Abstract: We prove that for every decision tree, the absolute values of the Fourier coefficients of given order $\ell\geq1$ sum to at most $c^{\ell}\sqrt{\binom{d}{\ell}(1+\log n)^{\ell-1}},$ where $n$ is the number of variables, $d$ is the tree depth, and $c>0$ is an absolute constant. This bound is essentially tight and settles a conjecture due to Tal (arxiv 2019; FOCS 2020). The bounds prior to our work degraded rapidly with $\ell,$ becoming trivial already at $\ell=\sqrt{d}.$ As an application, we obtain, for every integer $k\geq1,$ a partial Boolean function on $n$ bits that has bounded-error quantum query complexity at most $\lceil k/2\rceil$ and randomized query complexity $\tilde{\Omega}(n^{1-1/k}).$ This separation of bounded-error quantum versus randomized query complexity is best possible, by the results of Aaronson and Ambainis (STOC 2015). Prior to our work, the best known separation was polynomially weaker: $O(1)$ versus $\Omega(n^{2/3-\epsilon})$ for any $\epsilon>0$ (Tal, FOCS 2020). As another application, we obtain an essentially optimal separation of $O(\log n)$ versus $\Omega(n^{1-\epsilon})$ for bounded-error quantum versus randomized communication complexity, for any $\epsilon>0.$ The best previous separation was polynomially weaker: $O(\log n)$ versus $\Omega(n^{2/3-\epsilon})$ (implicit in Tal, FOCS 2020).

February 24th, 2021: Private Mean Estimation of Heavy-Tailed Distributions

A&C Seminar [video]

Speaker: Vikrant Singhal, NEU

Abstract: We give new upper and lower bounds on the minimax sample complexity of differentially private mean estimation of distributions with bounded k-th moments. Roughly speaking, in the univariate case, we show that n = \Theta(1/\alpha^2 + 1/\epsilon\alpha^{k/(k-1)}) samples are necessary and sufficient to estimate the mean to \alpha-accuracy under \epsilon-differential privacy, or any of its common relaxations. This result demonstrates a qualitatively different behavior compared to estimation absent privacy constraints, for which the sample complexity is identical for all k >= 2. We also give algorithms for the multivariate setting whose sample complexity is a factor of O(d) larger than the univariate case.

February 19th, 2021: Optimal Sub-Gaussian Mean Estimation in R

A&C Seminar [video]

Speaker: Jasper Lee, Brown University

Abstract: We revisit and settle one of the most fundamental problems in statistics: given access to independent samples from a 1D random variable (with finite but unknown mean and variance), what is the best way to estimate the mean in the high probability regime,in terms of error convergence with respect to sample size? The conventional wisdom is to use the empirical mean as our estimate. However, it is known that the empirical mean can in fact have exponentially sub-optimal convergence for certain heavy-tailed distributions.On the other hand, the median-of-means estimator (invented and reinvented in various literature) does have sub-Gaussian convergence for all finite-variance distributions, albeit only in the big-O sense with a sub-optimal multiplicative constant. The natural remaining question then, is whether it is possible to bridge the gap, to have an estimator that has optimal sub-Gaussian concentration with an optimal constant, for all finite-variance distributions.

In this talk, we answer the question affirmatively by giving an estimator that converges with the optimal constant inside the big-O, up to a 1+o(1) multiplicative factor. Our estimator is furthermore computable in time linear in the sample size. The convergence analysis involves deriving tail bounds using linear and convex-concave programming dualities, which may be of independent interest.

This is joint work with Paul Valiant.

February 17th, 2021: Title not recorded

A&C Seminar

Speaker: Jason Li

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2020

November 12th, 2020: Title not recorded

A&C Seminar

Speaker: Hong Zhou, University of Waterloo

Abstract: In this talk, we will present a local search framework to design and analyze both combinatorial algorithms and rounding algorithms for experimental design problems. This framework provides a unifying approach to match and improve all known results in D/A/E-design and to obtain new results in previously unknown settings. For combinatorial algorithms, we provide a new analysis of the classical Fedorov’s exchange method. We prove that this simple local search algorithm works well as long as there exists an almost optimal solution with good condition number. Moreover, we design a new combinatorial local search algorithm for E-design using the regret minimization framework. For rounding algorithms, we provide a unified randomized exchange algorithm to match and improve previous results for D/A/E-design. Furthermore, the algorithm works in the more general setting to approximately satisfy multiple knapsack constraints, which can be used for weighted experimental design and for incorporating fairness constraints into experimental design. Joint work with Lap Chi Lau, to appear in SODA 2021. Link to paper on ArXiv: https://arxiv.org/abs/2010.15805

October 21st, 2020: Lazy Search Trees

A&C Seminar [video] [slides]

Speaker: Bryce Sandlund

Abstract: We introduce the lazy search tree data structure. The lazy search tree is a comparison-based data structure on the pointer machine that supports order-based operations such as rank, select, membership, predecessor, successor, minimum, and maximumwhile providing dynamic operations insert, delete, change-key, split, and merge. We analyze the performance of our data structure based on a partition of current elements into a set of gaps {Delta_i} based on rank. A query falls into a particular gap and splitsthe gap into two new gaps at a rank r associated with the query operation. If we define B = sum_i |Delta_i| log_2(n/|Delta_i|), our performance over a sequence of n insertions and q distinct queries is O(B + min(n log log n, n log q)). We show B is a lowerbound.

Effectively, we reduce the insertion time of binary search trees from Theta(log n) to O(min(log(n/|Delta_i|) + log log |Delta_i|, log q)), where Delta_i is the gap in which the inserted element falls. Over a sequence of n insertions and q queries, a time boundof O(n log q + q log n) holds; better bounds are possible when queries are non-uniformly distributed. As an extreme case of non-uniformity, if all queries are for the minimum element, the lazy search tree performs as a priority queue with O(log log n) timeinsert and decrease-key operations. The same data structure supports queries for any rank, interpolating between binary search trees and efficient priority queues.

Lazy search trees can be implemented to operate mostly on arrays, requiring only O(min(q, n)) pointers, suggesting smaller memory footprint, better constant factors, and better cache performance compared to many existing efficient priority queues or binarysearch trees. Via direct reduction, our data structure also supports the efficient access theorems of the splay tree, providing a powerful data structure for non-uniform element access, both when the number of accesses is small and large.

Joint work with Sebastian Wild. Accepted for publication in FOCS 2020; arXiv version pending.

September 30th, 2020: From high-dimensional random landscapes to statistical inference

A&C Seminar [video]

Speaker: Aukosh Jagannath, University of Waterloo

Abstract: I will report on recent work with G. Ben Arous (NYU) and R. Gheissari (Berkeley) on the performance of online stochasticgradient in high-dimensional inference tasks. We develop a classification of loss landscapes for the difficulty of such problems, namely whether for a given loss function and typical realizations of the data, SGD requires linear, quasilinear, or polynomiallymany samples in the dimension to perform the inference task. This classification depends on an intrinsic property of the population loss which we call the ``information exponent’’ as opposed to almost sure properties of the loss landscape (e.g., quasi-convexitysaddle-type properties). We find that from uniform at random starts, the majority of the data is used in the initial “search” phase (where the landscape is highly non-convex) as comparedto the final “descent” phase (where the algorithm is in a trust region).

In this talk, I will illustrate our methods on a simple class of problems, namely supervised learning with a single-layernetwork in the case of Gaussian patterns. Here we obtain a classification of the sample complexity as one varies the activation function. If there is time remaining, I will illustrate how this approach can be extended to analyze gradient descent in the caseof in Tensor PCA.

September 9th, 2020: New Nearly-Optimal Coreset for Kernel Density Estimation

A&C Seminar [video] [slides]

Speaker: Wai Ming Tai, University of Utah

Abstract: Given a point set $P \subset \mathbb{R}^d$, kernel density estimation for Gaussian kernel is defined as $\overline{\mathcal{G}}_P(x) = \frac{1}{\left|P\right|}\sum_{p\inP}e^{-\left\lVert x-p \right\rVert^2}$ for any $x\in\mathbb{R}^d$. We study how to construct a small subset $Q$ of $P$ such that the kernel density estimation of $P$ can be approximated by the kernel density estimation of $Q$. This subset $Q$ is called \emph{coreset}. The primary technique in this work is to construct $\pm 1$ coloring on the point set $P$ by the discrepancy theory and apply this coloring algorithm recursively. Our result leverages Banaszczyk’s Theorem. When $d>1$ is constant, our construction gives a coreset of size $O\left(\frac{1}{\varepsilon}\sqrt{\log\log\frac{1}{\varepsilon}}\right)$as opposed to the best-known result of $O\left(\frac{1}{\varepsilon}\sqrt{\log\frac{1}{\varepsilon}}\right)$. It is the first to give a breakthrough on the barrier of $\sqrt{\log}$ factor even when $d=2$.

July 2nd, 2020: Factoring Polynomials given as Arithmetic Branching Programs

A&C Seminar

Speaker: Amit Sinhababu, Aalen university, Germany

Abstract: A basic problem in computational algebra is polynomial factoring: Given a polynomial, compute all its irreducible factors. In a classic result, Kaltofen proved that if a polynomial P(x_1,…,x_n) of degree d is given as an arithmetic circuit of size s, then all its factors can be computed by arithmetic circuits of size poly(s,d). In other words, the algebraic complexity class VP is closed under factors. This result has applications in algebraic complexity, for example in derandomization of polynomial identity testing using explicit hard polynomials (lower bounds).

A natural direction to extend Kaltofen’s result is to prove analogous factor size bounds in more restricted / more general models than arithmetic circuits. Kaltofen’s proofs do not extend to the restricted models, as reusing previous computations is costly for these models. Recent works in this area focused on models like constant depth circuits, arithmetic formulas, arithmetic branching programs. In a beautiful work, Oliveira (CCC 2015) proved that factors of a low-depth circuit/formula can be computed by small low-depth circuits/formulas, if we assume that the individual degree of the given polynomial is bounded by constant. In a follow-up work, Dutta, Saxena and Sinhababu (DSS, STOC 18) showed poly (s^log d) factor size upper bound for arithmetic formulas and branching programs.

In this work, we show poly (s,d) factor size upper bound for the model of arithmetic branching programs (ABP). Whereas the works of Oliveira and DSS used different versions of Newton iteration, we use the classic technique of Hensel lifting. Although Newton iteration and Hensel lifting are related techniques, the latter gives better bound for ABPs. We also use that the determinant of a symbolic matrix can be computed by a small ABP. In this talk, we give a brief survey of the earlier works and present the main ideas of our proof.

Paper: https://image.informatik.htw-aalen.de/~thierauf/Papers/ABP-factors.pdf

June 24th, 2020: Testing Noisy Linear Equations for Sparsity

A&C Seminar [video]

Speaker: Anindya De, University of Pennsylvania

Abstract: Consider the following basic problem in sparse linear regression – an algorithm gets labeled samples of the form (x, <w.x> + \eps) where w is an unknown n-dimensional vector, x is drawn from a background distribution D and \eps is some independent noise. Given the promise that w is k-sparse, the breakthrough work of Candes, Rhomberg and Tao (2005) shows that w can be recovered with samples and time which scales as O(k log n). This should be contrasted with general linear regression where O(n) samples are information theoretically necessary.

In this talk, we look at this question from the vantage point of property testing and study the decision variant of the following question – namely, what is the complexity of deciding if the unknown vector w is k-sparse (or at least say 0.01 far from k-sparse in \ell_2 distance). We show that the decision version of the problem can be solved with samples which are independent of n as long as the background distribution D is i.i.d. and the components are not Gaussian. We further show that weakening any of the conditions in this result necessarily makes the complexity scale as log n (thus showing our results are tight).

Joint work with Xue Chen (Northwestern) and Rocco Servedio (Columbia).

June 18th, 2020: A Simple Algorithm for Minimum Cuts in Near-Linear Time and Space-Efficient Data Structures for Lattices (two papers)

A&C Seminar

Speaker: Bryce Sandlund, University of Waterloo

Abstract: 1. A Simple Algorithm for Minimum Cuts in Near-Linear Time: We consider the minimum cut problem in undirected, weighted graphs. We give a simple algorithm to find a minimum cut that 2-respects (cuts two edges of) a spanning tree T of a graph G.This procedure can be used in place of the complicated subroutine given in Karger’s near-linear time minimum cut algorithm (J. ACM, 2000). We give a self-contained version of Karger’s algorithm with the new procedure, which is easy to state and relatively simpleto implement. It produces a minimum cut on an m-edge, n-vertex graph in O(m log^3 n) time with high probability, matching the complexity of Karger’s approach. 2. Space-Efficient Data Structures for Lattices: A lattice is a partially-ordered set in which every pair of elements has a unique meet (greatest lower bound) and join (least upper bound). We present new data structures for lattices that are simple,efficient, and nearly optimal in terms of space complexity.

Our first data structure can answer partial order queries in constant time and find the meet or join of two elements in O(n^(3/4)) time, where n is the number of elements in the lattice. It occupies O(n^(3/2) logn) bits of space, which is only a Θ(logn) factor from the Θ(n^(3/2))-bit lower bound for storing lattices. The preprocessing time is O(n^2). This structure admits a simple space-time tradeoff so that, for any c ∈ [1/2,1], the data structure supports meet and join queries in O(n^(1−c/2)) time, occupiesO(n^(1+c) log n) bits of space, and can be constructed in O(n^2+n^(1+3c/2)) time.

Our second data structure uses O(n^(3/2) log n) bits of space and supports meet and join in O(d log n /log d) time, where d is the maximum degree of any element in the transitive reduction graph of the lattice. This structure is much faster for latticeswith low-degree elements.

This paper also identifies an error in a long-standing solution to the problem of representing lattices. We discuss the issue with this previous work.

June 17th, 2020: A Spectral Approach to Network Design

A&C Seminar [video] [slides]

Speaker: Hong Zhou, University of Waterloo

Abstract: In this talk, I will present a spectral approach to design approximation algorithms for network design problems. We observe that the underlying mathematical questions are the spectral rounding problems, which were studied in spectral sparsification and in discrepancy theory. We extend these results to incorporate additional non-negative linear constraints, and show that they can be used to significantly extend the scope of network design problems that can be solved. Our algorithm for spectral rounding is an iterative randomized rounding algorithm based on the regret minimization framework. In some settings, this provides an alternative spectral algorithm to achieve constant factor approximation for the classical survivable network design problem.

Joint work with Lap Chi Lau. Arxiv link of the manuscript: https://arxiv.org/abs/2003.07810

June 3rd, 2020: Forecasting algorithms: a new approach to randomized lower bounds

A&C Seminar [video] [slides]

Speaker: Shalev Ben-David, University of Waterloo

Abstract: In this talk, I’ll present a new approach to randomized lower bounds, particularly in a setting in which we wish to give a fine-grained analysis of randomized algorithms that achieve small bias. The approach is as follows: instead of considering ordinary randomized algorithms which give an output in {0,1} and may err, we switch models to look at “forecasting” randomized algorithms which output a confidence in [0,1] for whether they think the answer is 1. When scored by a proper scoring rule, the performance of the best forecasting algorithm is closely related to the bias of the best (ordinary) randomized algorithm; however, the forecasting algorithms (when analyzed using the right scoring rule) have elegant mathematical properties that make them more amenable to analysis.

As an application, I’ll present a new minimax theorem for randomized algorithms, which can be viewed as a strengthening of Yao’s minimax theorem. Yao’s minimax theorem guarantees the existence of a hard distribution for a function f such that solving f against this distribution (to a desired error level) is as hard as solving f in the worst case (to that same error level). However, the hard distribution provided by Yao’s theorem depends on the chosen error level. Our minimax theorem removes this dependence,giving a distribution which certifies the hardness of f against all bias levels at once. In recent work, we used this minimax theorem to give a tight composition theorem for randomized query complexity.

Joint work with Eric Blais.

June 1st, 2020: Random Restrictions of High Dimensional Distributions and Uniformity Testing with Subcube Conditioning

A&C Seminar

Speaker: Amit Levi, University of Waterloo

Abstract: Given a distribution p supported on {-1,1}^n, we want to test whether p is uniform or ϵ-far from uniform in total variation distance. The fact that p is a high dimensional distribution is inconsequential for the sample complexity, which is well-known tobe Θ(2^{n/2} / ε^2). To benefit from the high dimensional structure, we study the subcube conditional sampling model, first considered in Bhattacharyya and Chakraborty (2018), and give a nearly optimal algorithm for testing uniformity on {-1,1}^n making Õ(√n/ ε^2) queries. The key ingredient is a natural notion of random restriction for distributions on {-1,1}^n, and an inequality describing the behavior of the mean vector after a random restriction. Joint work with Clement Canonne, Xi Chen, Gautam Kamath and Erik Waingarten.

May 27th, 2020: Can algebraic circuit lower bounds have easy proofs?

A&C Seminar [slides]

Speaker: Anamay Tengse, Tata Institute of Fundamental Research (TIFR)

Abstract: The field of Algebraic Circuit Complexity studies how succinctly one can represent explicit multivariate polynomials. Algebraic circuits (analogous to boolean circuits) are the most natural model for computing polynomials. A central question in this field istherefore to find explicit polynomials that require “large” algebraic circuits. The best known lower bounds for general circuits are due to Baur and Strassen (1983) and Smolenksy (1997), both of which give (tight) Omega(n log d) lower bounds for an explicitn-variate polynomial of degree d. Although there has been considerable progress in proving lower bounds for simpler models, there has not been much progress for more general computational classes. This lack of progress over the last few decades leads to thefollowing question. Are most of the current techniques incapable of proving a superpolynomial lower bound for general circuits?

Almost all the current techniques for showing a lower bound against a class of circuits C, can also be used to obtain what is called a “defining equation” for the class C that is also easy to compute. Thus, the above question is equivalent to asking whetherthe class of polynomials computable by polynomial-sized circuits has efficiently computable defining equations.

Based on the important work of Razborov and Rudich (1997) in the boolean world, Forbes, Shpilka, and Volk (2018), and Grochow, Kumar, Saks and Saraf (2017) introduced the framework of Algebraically Natural Proofs. They showed that under a certain derandomizationassumption, such defining equations do not exist; thereby hinting towards the existence of a barrier for proving lower bounds. We show that there \emph{are} efficiently computable defining equations for the class of polynomials with bounded coefficients thatare computable by polynomial-sized circuits. This provides evidence \emph{against} the existence of a barrier for proving lower bounds.

Joint work with Prerona Chatterjee (TIFR, Mumbai), Mrinal Kumar (IITB, Mumbai), C. Ramya (TIFR, Mumbai) and Ramprasad Saptharishi (TIFR, Mumbai).

May 13th, 2020: Robustness in unsupervised and supervised machine learning

A&C Seminar [video] [slides]

Speaker: Gautam Kamath, University of Waterloo

Abstract: Recently, the need for robust machine learning algorithms has become apparent. Whether due to errors in data collection, model misspecification, or adversarial attacks, contaminated datasets arise in many areas. This is an issue, as existing methods appear to be quite brittle to small amounts of errors. Even more worryingly, these models are being deployed in many security-critical settings, such as self-driving cars, where reliability is an absolute must.

In this talk, I will describe a line of work in which we provide provable guarantees for robust machine learning in several fundamental settings. I’ll begin by discussing the problem of robust estimation of mean and covariance of a Gaussian distribution, and how to relax this to distributions with weaker assumptions on the moments. I will then describe how these methods can be used to “robustify” supervised learning algorithms by applying robust mean estimation algorithms to the gradients of the dataset. While theoretically sound, the algorithms are also realizable and efficient, and I will present experimental results on both synthetic and real-world data.

Based on joint works with Ilias Diakonikolas, Daniel M. Kane, Jerry Li, Ankur Moitra, Jacob Steinhardt, and Alistair Stewart.

May 7th, 2020: Decision Algorithms for Ostrowski-Automatic Sequences

Thesis Presentation

Speaker: Aseem Baranwal, University of Waterloo

Abstract: We extend the notion of automatic sequences to Ostrowski-automatic sequences. Our primary contributions include a decision procedure for computationally deciding certain combinatorial and enumeration questions about these sequences, partial resolutionof a conjecture on balanced words, and several new results pertaining to rich words, Lucas words, and antisquare avoidance. We also provide an implementation of these procedures as part of the software package Walnut.

May 6th, 2020: Building a Larger Class of Graphs for Efficient Reconfiguration of Vertex Colouring

Thesis Presentation

Speaker: Owen Merkel, University of Waterloo

Abstract: For a graph G, the reconfiguration graph of the k-colourings is the graph whose vertices are the k-colourings of G and two colourings are joined by an edge if they differ in colour on exactly one vertex. For a k-colourable graph, we investigate the connectivity and diameter of the reconfiguration graph of the (k+1)-colourings. We introduce a new class of graphs called OAT graphs that are built from four simple operations, disjoint union, join, and the addition of a clique or comparable vertex. We prove that if G is a k-colourable OAT graph, then the reconfiguration graph of the (k+1)-colourings is connected with diameter O(n^2). Furthermore, we give polynomial time algorithms to recognize OAT graphs and to find a path between any two colourings in the reconfiguration graph.

April 29th, 2020: Local Graph Clustering

A&C Seminar [slides]

Speaker: Kimon Fountoulakis, University of Waterloo

Abstract: Graphs, long popular in computer science and discrete mathematics, have received renewed interest because they provide a useful way to model many types of relational data. In biology, e.g., graphs are routinely used to generate hypotheses for experimental validation; in neuroscience, they are used to study the networks and circuits in the brain; and in social networks, they are used to find common behaviors of users. These modern graph applications require the analysis of large graphs, and this can be computationally expensive. Graph algorithms have been developed to identify and interpret small-scale local structure in large-scale data without the requirement to access all the data.

In this talk, we will discuss state-of-the-art local spectral-, flow-, and Lp-norm-based algorithms that are specialized in finding small-scale clusters in large graphs without accessing the whole graph. We will discuss worst- and average-case theoretical results of these algorithms and we will demonstrate their empirical performance.

Based on joint work with Shenghao Yang, Di Wang, Wooseok Ha, Julian Shun, Farbod Roosta Khorasani, Xiang Cheng.

February 28th, 2020: Improved Analysis of Higher Order Random Walks and Applications

A&C Seminar

Speaker: Vedat Levi Alev, University of Waterloo

Abstract: Local spectral expansion is a very useful method for arguing about the spectral properties of severalrandom walk matrices over simplicial complexes. The motivation of this work is to extend this method to analyze the mixing times of Markov chains for combinatorial problems. Our main result is a sharp upper bound on the second eigenvalue of the down-up walkon a pure simplicial complex, in terms of the second eigenvalues of its links. We show some applications of this result in analyzing mixing times of Markov chains including sampling independent sets of a graph.

(https://arxiv.org/abs/2001.02827) Joint work with: Lap Chi Lau

February 26th, 2020: On the Complexity of BWT-runs Minimization via Alphabet Reordering

A&C Seminar

Speaker: Daniel Gibney, University of Central Florida

Abstract: AbstractFirst introduced in1994, the Burrows-Wheeler Transform (BWT) went on to provide the backbone for thefirst encoding of the classic suffix tree data structure in spaceclose to the entropy-based lowerbound. Within the last decade, it has seen its role further enhancedwith the development ofsuffix trees in space proportional to“r”, the number of runs inthe BWT of the text. Moreover,r is appearing increasingly often in the time complexity of newalgorithms, making having thesmallest value of r of great importance. Unlike other popularmeasures of compression, theparameter r is sensitive to the lexicographic ordering given to thetext’s alphabet. Despiteseveral past attempts, a provably efficient algorithm for finding anoptimal alphabet orderingminimizing r has been an open problem for many years. We help to explainthis lack of progress by presenting the first set of results on thecom- putationalcomplexity of minimizing BWT-runs via alphabet reordering. We provethat the decision version ofthis problem is NP-complete and cannot be solved in time (2^o(σ))nunless the Exponential TimeHypothesis fails, where σ is the size of the alphabet. Additionally,we provide an efficientalgorithm for the more restricted problem of finding an optimalordering on a subset of symbols(occurring only once) under ordering constraints. The algorithm runsin optimal time for small values ofσ. We also look at a version of the problem on the newly discoveredclass of graphs with BWT likeproperties called Wheeler graphs. Here also we show NP-hardnessresults

on a related problemwhich we call Source Ordering. This is joint work with Jason Bentley and Sharma V. Thankachan

February 21st, 2020: Motion Planning with Local Interaction: A Framework for Proving Hardness

A&C Seminar

Speaker: Jayson Lynch, MIT

Abstract: This talk describes a general theory for characterizing the computational complexity of motion planning of robot(s) through a graph of “gadgets”, where each gadget has its own state defining a set of allowed traversals which in turn modify the gadget’s state. We study two general families of such gadgets within this theory, one which naturally leads to motion planning problems with polynomially bounded solutions, and another which leads to polynomially unbounded (potentially exponential) solutions. We also study a range of competitive game-theoretic scenarios, from one player controlling one robot to teams of players each controlling their own robot and racing to achieve their team’s goal. Under certain restrictions on these gadgets, we fully characterize the complexity of bounded 1-player motion planning (NL vs. NP-complete), unbounded 1-player motion planning (NL vs. PSPACE-complete), and bounded 2-player motion planning (P vs. PSPACE-complete), and we partially characterize the complexity of unbounded 2-player motion planning (P vs. EXPTIME-complete), bounded 2-team motion planning (P vs. NEXPTIME-complete), and unbounded 2-team motion planning (P vs. undecidable). This framework then used to simplify existing proofs and show new results about video games and models of micro-assembly.

Bio: Jayson Lynch is a PhD student at MIT under Erik Demaine. Jayson’s work includes the computational complexity of geometric and motion planning problems; designing reversible algorithms for more energy efficient computing; and exploring the capability and limitations of cache based and parallel algorithms.

February 19th, 2020: Network Design for s-t Effective Resistance

A&C Seminar

Speaker: Hong Zhou, University of Waterloo

Abstract: Weconsider a problem of designing a network with small s-t effective resistance. In the problem, we are given an undirected graph G=(V,E), two designated vertices s,t in V, and a budget k. Thegoal is to choose a subgraph of G with at most k edges to minimize the s-t effective resistance. This problem is an interpolation between the shortest path problem and the minimum cost flow problemand has applications in electrical network design. Unlike the classic shortest path problem and min-cost flow problem, we show that the s-teffective resistance network design problem is NP-hard. On the algorithmic side, we analyze a convex programming relaxation of the problem and design a constant factor approximation algorithm. Thekey of the rounding algorithm is a randomized path-rounding procedure based on the optimality conditions and a flow decomposition of the fractional solution.

Joint work withPak Hay Chan, Lap Chi Lau, Aaron Schild and Sam Chiu-wai Wong. Arxiv link of the manuscript: https://arxiv.org/abs/1904.03219.

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2019

December 4th, 2019: Universal Communication, Universal Graphs, and Graph Labeling

A&C Seminar

Speaker: Nathan Harms, University of Waterloo

Abstract: We introduce a communication model called Universal SMP, in which Alice and Bob receive a function f belonging to a family F, and inputs x and y. Alice and Bob use shared randomness to send a message to a third partywho cannot see f, x, y, or the shared randomness, and must decide f(x,y). Our main application is to relate communication complexity to graph labeling, where the goal is to give a short label to each vertex in a graph, so that adjacency or other functions oftwo vertices and can be determined from the labels. We give a universal SMP protocol for deciding dist(x,y) < k in distributive lattices, with cost independent of the lattice size, and explain how this implies a labeling scheme. We demonstrate that many graphfamilies known to have efficient labeling schemes also admit constant-cost communication protocols, such as trees, low-arboricity graphs, and planar graphs. We also give protocols for deciding dist(x,y) < k in trees and dist(x,y) <= 2 in planar graphs, whichimplies a new labeling scheme for planar graphs.

November 7th, 2019: Dynamic Planar Point Location in External Memory

A&C Seminar

Speaker: Yakov Nekrich, Michigan Technological University

Abstract: We describe a fully-dynamic data structure for the planar point location problem in the external memory model. Our data structure supports queries in O(logBn(loglogBn)3)) I/Os and updates in O(logBn(loglogBn)2)) amortized I/Os, where n is the number of segments in the subdivision and B is the block size. This is the first dynamic data structure with almost-optimal query cost. For comparison all previously known results for this problem require O(log2Bn) I/Os to answer queries. Our result almost matches the best known upper bound in the internal-memory model.

Joint work with Ian Munro.

November 6th, 2019: Polylogarithmic Approximation for k-Connected Directed Steiner Tree in Quasi-Bipartite Graphs

A&C Seminar

Speaker: Speaker not recorded

Abstract: In the k-Connected Directed Steiner Tree problem (k-DST), we are given a directed graph G=(V,E) with edge (or vertex) costs, a root vertex r, a set of q terminals T, and a connectivity requirement k>0; the goal is to find a minimum-cost subgraph H of G such that H has k edge-disjoint paths from the root r to each terminal t in T. The k-DST problem is a natural generalization of the classical Directed Steiner Tree problem (DST) in the fault-tolerant setting in which the solution subgraph is required to have an r,t-path, for every terminal t, even after removing k-1 vertices or edges.

Despite being a classical problem, there are not many positive results on the problem, especially for the case k >= 3. In this talk, we will present an O(log k log q)-approximation algorithm for k-DST when an input graph is quasi-bipartite, i.e., when there is no edge joining two non-terminal vertices. To the best of our knowledge, our algorithm is the only known non-trivial approximation algorithm for k-DST, for k >= 3, that runs in polynomial-time regardless of the structure of the optimal solution. In addition, our algorithm is tight for every constant k, due to the hardness result inherited from the Set Cover problem.

This is a joint work Chun-Hsiang (Kenny) Chan, Hao-Ting Wei and Yuhao Zhang.

October 23rd, 2019: Online Computation with Untrusted Advice

A&C Seminar

Speaker: Spyros Angelopoulos, CNRS, LIP6, Sorbonne Université

Abstract: The advice model of online computation captures the setting in which an online algorithm is given some partial information concerning the request sequence. This paradigm allows to establish tradeoffs between the amount of this additional information and the performance of the online algorithm. However, unlike real life in which advice is a recommendation that we can chose to follow or to ignore based on trustworthiness, in the current advice model, the online algorithm typically treats it as infallible. This means that if the advice is corrupt or, worse, if it comes from a malicious source, the algorithm may perform poorly. In this work, we study online computation in a setting in which the advice is provided by an untrusted source. Our objective is to quantify the impact of untrusted advice so as to design and analyze robust online algorithms that are resilient and perform well even when the advice is generated in a malicious, adversarial manner. We show how the new paradigm can be applied to well-studied online problems such as ski rental, online bidding, bin packing, and list update.

Joint work with Christoph Dürr, Shendan Jin, Shahin Kamali and Marc Renault.

July 31st, 2019: Optimal Offline Dynamic 2,3-Edge/Vertex Connectivity

A&C Seminar

Speaker: Bryce Sandlund, University of Waterloo

Abstract: We give offline algorithms for processing a sequence of 2- and 3-edge and vertex connectivity queries in a fully-dynamic undirected graph. While the current best fully-dynamic online data structures for 3-edge and 3-vertex connectivity require O(n^(2/3)) and O(n) time per update, respectively, our per-operation cost is only O(log n), optimal due to the dynamic connectivity lower bound of Patrascu and Demaine. Our approach utilizes a divide and conquer scheme that transforms a graph into smaller equivalents that preserve connectivity information. This construction of equivalents is closely-related to the development of vertex sparsifiers, and shares important connections to several upcoming results in dynamic graph data structures, including online models. Joint work with Richard Peng and Daniel D. Sleator.

July 10th, 2019: Title not recorded

A&C Seminar

Speaker: Jan Gorzny, University of Waterloo

Abstract: We show that both Cutwidth and Imbalance are fixed-parameter tractable when parameterized by the twin-cover number of the input graph. We further show that Imbalance is NP-complete for split graphs and linear-time solvable for proper interval (bipartite) graphs, which equals the complexity of Cutwidth on these classes. Both results follow from a new structural theorem, that every instance of Cutwidth or Imbalance has an optimal ordering of a restricted form. This is joint work with Jonathan Buss.

June 19th, 2019: State Complexity of Square of Star-Free Languages

A&C Seminar

Speaker: Sylvie Davies, University of Waterloo

Abstract: The state complexity of a regular language is the minimal number of states needed to recognize that language with a DFA. Then given a unary operation on regular languages, we may ask: if the input language is recognized by a DFA with n states, what is the worst-case state complexity of the output language, as a function of n? This worst-case value is called the state complexity of the operation. For the square operation (concatenation of a language with itself), the state complexity is known if the input is an arbitrary regular language, but not if the input is a star-free language. The state complexity of many common operations on star-free languages (union, intersection, concatenation, star, reversal) is known, but square remains elusive. We describe recent progress on this still-open problem, building on some work of Brzozowski and Szykuła on large aperiodic semigroups.

June 13th, 2019: Cryptography for Safe Machine Learning

Joint A&C Seminar and Distinguished Lecture Series [video]

Speaker: Shafi Goldwasser, MIT

May 22nd, 2019: The Combinatorics of Furthest and Nearest Values

A&C Seminar

Speaker: Lily Wang, University of Waterloo

Abstract: A classical problem asks us to find, for each element $A[i]$ of an array of integers, the position of the nearest smallest element. Similarly, we can ask about the dual problem: for each element of an array of integers $A[i]$, what is the position of the furthest smaller element? In our paper, we discussed both these problems from a combinatorial perspective and considered algorithms to solve them. By examining results of permutations of distinct integers and behaviour of the algorithms, we find many classical combinatorial sequences such as the Stirling numbers, the Catalan numbers, the Bell numbers, and the harmonic numbers.

May 8th, 2019: Lower Bounds for Oblivious Near-Neighbor Search

A&C Seminar

Speaker: Kevin Yeo, Google Research

Abstract: We prove an Ω(dlgn/(lglgn)^2) lower bound on the dynamic cell-probe complexity of statistically oblivious approximate-near-neighbor search (ANN) over the d-dimensional Hamming cube. For the natural setting of d=Θ(lgn), our result implies an Ω~(lg^2 n) lower bound, which is a quadratic improvement over the highest (non-oblivious) cell-probe lower bound for ANN. This is the first super-logarithmic unconditional lower bound for ANN against general (non black-box) data structures. We also show that any oblivious static data structure for decomposable search problems (like ANN) can be obliviously dynamized with O(lgn) overhead in update and query time, strengthening a classic result of Bentley and Saxe (Algorithmica, 1980).

Joint work with Kasper Green Larsen, Tal Malkin and Omri Weinstein.

March 13th, 2019: Lower Bounds for Tolerant Junta and Unateness Testing via Rejection Sampling of Graphs

A&C Seminar

Speaker: Amit Levi, University of Waterloo

Abstract: We introduce a new model for testing graph properties which we call the{rejection sampling model}. We show that testing bipartiteness of n-nodes graphs using rejection sampling queries requires complexity Ω(n^2). Via reductions from the rejection sampling model, we give three new lower bounds for tolerant testing of Boolean functions of the form f:{0,1}^n→{0,1}: ∙Tolerant k-junta testing with {non-adaptive} queries requires Ω(k^2) queries. ∙Tolerant unateness testing requires Ω(n) queries. ∙Tolerant unateness testing with {non-adaptive} queries requires Ω(n^3/2) queries. Given the O(k^3/2)-query non-adaptive junta tester of Blais, we conclude that non-adaptive tolerant junta testing requires more queries than non-tolerant junta testing. In addition, given the O(n^3/4)-query unateness tester of Chen, Waingarten, and Xie and the O(n)-query non-adaptive unateness tester of Baleshzar, Chakrabarty, Pallavoor, Raskhodnikova, and Seshadhri, we conclude that tolerant unateness testing requires more queries than non-tolerant unateness testing, in both adaptive and non-adaptive settings. These lower bounds provide the first separation between tolerant and non-tolerant testing for a natural property of Boolean functions.

March 6th, 2019: Dynamic Low-Stretch Trees via Dynamic Low-Diameter Decompositions

A&C Seminar

Speaker: Gramoz Goranci, University of Vienna

Abstract: Spanning trees of low average stretch on the non-tree edges are natural graph-theoretic objects that have found applications in fast solvers for symmetric diagonally dominant (SDD) linear systems, construction of competitive oblivious routing schemes and approximation algorithms.

In this talk, I will present the first non-trivial algorithm for maintaining such trees under edge insertions and deletions to the input graph. Our algorithm has update time n^{1/2+o(1)} and the average stretch of the maintained tree is n^{o(1)}, which matches the stretch in the seminal result of Alon et al. [SICOMP’ 95]. The key ingredients to our result are (1) dynamic maintenance of low-diameter decompositions (LLDs), (2) controlling the propagation of updates within a hierarchy of dynamic LDDs and (3) incorporation of dynamic cut sparsifiers to improve the update time.

This is joint work with Sebastian Forster, and will appear at STOC 2019.

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2018

December 12th, 2018: Counting, Adding, and Regular Languages

A&C Seminar

Speaker: Finn Lidbetter

Abstract: In this talk we consider two mostly disjoint topics in formal language theory that both involve the study and use of regular languages. The first topic lies in the intersection of automata theory and additive number theory. We introduce a method of producing results in additive number theory, relying on theorem-proving software and an approximation technique. As an example of the method, we prove that every natural number greater than 25 can be written as the sum of at most 3 natural numbers whose canonical base-2 representations have an equal number of 0’s and 1’s. The second topic is the study of languages defined by criteria involving the number of occurrences of a particular pair of words within other words. That is, we consider languages of words z defined with respect to words x, y where z has the same number of occurrences (resp., fewer occurrences), (resp., fewer occurrences or the same number of occurrences) of x as a subword of z and y as a subword of z. We give a necessary and sufficient condition on when such languages are regular, and show how to check this condition efficiently. This work is the result of collaborations with Jason Bell, Charles Colbourn, Ryan Dougherty, and Jeffrey Shallit.

December 5th, 2018: Local flow-based methods for graph clustering

A&C Seminar

Speaker: Di Wang

Abstract: We study the problem of graph clustering where the goal is to partition a graph into clusters, i.e. disjoint subsets of vertices, such that each cluster is well connected internally while sparsely connected to the rest of the graph. In particular, we use a natural bicriteria notion motivated by Kannan, Vempala, and Vetta [KVV00] which we refer to as expander decomposition. Expander decomposition has become one of the building blocks in the design of fast graph algorithms, most notably in the nearly linear time Laplacian solver by Spielman and Teng [ST04], and it also has wide applications in practice.

For the global version, given graph $G$ and parameter $\phi$, we design $\tilde{O}{m/\phi)$ time algorithm to partition the vertices into clusters such that each cluster induces a subgraph of conductance at least $\phi$, while only a $\tilde{O}(\phi)$ fraction of the edges in the graph have endpoints across different clusters. This is the first nearly linear time algorithm when $\phi$ is at least $1/ polylog m$, which is the case in most practical settings and theoretical applications, and only relies on simple and basic flow-based techniques. Previous results either take $m^{1+o(1)}$ time (e.g. [NS17, Wul17]), or attain nearly linear time but with a weaker expansion guarantee where each output cluster is guaranteed to be contained inside some unknown expander (e.g. [ST13, ACL06]).

In the local version of the problem, we are given a seed node $s$, and want to find a good cluster containing $s$ (if there exists one) in time proportional to the size of the (unknown) cluster. While flow and probability mass diffusion (or more generally, spectral methods) have a long history of competing to provide good graph decomposition, local methods are predominantly based on diffusion. We design the first primarily flow-based local method for locating low conductance cuts, and it has exhibited improved theoretical and empirical behavior over classical diffusion methods, e.g. PageRank.

November 21st, 2018: Revisiting the simulation of quantum Turing machines by quantum circuits

A&C Seminar

Speaker: Abel Molina

Abstract: Yao (1993) proved that quantum Turing machines and uniformly generated quantum circuits are polynomially equivalent computational models: t >= n steps of a quantum Turing machine running on an input of length n can be simulated by a uniformly generated family of quantum circuits with size quadratic in t, and a polynomial-time uniformly generated family of quantum circuits can be simulated by a quantum Turing machine running in polynomial time. We revisit the simulation of quantum Turing machines with uniformly generated quantum circuits, which is the more challenging of the two simulation tasks, and present a variation on the simulation method employed by Yao together with an analysis of it. This analysis reveals that the simulation of quantum Turing machines can be performed by quantum circuits having depth linear in t, rather than quadratic depth, and can be extended to variants of quantum Turing machines, such as ones having multi-dimensional tapes. Our analysis is based on an extension of a method of Arrighi, Nesme, and Werner (2011) that allows for the localization of causal unitary evolutions. This talk will not require a background in quantum computing.

October 24th, 2018: The power of constructive proofs

A&C Seminar

Speaker: Antonina Kolokolova

Abstract: A unifying theme in complexity theory in the past few years has been the duality between lower bounds and algorithms. Indeed, some of the main recent lower bounds have been proven by developing better algorithms.

In this talk, I would like to focus on the other direction of this duality: obtaining new algorithms from (constructive) proofs. I will talk about a general method for extracting learning algorithms from natural proofs of lower bounds, which gave us the first algorithm for learning bounded-depth Boolean circuits with parity gates. Moreover, we show that natural proofs based on random restrictions contain new algorithms for compression and for counting satisfying assignments.

But what makes a proof constructive? One way to formalize this is to show that the proof can be carried out in a reasoning system of a given complexity: for example, natural proofs discussed above can be formalized in the theory of polynomial-time reasoning. Pinpointing the complexity of reasoning needed to prove complexity-theoretic statements can lead to more constructive proofs, and in turn to new results: time permitting, I will discuss one such example, where a low-complexity proof of existence of expander graphs allowed us to show that in the proof complexity setting, monotone reasoning is as powerful as non-monotone, in stark contrast with circuit complexity.

Hosted by Professor Shai Ben-David

October 10th, 2018: The Complexity of Graph Drawings and Representations

A&C Seminar

Speaker: Anna Lubiw

Abstract: In this talk I will look at geometric graph representations from the perspective of three issues: the algorithmic complexity of finding a representation; the bit complexity of the representation; and whether there is a morph between any two combinatorially equivalent representations.

The case of straight-line drawings of planar graphs is nice for all three issues: every planar graph has such a representation; the bit complexity is log n since vertices can be placed on an nxn grid; and morphs always exist.

By contrast, recognizing intersection graphs of unit discs is hard for existential theory of the reals; coordinates may require an exponential number of bits; and morphs are not always possible. These negative results are related.

I will show a version of straight-line planar drawing that lies on the “hard” end of this spectrum. It is the problem of drawing a planar graph with straight-line edges inside a given polygonal region, when some vertices are fixed on the boundary of the region. In other words, the extension of Tutte’s graph drawing algorithm when his convex outer boundary is generalized to a polygonal region.

Joint work with Tillmann Miltzow and Debajyoti Mondal.

August 29th, 2018: Counting Subwords and Regular Languages

A&C Seminar

Speaker: Finn Lidbetter

Abstract: Let x and y be words. We consider the languages whose words z are those for which the numbers of occurrences of x and y, as subwords of z, are the same (resp., the number of x’s is less than the number of y’s, resp., is less than or equal). In this talk we will give a necessary and sufficient condition on x and y for these languages to be regular, and we show how to check this condition efficiently. This is a joint work with Charles Colbourn, Ryan Dougherty, and Jeffrey Shallit. Note: This is a practice talk for DLT18 and so should be around 30 minutes.

August 1st, 2018: Minimum Shared-Power Edge Cut

A&C Seminar

Speaker: Kshitij Jain

Abstract: We introduce a problem called the Minimum Shared-Power Edge Cut (MSPEC). The input to the problem is an undirected edge-weighted graph with distinguished vertices s and t, and the goal is to find an s-t cut by assigning “powers” at the vertices and removing an edge if the sum of the powers at its endpoints is at least its weight. The objective is to minimize the sum of the assigned powers.

MSPEC is a graph generalization of a barrier coverage problem in a wireless sensor network: given a set of unit disks with centers in a rectangle, what is the minimum total amount by which we must shrink the disks to permit an intruder to cross the rectangle undetected, i.e. without entering any disc. This is a more sophisticated measure of barrier coverage than the minimum number of disks whose removal breaks the barrier.

We develop a fully polynomial time approximation scheme (FPTAS) for MSPEC. We give polynomial time algorithms for the special cases where the edge weights are uniform, or the power values are restricted to a bounded set. Although MSPEC is related to network flow and matching problems, its computational complexity (in P or NP-hard) remains open.

July 25th, 2018: Title not recorded

A&C Seminar

Speaker: Philipp Kindermann

Abstract: The visual complexity of a graph drawing is defined as the number of geometric objects needed to represent all its edges. In particular, one object may represent multiple edges, e.g., one needs only one line segment to draw two collinear incident edges. We investigate whether drawings with few segments have a better aesthetic appeal and help the user to assess the underlying graph. We develop algorithms for drawing planar graphs with few segments. Then we design a user study that investigates two different graph types (trees and sparse graphs), three different layout algorithms for trees, and two different layout algorithms for sparse graphs. We asked the participants to give an aesthetic ranking on the layouts and to perform a furthest-pair or shortest-path task on the drawings.

July 18th, 2018: Additive Number Theory via Approximation by Regular Languages

A&C Seminar

Speaker: Finn Lidbetter

Abstract: The fundamental problem of additive number theory is to determine whether there exists an integer m such that every nonnegative integer (resp., every sufficiently large nonnegative integer) is the sum of at most m elements of S. If so, we call S an additive basis of order m (resp., an asymptotic additive basis of order m). If such an m exists, we also want to find the smallest such m. In this talk we will prove some new theorems concerning this fundamental problem in additive number theory, using novel techniques from automata theory and formal languages. As an example of our method, we prove that every natural number > 25 is the sum of at most three natural numbers whose base-2 representation has an equal number of 0’s and 1’s. This is joint work with Jeffrey Shallit and Jason Bell.

Note: This work will be presented at DLT18 (Developments in Language Theory) in September, and so the presentation will be around 30 minutes.

May 2nd, 2018: Approximate Shortest Paths and Distance Oracles in Weighted Unit-Disk Graphs

A&C Seminar

Speaker: Dimitrios Skrepetos, University of Waterloo

Abstract: We give the first near-linear-time (1+epsilon)-approximation algorithm for the diameter of a weighted unit-disk graph of n vertices, running in O(nlog^2 n) time for any constant epsilon>0, considerably improving the near-O(n^{3/2})-time algorithm of Gao and Zhang [STOC 2003]. We can also construct a (1+epsilon)-approximate distance oracle for weighted unit-disk graphs with O(1) query time, with a similar improvement in the preprocessing time, from near O(n^{3/2}) to O(n\log^3 n). We obtain similar new results for a number of other related problems in the weighted unit-disk graph metric, such as the radius and bichromatic closest pair.

As a further application, we use our new distance oracle, along with additional ideas, to solve the (1+epsilon)-approximate all-pairs bounded-leg shortest paths problem for a set of n planar points, with near O(n^{2.579}) preprocessing time, O(n^2\log n) space, and O(\log{\log n}) query time, improving the near-cubic preprocessing bound by Roditty and Segal [SODA 2007].

It is a joint work with my supervisor Timothy M. Chan, and it will be presented in SoCG 2018.

April 11th, 2018: Using Automata Theory to Solve Problems in Additive Number Theory

A&C Seminar

Speaker: Speaker not recorded

Abstract: Additive number theory is the study of the additive properties of integers. Perhaps the best-known theorem is Lagrange’s result that every natural numberis the sum of four squares. We study numbers whose base-k representationshave certain interesting properties. In particular, we look at palindromes,which are numbers whose base-k representations read the same forward and backward,and binary squares, which are numbers whose binary representation is some block repeated twice (like (36)_2 = 100100). We show that all natural numbers are the sum of four binary palindromes. We alsoshow that all natural numbers are the sum of three base-3 palindromes, and are alsothe sum of three base-4 palindromes. We also show that every sufficiently large naturalnumber is the sum of four binary squares. We establish these results using virtually no number theory at all. Instead, we construct automated proofs using automata. The general proof technique isto build an appropriate machine, and then run decision algorithms to establish our theorems.

April 4th, 2018: Succinct Color Searching in One Dimension

A&C Seminar

Speaker: Speaker not recorded

Abstract: We present succinct data structures for one-dimensional color reporting and color counting problems. We are given a set of $n$ points with integer coordinates in the range $[1,m]$ and every point is assigned a color from the set $\{\,1,\ldots,\sigma\,\}$. A color reporting query asks for the list of distinct colors that occur in a query interval $[a,b]$ and a color counting query asks for the number of distinct colors in $[a,b]$.We describe a succinct data structure that answers approximate color counting queries in $O(1)$ time and uses $\mathcal{B}(n,m) + O(n) + o(\mathcal{B}(n,m))$ bits, where $\mathcal{B}(n,m)$ is the minimum number of bits required to represent an arbitrary set of size $n$ from a universe of $m$ elements. Thus we show, somewhat counterintuitively, that it is not necessary to store colors of points in order to answerapproximate color counting queries. In the special case when points are in the rank space (i.e., when $n=m$),our data structure needs only $O(n)$ bits. Also, we show that $\Omega(n)$ bits are necessary in that case.Then we turn to succinct data structures for color reporting. We describe a data structure that uses $\mathcal{B}(n,m) + nH_d(S) + o(\mathcal{B}(n,m)) + o(n\lg\sigma)$ bits and answers queries in $O(k+1)$time, where $k$ is the number of colors in the answer, and $nH_d(S)$ ($d=\log_{\sigma} n$) is the $d$-th order empirical entropy of the color sequence. Finally, we consider succinct color reporting under restricted updates.Our dynamic data structure uses $nH_d(S)+o(n\lg\sigma)$ bits and supports queries in $O(k+1)$ time.

March 21st, 2018: Raising Permutations to Powers In Place

A&C Seminar

Speaker: Speaker not recorded

Abstract: Given a permutation of $n$ elements, stored as an array, we address theproblem of replacing the permutation by its $k^{\mathrm{th}}$ power.We aim to perform this operation quickly using $o(n)$ bits of extra storage.To this end, we first present an algorithm for inverting permutations thatuses $O(\lg^2 n)$ additional bits and runs in $O(n\lg n)$ worst case time.This result is then generalized to the situation in which the permutationis to be replaced by its $k^{\mathrm{th}}$ power.An algorithm whose worst case running time is $O(n\lg n)$ and uses$O(\lg^2 n + \min\{k\lg n,n^{\rfrac{3}{4}+\epsilon}\})$ additional bits ispresented.

February 28th, 2018: Dependent Random Graphs and Multiparty Pointer Jumping

A&C Seminar

Speaker: Speaker not recorded

Abstract: We initiate a study of a relaxed version of the standard Erdos-Renyi random graph model, where each edge may depend on a few other edges. We call such graphs *dependent random graphs* and give tight bounds on the clique and chromatic numbers of such graphs. Surprisingly, some of the bounds in the standard random graph model continue to hold in this relaxed setting. For example, the size of the largest clique in a dependent random graph remains roughly log(n)/log(1/p).

As an application, we give a new upper bound on communication complexity of the Multiparty Pointer Jumping (MPJ) problem in the number-on-the-forehead (NOF) model. NOF communication lies at the current frontier of our understanding of communication complexity, and MPJ is one of the canonical problems in this setting. Furthermore, sufficiently strong bounds for MPJ would have important consequences for circuit complexity.

No prior knowledge is assumed aside from basic discrete probability. I hope to motivate both random graphs and the application and demonstrate why NOF communication is an important active research area.

February 21st, 2018: Faster Approximate Diameter and Distance Oracles in Planar Graphs

A&C Seminar

Speaker: Speaker not recorded

Abstract: We present an algorithm that computes a (1+\varepsilon)-approximation of the diameter of a weighted, undirected planar graph of n vertices with non-negative edge lengths in O(n \log n (\log n + (1/\varepsilon)^5)) expected time, improving upon the O(n ((1/\varepsilon)^4 \log^4 n + 2^{O(1/\varepsilon)}))-time algorithm of Weimann and Yuster [ICALP 2013]. Our algorithm makes two improvements over that result: first and foremost, it replaces the exponential dependency on 1/\varepsilon with a polynomial one, by adapting and specializing Cabello’s recent abstract-Voronoi-diagram-based technique [SODA 2017] for approximation purposes; second, it shaves off two logarithmic factors by choosing a better sequence of error parameters during recursion.

Moreover, using similar techniques, we improve the (1+\varepsilon)-approximate distance oracle of Gu and Xu [ISAAC 2015] by first replacing the exponential dependency on 1/\varepsilon on the preprocessing time and space with a polynomial one and second removing a logarithmic factor from the preprocessing time.

February 7th, 2018: Evolutionary graph theory

A&C Seminar

Speaker: Speaker not recorded

Abstract: Let $G$ be a graph on $n$ vertices with one random vertex colored black and all the other vertices colored white. In the language of population genetics, this corresponds to a single (advantageous) mutant appearing in a structured population of indistinguishable residents. Moran process is a certain discrete-time stochastic process that changes the coloring of the vertices in such a way that eventually the vertices become either all white (the mutant went extinct) or all black (the mutant “fixated”). Evolutionary graph theory studies how the fixation probability $\fp(G)$ depends on the underlying graph structure. For example it is known that $\fp(K_n)=\fp(C_n)<\fp(S_n)$ where $K_n$, $C_n$, $S_n$ are the complete graph, the cycle, and the star, respectively.

In this talk we will give an overview of the research done in the field and present some recent results concerning graphs for which the fixation probability tends to 1 as the number of vertices tends to infinity.

January 31st, 2018: A basic learning problem that is independent of the set theory ZFC axioms

A&C Seminar

Speaker: Speaker not recorded

Abstract: We consider the Expectation Maximization (EMX) problem that captures many well studied learning problems. We study the interaction between the statistical sample complexity of a class of functions and its combinatorial structure. Surprisingly, we show that the EMX learnability, as well as the learning rates of some basic class of functions F, depend on the cardinality of the continuum in the model of set theory one “lives in” and is therefore independent of the set theory ZFC axioms (that are widely accepted as a formalization of the notion of a mathematical proof).

This result implies that there exist no “finitery” combinatorial parameter that characterizes EMX learnability in a way similar to the VC-dimension characterization of binary valued classification problems.

The talk is based on joint work with Pavel Hrubeˇs, Shay Moran, Amir Shpilka, and Amir Yehudayoff

January 24th, 2018: Bio-inspired Operations on Formal Languages, and Their Properties

A&C Seminar

Speaker: Speaker not recorded

Abstract: In molecular biology, gene sequences can undergo several types of rearrangements, such as inversions, deletions, insertions, and duplications. From a formal-language viewpoint, our research focusses on (1) modelling versions of such gene arrangements that allow for some errors, and (2) modeling laboratory techniques that produce rearrangements at specific sites of gene sequences, under enzymatic activities. The goal is to understand computational properties of such gene rearrangements and to identify efficient ways to carry out molecular tasks, via their formal language and automata theory models. We introduce several bio-inspired operations, motivated by biological phenomena, and we study typical formal languages problems regarding these operations, such as their state complexity, and closure and decision properties of various language families. This approach could lead to a fruitful link between theoretical and experimental work with DNA sequences and, more generally, between theoretical bio-computing and molecular biology.

January 17th, 2018: Lower bounds on regular expression size

A&C Seminar

Speaker: Speaker not recorded

Abstract: Given a regular language, it is PSPACE-hard to determine the size of its shortest regular expressions. Also, there are very few techniques known for bounding the size of the regular expressions for a given language. One such technique uses bounds from circuit complexity. In this talk, we see a technique that resembles the semidefinite programming duality of the adversary method used in quantum query complexity.

We show that to every finite language there corresponds a linear program with the following properties. The optimum value of the primal problem is a lower bound on the size of the regular expressions of the language. Moreover, every regular expression can be turned into a dual feasible solution with an objective value that is equal to the size of the regular expression. We use this construct and prove lower bounds on the size of regular expressions of binomial languages, $B(n,k)={x \in {0,1}^n : |x|_1 = k}$, and threshold languages, $T(n,k)={x \in {0,1}^n : |x| >= k}$.

January 15th, 2018: Orbit Theorem for Flipping Edge-Labelled Triangulations

A&C Seminar

Speaker: Speaker not recorded

Abstract: Given a triangulation of a point set in the plane, a flip deletes an edge e whose removal leaves a convex quadrilateral, and replaces e by the opposite diagonal of the quadrilateral. It is well known that any triangulation of a point set can be reconfigured to any other triangulation by some sequence of flips. We explore this question in the setting where each edge of a triangulation has a label, and a flip transfers the label of the removed edge to the new edge. It is not true that every labelled triangulation of a point set can be reconfigured to every other labelled triangulation via a sequence of flips. We characterize when this is possible by proving the Orbit Conjecture [Bose, Lubiw, Pathak and Verdonschot].

Our proof uses the topological result that the sets of pairwise non-crossing edges on a planar point set form a simplicial complex that is homeomorphic to a high-dimensional ball. The dual cell complex of this simplicial ball, called the flip complex, has the usual flip graph as its 1-skeleton. We use properties of the 2-skeleton of the flip complex to prove the Orbit Conjecture. This is joint work with Anna Lubiw and Uli Wagner.

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2017

November 29th, 2017: Graph Thickness, Planar Layers, and Bend Complexity

A&C Seminar

Speaker: Speaker not recorded

Abstract: The thickness of a graph G with n vertices is the minimum number of planar subgraphs of G whose union is G. By Fáry’s theorem, every planar graph (thickness-one graph) has a planar straight-line drawing on the Euclidean plane, i.e., a planar drawing where every vertex is mapped to a point and every edge is mapped to a straight line segment. A natural way to extend this result to graphs with higher thickness is to leverage polyline drawings, where edges are drawn using polygonal chains. For a graph of thickness t, we seek for such a drawing where the polyline edges can be partitioned into t disjoint sets, each corresponding to a planar drawing with a small number of bends per edge. Geometric thickness is another related concept, where the edges must be drawn using straight line segments.

This talk will review some recent developments that relate graph thickness to planar layers and edge bends. In addition, it will highlight some interesting directions for future research.

Biography: Debajyoti Mondal is an assistant professor at the Department of Computer Science, University of Saskatchewan. His research interests include graph drawing, network visualization, and computational geometry. Before joining the University of Saskatchewan, he was an NSERC Postdoctoral Fellow at the University of Waterloo (supervised by Anna Lubiw). His PhD is from the University of Manitoba in 2016 (supervised by Stephane Durocher).

November 22nd, 2017: Graph clustering using effective resistance

A&C Seminar

Speaker: Speaker not recorded

Abstract: We show a new connection between graph expansion and effective resistance distance – the effective resistance between two vertices when we interpret the graph as an electrical network – by showing that in a graph where every set exhibits a mild expansion property, the effective resistance between every pair of vertices is small. We use this connection to design an efficient algorithm to partition a graph into components with small effective resistance diameters. In particular, we show that it is always possible to delete a constant fraction of the edges of a graph, so that each remaining component is an “electrical expander” – a graph where the resistance distance between any two vertices is smallest possible.

No background will be assumed from the audience. We will start the talk by surveying the known results about effective resistance, and we will explain the motivation of the decomposition by comparing it to the well-known low diameter graph decomposition and expander decomposition.

Joint work with Nima Anari, Lap Chi Lau and Shayan Oveis Gharan

November 10th, 2017: Fast Approximate Gaussian Elimination for Laplacians

A&C Seminar

Speaker: Sushant Sachdeva, University of Toronto

Abstract: Solving systems of linear equations in graph Laplacians is afundamental primitive in scientific computing. Starting with the seminal work of Spielman-Teng that gave the first nearly-linear time algorithm for solving Laplacian systems, there has been a long line of work giving faster Laplacian solvers. These solvers have had a large impact on the design of fast graph algorithms.

In this talk, I’ll present a very simple, nearly-linear time Laplacian solver that is based purely on random sampling, and does not use any graph theoretic constructions such as low-stretch trees, sparsifiers, or expanders. Our solver builds a sparse Cholesky factorization for Laplacians — the symmetric version of Gaussian elimination. More precisely, it approximates a Laplacian L as U’U, where U is a sparse upper triangular matrix. Since triangular matrices are easy to invert, this immediately implies a fast Laplacian solver via iterative refinement. The analysis is based on concentration of matrix martingales.

This is joint work with Rasmus Kyng.

About the Speaker: Sushant Sachdeva is a faculty member at the CS dept. at the University of Toronto. He is interested in Algorithms, and its connections to optimization, machine learning, and statistics. His recent research focus has been the design of fast algorithms for graph problems.

Before joining UofT, he was a research scientist at Google. He completed his postdoc at Yale with Dan Spielman (2016), his PhD from Princeton (2013) under Sanjeev Arora, and his BTech from IIT Bombay (2008). He is the recipient of Simons Berkeley Research Fellowship (2013), and the IITB President of India Gold Medal (2008).

November 8th, 2017: Testing Linearity against No-Signaling Strategies

A&C Seminar

Speaker: Speaker not recorded

Abstract: No-signaling strategies are collections of distributions with certain non-local correlations. In this talk, we study the classical problem of linearity testing (Blum, Luby, and Rubinfeld; JCSS 1993) against no-signaling strategies. We prove that any no-signaling strategy that passes the linearity test with high probability must be close to a quasi-distribution over linear functions.

Quasi-distributions generalize the notion of probability distributions over functions by allowing negative probabilities, while at the same time requiring that “local views” follow standard distributions (with non-negative probabilities). As part of our analysis, we also establish a general equivalence between no-signaling strategies and quasi-distributions, which, we believe, provides a useful perspective on the study of no-signaling strategies beyond property testing.

Joint work with Alessandro Chiesa and Peter Manohar (UC Berkeley).

November 1st, 2017: Dual-Pivot Quicksort and Beyond

A&C Seminar

Speaker: Speaker not recorded

Abstract: Quicksort is one of most well-understood algorithms both theoretically and in practice. An implementation of Quicksort is part of almost every programming library. After excessive experimenting and engineering in the 1970s, the tuning efforts seemed to have converged to a stable state; but now, there is again excitement within the algorithms community, triggered by the success of a new dual-pivot Quicksort used in the Java 7 runtime library.

I will introduce the new algorithm and present analytical evidence for the hypothesis why (a) dual-pivot Quicksort is faster than the previously used (classic) Quicksort and (b) why this basic improvement was not already found much earlier. In passing, I try to give the intuition behind my favorite mathematical tools for the analysis of algorithms.

October 18th, 2017: The Art Gallery Problem is $\exists \mathbb{R}$-complete

A&C Seminar

Speaker: Speaker not recorded

Abstract: We prove that the \emph{art gallery problem} is equivalent under polynomial time reductions to deciding whether a system of polynomial equations over the real numbers has a solution.

The art gallery problem is a classical problem in computational geometry, introduced in 1973 by Viktor Klee. Given a simple polygon $P$ and an integer $k$, the goal is to decide if there exists a set $G$ of $k$ \emph{guards} within $P$ such that every point $p\in P$ is seen by at least one guard $g\in G$. Each guard corresponds to a point in the polygon $P$, and we say that a guard $g$ \emph{sees} a point $p$ if the line segment $pg$ is contained in $P$.

The art gallery problem has stimulated a myriad of research in geometry and in algorithms. However, despite extensive research, the complexity status of the art gallery problem has not been resolved. It has long been known that the problem is NP-hard, but no one has been able to show that it lies in NP. Recently, the computational geometry community became more aware of the complexity class $\exists \mathbb{R}$ (ER). The class ER consists of problems that can be reduced in polynomial time to the problem of deciding whether a system of polynomial equations with integer coefficients and any number of real variables has a solution. It can be easily seen that $NP \subseteq ER$.

We prove that the art gallery problem is ER-complete, implying that (1) any system of polynomial equations over the real numbers can be encoded as an instance of the art gallery problem, and (2) the art gallery problem is not in the complexity class NP unless $NP=ER$.

As a corollary of our construction, we prove that for any real algebraic number $\alpha$ there is an instance of the art gallery problem where one of the coordinates of the guards equals $\alpha$ in any guard set of minimum cardinality.

That rules out many geometric algorithmic approaches to the problem.

To illustrate further the connections between real algebra and the art gallery problem we show the following statement.

Given any compact semi-algebraic set $S \subset [0,1]^2$ in the plane. There is a polygon $\poly$ with vertices at rational coordinates such that for any point $p \in [0,1]^2$ the following two statements are equivalent

\begin{itemize} \item There exists an optimal guard set $G$ with $p\in G$. \item $p \in S$. \end{itemize}

In the talk, we will explain in detail the underlying definitions and the results. If time permits, we will give some simple key ideas of the proof.

Joint work with Anna Adamaszek and Mikkel Abrahamsen

September 20th, 2017: Distances Between Languages: Algorithms and Descriptional Complexity

A&C Seminar

Speaker: Speaker not recorded

Abstract: Distance measures are defined on words to describe their similarity. These measures can be extended to languages. We first consider the descriptional complexity of neighbourhoods of regular languages. The neighbourhood of a language L is the set of words within some fixed distance of a word in L. We consider the deterministic and nondeterministic state complexity of prefix, suffix, and subword distance neighbourhoods.

We then consider the relative distance between two languages. The relative distance from a language L_1 to a language L_2, if finite, is the smallest integer k such that for every word in L_1, there is a word in L_2 with distance at most k. We study the relative prefix distance between regular, visibly pushdown, deterministic context-free, and context-free languages. We show how to compute the distance between regular languages and determine whether the distance is bounded. For deterministic context-free languages and visibly pushdown languages, we show that the relative prefix distance to and from regular languages is decidable.

August 9th, 2017: Approximating Unique Games Using Low Diameter Graph Decomposition

A&C Seminar

Speaker: Speaker not recorded

Abstract: We design approximation algorithms for Unique Gmeas when the constraint graph admits good low diameter graph decomposition. For the M2Lin(k) problem in K(r)-minor free graphs, when there is an assignment satisfying 1-eps fraction of constraints, we present an algorithm that produces an assignment satisfying 1-O(r*eps) fraction of constraints, with the approximation ratio independent of the alphabet size. A corollary is an improved approximation algorithm for the Min-UnCut problem for K(r)-minor free graphs. For general Unique Games in K(r)-minor free graphs, we provide another algorithm that produces an assignment satisfying 1-O(r *sqrt(eps)) fraction of constraints.

Our approach is to round a linear programming relaxation to find a minimum subset of edges that intersects all the inconsistent cycles. We show that it is possible to apply the low diameter graph decomposition technique on the constraint graph directly, rather than to work on the label extended graph as in previous algorithms for Unique Games. The same approach applies when the constraint graph is of genus g, and we get similar results with r replaced by log g in the M2Lin(k) problem and by sqrt(log g) in the general problem. The former result generalizes the result of Gupta-Talwar for Unique Games in the M2Lin(k) case, and the latter result generalizes the result of Trevisan for general Unique Games.

August 2nd, 2017: Sums of Palindromes: An Approach via Automata

A&C Seminar

Speaker: Speaker not recorded

Abstract: We prove, using a decision procedure based on automata, that every natural number is the sum of at most 9 natural numbers whose base-2 representation is a palindrome. We consider some other variations on this problem, and prove similar results. We argue that heavily case-based proofs are a good signal that a decision procedure may help to automate the proof.

June 29th, 2017: ETR, or the Real Logic of Drawing Graphs

A&C Seminar

Speaker: Speaker not recorded

Abstract: What is harder to compute: The crossing number, or the rectilinear crossing number of a graph? We know that the two numbers are different, but maybe computing them has the same complexity? It turns out that the two problems differ significantly in complexity, one is NP-complete, the other is complete for the existential theory of the reals (ETR), due to an old result of Bienstock’s. Recently, the visibility graph recognition problem, and the art gallery problem have been shown ETR-complete. In this talk we survey the existential theory of the reals, and the role it has (and can) play in graph drawing.

June 28th, 2017: Settling the query complexity of non-adaptive junta testing

A&C Seminar

Speaker: Speaker not recorded

Abstract: In this talk, I will show that any non-adaptive algorithm that tests whether an unknown Boolean function f:{0,1}^n→{0,1} is a k-junta or ϵ-far from every k-junta must make Ω˜(k^(3/2)/ϵ) many queries. This result is essentially optimal given Blais’s non-adaptive junta tester, which makes O˜(k^(3/2))/ϵ queries and shows that adaptivity enables polynomial savings in query complexity for junta testing. At a very high level, the proof proceeds by reducing the non-adaptive junta testing of a new class of random Boolean functions to a problem of distinguishing two binomial distributions with a specific kind of noisy query.

This is joint work with Xi Chen, Rocco Servedio, Li-Yang Tan, and Jinyu Xie.

June 27th, 2017: Fluctuations in the asymptotic behavior of partial sums of $k$-regular sequences

A&C Seminar

Speaker: Daniel Krenn

Abstract: In this talk, we focus on the asymptotic behavior of $k$-regular sequences, in particular on the example of the number of ones in Pascal’s rhombus. A combination of Dirchlet series together with Mellin–Perron summation of orders one and two, and asymptotic results on $k$-regular sequences is used to analyze the fluctuation of the main term in the asymptotics of the sequence.

June 13th, 2017: ALGORAND, A New Public Ledger

Joint A&C Seminar and Distinguished Lecture Series [video]

Speaker: Silvio Micali, MIT

Abstract: A public ledger is a tamper proof sequence of data that can be read and augmented by everyone. Shared public ledgers stand to revolutionize the way a democratic society operates. They secure all kinds of traditional transactions— such as payments, asset transfers, titling — in the exact order in which they occur; and enable totally new transactions such as cryptocurrencies and smart contracts. They can remove intermediaries and usher in a new paradigm for trust. As currently implemented, however, public ledgers cannot achieve their enormous potential.

Algorand is a quite alternative, truly democratic, and very efficient way to implement a public ledger. Unlike prior implementations based on proof of work, it requires a negligible amount of computation, and generates a transaction history that will not “fork” with overwhelmingly high probability.

May 17th, 2017: Genome Matrices and the Median Problem

A&C Seminar

Speaker: Speaker not recorded

Abstract: The genome median problem is an important problem in phylogenetic reconstruction under rearrangement models. It can be stated as follows: Given three genomes, find a fourth that minimizes the sum of the pairwise rearrangement distances between it and the three input genomes. In this paper, we model genomes as matrices and study the matrix median problem using the rank distance. It is known that, for any metric distance, at least one of the corners is a 4/3-approximation of the median. Our results allow us to compute up to three additional matrix median candidates, all of them with approximation ratios at least as good as the best corner, when the input matrices come from genomes. We also show a class of instances where our candidates are optimal. From the application point of view, it is usually more interesting to locate medians farther from the corners, and therefore, these new candidates are potentially more useful. In addition to the approximation algorithm, we suggest a heuristic to get a genome from an arbitrary square matrix. This is useful to translate the results of our median approximation algorithm back to genomes, and it has good results in our tests. To assess the relevance of our approach in the biological context, we ran simulated evolution tests and compared our solutions to those of an exact DCJ median solver. The results show that our method is capable of producing very good candidates.

May 9th, 2017: Contact Representation of Planar Graphs in 2D and 3D

A&C Seminar

Speaker: Stephen Kobourov, University of Arizona

Abstract: In a proportional contact representation of a planar graph, each vertex is represented by a simple polygon with area proportional to a given weight, and edges are represented by adjacencies between the corresponding pairs of polygons. We show how to use Schnyder realizers and canonical orders for planar graphs to obtain different types of contact representations. Specifically, we describe an algorithm that constructs proportional contact representation for arbitrary planar graphs using 10-sided rectilinear polygons. We also describe a construction with 8-sided polygons, which is optimal in terms of polygonal complexity, as 8-sided polygons are sometimes necessary. In 3D vertices are represented by polytopes and edges by contacts between the corresponding polytopes contacts. We show that planar 3-trees have contact representations with cubes and proportional contact representations with boxes.

April 26th, 2017: Simplification of a polygonal chain

A&C Seminar

Speaker: Sweta Barman

Abstract: Speaker: Sweta Barman

We will survey some results on polygonal chain simplification. Given a polygonal chain, we wish to simplify it by reducing the number of vertices while staying “close to” the original. We concentrate on the version of the problem where the vertices of the simplified chain must be a subset of the original vertices. We will describe: a good heuristic due to Douglas and Peucker that has fast implementations; an exact algorithm by Imai and Iri with improvements that achieve almost quadratic running time; and a sub-quadratic algorithm of Agarwal and Varadarajan, which, however, does not work for the standard Euclidean norm.

April 24th, 2017: Small bias requires large formulas

A&C Seminar

Speaker: Speaker not recorded

Abstract: A small-biased function is a randomized function whose distribution of truth-tables is small-biased. We show that known explicit lower bounds on the size of (1) general Boolean formulas, (2) Boolean formulas of fan-in two, (3) de Morgan formulas, as well as (4) correlation lower bounds against small de Morgan formulas apply to small-biased functions. As a consequence, any strongly explicit small-biased generator is subject to the best known explicit formula lower bounds in all these models.

On the other hand, we give a construction of a small-biased function that is tight with respect to lower bounds (1) and (2) for the relevant range of parameters. We interpret this construction as a natural-like barrier against substantially stronger lower bounds for general formulas.

April 12th, 2017: On the Planar Split Thickness of Graphs

A&C Seminar

Speaker: Speaker not recorded

Abstract: Motivated by applications in graph drawing and information visualization, we examine the planar split thickness of a graph, that is, the smallest k such that the graph is k-splittable into a planar graph. A k-split operation substitutes a vertex v by at most k new vertices such that each neighbor of v is connected to at least one of the new vertices. We first examine the planar split thickness of complete and complete bipartite graphs. We then prove that it is NP-hard to recognize graphs that are 2-splittable into a planar graph, and show that one can approximate the planar split thickness of a graph within a constant factor. If the treewidth is bounded, then we can even verify k-splittability in linear time, for a constant k. This is a joint work with the following people and published in LATIN'16 symposium: David Eppstein, Philipp Kindermann, Stephen Kobourov, Giuseppe Liotta, Anna Lubiw, Aude Maignan, Debajyoti Mondal, Sue Whitesides, Stephen Wismath.

March 23rd, 2017: The Laplacian Matrices of Graphs: Algorithms and Applications

Joint A&C Seminar and Distinguished Lecture Series [video]

Speaker: Daniel Alan Spielman, Yale University

Abstract: The Laplacian matrices of graphs arise in many fields, including machine learning, computer vision, optimization, computational science, and of course network analysis. We will explain what these matrices are and why they appear in so many applications. We then survey recent ideas that allow us to solve systems of linear equations in Laplacian matrices in nearly linear time, emphasizing the utility of graph sparsification — the approximation of a graph by a sparser one — and a recent algorithm of Kyng and Sachdeva that uses random sampling to accelerate Gaussian Elimination.

March 22nd, 2017: Undecidability and Finite Automata

A&C Seminar

Speaker: Tim Smith

Abstract: Using a novel rewriting problem, we show that two natural decision problems about finite automata are undecidable (i.e., recursively unsolvable). The problems involve cyclic shifts over a two-track input alphabet, i.e. an alphabet consisting of pairs of letters. One of the problems asks whether a finite automaton accepts any input of the form uv × vu, and the other asks about the restricted case in which v consists of repetitions of a single letter not in u. In contrast, we prove that several related problems are decidable. We also apply one result to prove the undecidability of a problem about k-automatic sets of rational numbers.

March 15th, 2017: Computational design of lace patterns

A&C Seminar

Speaker: Veronika Irvine

Abstract: Bobbin lace, created by braiding together hundreds of threads, is a fibre art form used for over 500 years as an ornament in fashion. A fundamental component of bobbin lace, called a ground, is a doubly periodic alternating braid which can be modeled as the pair (Delta(G), z(v)). Here Delta(G) is a 2-regular directed graph drawn on a torus and z(v) is a mapping from the vertices of G to a set of braid words. This model bears some resemblance to the tangle method for knot generation introduced by Conway and is heavily influenced by traditional lacemaking techniques.

In this talk, I will introduce how bobbin lace is made, discuss some interesting properties of the associated directed graphs and demonstrate how lattice paths can be used to exhaustively generate graph drawings of digraphs satisfying the criteria of workable lace. Using this approach, I have been able to generate hundreds of known ground patterns as well as millions of original patterns. To extract some of the more aesthetically pleasing patterns from this large collection, I have applied a symmetry filter and identified patterns from each of the 17 crystallographic groups.

The talk will include material done in collaboration with Frank Ruskey at the University of Victoria which appeared in an article of the Journal of Mathematics and the Arts, as well as some new ideas being explored in collaboration with Therese Biedl, Craig Kaplan and Anna Lubiw.

Photographs and additional information can be found at tesselace.com

March 8th, 2017: Maximal bifix decoding

A&C Seminar

Speaker: Francesco Dolce, UQAM

Abstract: An Arnoux-Rauzy set is the set of words obtained as the language (set of factors) of a strictly episturmian word. An Interval exchange set is the language of the natural codings of a regular interval exchange transformation. In this talk, we consider two classes of sets of words which extend in a natural way both families of Arnoux-Rauzy sets and interval exchange sets. These classes of sets consist of recurrent Tree sets and recurrent Neutral sets. Both are defined by a condition on the possible extensions of bispecial factors in the set. We prove that these classes are closed under maximal bifix decoding. We also show some results about maximal bifix codes in a Tree set connecting the theories of Words, Codes and Free Subgroups.

March 1st, 2017: Compressed Indexes in Deterministic Linear Time and Small Space

A&C Seminar

Speaker: Yakov Nekrich

Abstract: We show that the compressed suffix array and the compressed suffix tree of a string $T$ can be built in $O(n)$ deterministic time using $O(n\log\sigma)$ bits of space, where $n$ is the string length and $\sigma$ is the alphabet size. Previously described deterministic algorithms either run in time that depends on the alphabet size or need $\omega(n\log \sigma)$ bits of working space. Our result has immediate applications to other problems, such as yielding the first deterministic linear-time LZ77 and LZ78 parsing algorithms that use $O(n \log\sigma)$ bits.

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2016

November 2nd, 2016: Title not recorded

A&C Seminar

Speaker: Igor Shinkar, Berkeley

Abstract: We study the computational hardness of problems whose inputs are obtained by applying random noise to worst-case instances. For an appropriate notion of noise we show that a number of classical NP-hard problems on graphs remain essentially as hard on the noisy instances as they are in the worst-case.

Focusing on the Graph Coloring problem, we establish the following result: Given any graph G, let H be a random subgraph of G obtained by deleting the edges of G independently with probability 0.5. We show that if $\chi(G)$ is large, then $\chi(H)$ is also large with high probability. This means that the chromatic number of any graph is ``robust’’ to random edge deletions.

Joint work with Huck Bennett and Daniel Reichman.

October 19th, 2016: Title not recorded

A&C Seminar

Speaker: Benjamin Moore, SFU

Abstract: The “reconfiguration problem” for circular colourings asks, given two $(p,q)$- colourings $f$ and $g$ of a graph $G$, is it possible to transform $f$ into $g$ by changing the colour of one vertex at a time such that every intermediate mapping is a $(p, q)$-colouring? We show that this problem can be solved in polynomial time for $2 \leq p/q < 4 and is PSPACE-complete for $p/q \geq 4$. This generalizes a known dichotomy theorem for reconfiguring classical graph colourings. Using reconfiguration techniques, we also show that if $G$ contains an edge $e$ such that $G \setminus \{e\}$ contains fewer than $\frac{(k−1)!}{2} cycles of length $0 \bmod{k}, then $\chi(G) \leq k$. This is joint with Rick Brewster (Thompson Rivers University), Sean McGuinness (Thompson Rivers University), and Jonathan Noel (ETH Zurich).

October 5th, 2016: Title not recorded

A&C Seminar

Speaker: Tomoyuki Yamakami

Abstract: We discuss the computational complexity of context-free languages, concentrating on two well-known structural properties—immunity and pseudorandomness. An infinite language is REG-immune (resp., CFL-immune) if it contains no infinite subset that is a regular (resp., context-free) language. We prove that (i) there is a context-free REG-immune language outside REG/n and (ii) there is a REG-bi-immune language that can be computed deterministically using logarithmic space. We also show that (iii) there is a CFL-simple set, where a CFL-simple language is an infinite context-free language whose complement is CFL-immune. Similar to the REG-immunity, a REG-primeimmune language has no polynomially dense subsets that are also regular. We further prove that (iv) there is a context-free language that is REG/n-bi-primeimmune. Concerning pseudorandomness of context-free languages, we show that (v) CFL contains REG/n-pseudorandom languages. Finally, we prove that (vi) against REG/n, there exists an almost 1-1 pseudorandom generator computable in nondeterministic pushdown automata equipped with a write-only output tape and (vii) against REG, there is no almost 1-1 weakly pseudorandom generator computable deterministically in linear time by a single-tape Turing machine.

October 5th, 2016: Title not recorded

A&C Seminar

Speaker: Ingo van Duijn, Aarhus University

Abstract: In the Line Cover problem a set of n points (in any dimension) is given and the task is to cover the points using at most k lines. We present an algorithm that solves the problem in O*((ck/log k)^k) for a universal constant c, whereas the previous fastest known algorithm solves the problem in O*((k/1.35)^k). The main idea of our algorithm is to try to cover rich lines first (lines covering many points). By the Szemerédi-Trotter theorem incidence bound, we can bound the number of rich lines, giving a low branching in the initial phase of the algorithm. When the incidence bounds are not tight enough to guarantee good branching behaviour, we have few enough points to efficiently switch to a base case algorithm running in O*(2^n).

September 28th, 2016: Title not recorded

A&C Seminar

Speaker: Neerja Pophli, McMaster

Abstract: A Generalization of Square-free Strings

The Study of Combinatorics on Words has seen great interest in recent years. This study which is an area of discrete mathematics, primarily deals with strings, which are ordered sequences of symbols from a finite set referred to as Alphabet. It dates back to early nineteen hundreds and the pioneering work done by Axel Thue. Thue’s work is primarily on repetitions in strings (strings consisting of adjacent repeating word blocks) over a fixed finite alphabet. He was the first to show the existence of square-free strings (strings with no repeating adjacent word blocks) over a ternary alphabet by giving a substitution function, termed as iterated morphism in the area.

A natural generalization of this problem is that only subsets of the alphabet with predefined cardinality are available, while selecting symbols of the square-free string. This problem has been studied by several authors, and the lowest possible bound on the cardinality of the subset given is four. The problem remains open for subset size three. We investigate this open problem and show that square-free strings exist in several specialized cases of the problem and propose approaches to solve the problem, ranging from patterns in strings to Proof Complexity.

September 21st, 2016: Title not recorded

A&C Seminar

Speaker: Li-Yang Tan, TTIC

Abstract: We consider the fundamental derandomization problem of deterministically finding a satisfying assignment to a CNF formula that has many satisfying assignments. We give a deterministic algorithm which, given an n-variable poly(n)-clause CNF formula F that has |F^{-1}(1)| \geq \eps 2^n, runs in time

n^{\tilde{O}(\log\log n)^2}

for \eps \ge 1/\polylog(n) and outputs a satisfying assignment of F. Prior to our work the fastest known algorithm for this problem was simply to enumerate over all seeds of a pseudorandom generator for CNFs; using the best known PRGs for CNFs [DETT10], this takes time n^{\tilde{\Omega}(\log n)} even for constant \eps.

Joint work with Rocco Servedio.

May 16th, 2016: The Rise and Fall of Boolean Functions

A&C Seminar

Speaker: Clément Canonne, Columbia university

Abstract: We initiate a systematic study of a natural generalization of monotone Boolean functions to k-monotone functions, in the Property Testing model. A Boolean k-monotone function defined over a finite poset domain D alternates between the values 0 and 1 at most k times on any ascending chain in D. We study sublinear-time algorithms that distinguish functions that are k-monotone (or are close to being k-monotone) from functions that are far from being k-monotone.

The study of k-monotone Boolean functions is well-motivated in the context of circuit complexity where it relates to the notion of negation complexity. Circuits with even few negation gates are notoriously hard to analyze and are the subject of intense interest in cryptography and complexity theory. Our work is spurred by the recent results of Blais et al. (RANDOM 2015) showing exponential lower bounds in the PAC learning model. However, in the property testing model, monotone functions can be tested efficiently, and tight bounds on the complexity of this task have by now been established in many settings.

We extend the study of monotonicity to k-monotonicity, on grids [n]^d and hypercube {0,1}^n domains, and discover some intriguing behavior and surprising connections. Our tolerant testers for functions f: [n]^d->{0,1} resolve a problem left open by Berman et al. (STOC 2014). For the special case d=1 we show that even n^{1-a}-monotonicity can be 2-sided tested with a constant number of queries, for any a > 0.

Our techniques exploit the testing-by-learning paradigm, use novel applications of Fourier analysis on the grid [n]^d, and draw connections to distribution testing techniques.

Joint work with Elena Grigorescu (Purdue University), Siyao Guo (Courant Institute), Akash Kumar (Purdue University), and Karl Wimmer (Duquesne University).

May 11th, 2016: Low-cost electrical monitoring: power domination in triangulations

A&C Seminar

Speaker: Claire Pennarun

Abstract: Power domination in graphs emerged from the problem of monitoring an electrical system by placing as few measurement devices in the system as possible. A vertex is monitored by a set of vertices $S$ if: $v$ is in $S$ or has a neighbor in $S$, or one monitored neighbor $u$ of $v$ has all its neighbors monitored except for $v$. Given a graph $G$, a set $S \in V(G)$ is said to be power dominating $G$ if all vertices of $G$ are monitored by it. The goal is then to find a power dominating set $S$ of minimum size for a given graph. I will present an algorithm producing a power dominating set of order at most $\dfrac{n-2}{4}$ for any maximal planar graph of order $n \geq 6$.

May 11th, 2016: Optimal Path-Decomposition of Tries

Thesis Presentation

Speaker: Alexandre Daigle

Abstract: In this thesis, we consider the path-decomposition representation of prefix trees. We show that given query probabilities for every word in the prefix tree, the heavy-path strategy produces the optimal trie with respect to the number of node accesses. We show how to implement the heavy-path strategy in O(N) time for a trie containing n words with total length N. To prove this result, we show a complete characterization of the choices made by the optimal decomposition strategy. Using this characterization, we describe how to efficiently support dynamic operations on the path-decomposed trie while preserving the optimality in O(sigma * log(n)) time for an alphabet of size sigma. We also give entropy-based bounds of the node accesses per query for their respective probabilities. Finally, we show theoretical and experimental results on the performance of heavy-path versus max-score, another popular path-decomposition strategy.

May 10th, 2016: Title not recorded

Thesis Presentation

Speaker: Zachary Frenette

Abstract: We examine the problem of representing integers modulo $L$ so that both increment and decrement operations can be performed efficiently. This problem is studied in the bitprobe model, where the complexity of the underlying problem is measured by the number of bit operations performed on the data structure. In this thesis, we will primarily be interested in constructing space-optimal data structures. That is, we would like to use exactly $n$ bits to represent integers modulo $2^n$. Brodal et al. gave such a data structure, which requires $n-1$ bit reads and $3$ bit writes, in the worst-case, to perform increment and decrement operations. We provide several improvements to their data structure. First, we give a data structure that requires $n-1$ bit reads and $2$ bit writes, in the worst-case, to perform increment and decrement operations. Then, we refine this result to obtain a data structure that requires $n-1$ bit reads and a single bit write to perform both operations. This disproves the conjecture that, when a space-optimal data structure uses only $1$ bit write to perform these operations, then every bit in the data structure must be inspected in the worst-case.

May 9th, 2016: Title not recorded

Thesis Presentation

Speaker: Oliver Grant

Abstract: We examine optimal and near optimal solutions to the classic binary search tree problem of Knuth. First, we re-examine a solution of G uttler, Melhorn and Schneider which was shown to have a worst case bound of c * H + 2 where c >= 1/(H(1/3, 2/3)) =~ 1.08. We give an improved worst case bound on the heuristic of H + 4. Next, we examine the optimum BST problem under a model of external memory. We use the Hierarchical Memory Model (HMM) of Aggarwal et al. and propose two approximate solutions which run in O(n) time where n is the number of words in our data set. Using these methods, we improve upon a bound given in Thite’s 2001 thesis under the related HMM2 model in the approximate setting. We also examine the related problem of binary trees on multisets of probabilities. We provide a simple O(n lg(n)) algorithm that is within n+1/(2n) of optimal on a multiset of n keys.

May 4th, 2016: Universal Locally Testable Codes

A&C Seminar

Speaker: Tom Gur, Weizmann Institute

Abstract: We initiate a study of ``universal locally testable codes" (ULTCs). These codes admit local tests for membership in numerous possible subcodes, allowing for testing properties of the encoded message. More precisely, a ULTC C:{0,1}^k \to {0,1}^n for a family of functions F = { f_i : {0,1}^k \to {0,1} }_{i \in [M]} is a code such that for every i in [M] the subcode ${ C(x) : f_i(x) = 1 \}$ is locally testable. We show a ``canonical" O(1)-local ULTC of length \tilde{O}(Ms) for any family F of M functions such that every f in F can be computed by a circuit of size s, and establish a lower bound of the form n=M^{1/O(k)}, which can be strengthened to n=M^{\Omega(1)} for any F such that every f,f’ in F disagree on a constant fraction of their domain.

We also consider a variant of ULTCs wherein the testing procedures are also given free access to a short proof, akin the MAPs of Gur and Rothblum (ITCS 2015). We call such codes ``universal locally verifiable codes" (ULVCs). We show ULVCs of length \tilde{O}(n^2)$ for t-ary constraint satisfaction problems (t-CSP) over k variables, with proof length and query complexity \tilde{O}(n^{2/3}), where t=O(1) and n\ge k is the number of constraints in the CSP instance. In addition, we prove a lower bound of pq = \tilde\Omega(k) for every polynomial length ULVC for CSPs (over k variables) having proof complexity p and query complexity q.

Lastly, we give an application for interactive proofs of proximity (IPP), introduced by Rothblum et al. (STOC 2013), which are interactive proof systems wherein the verifier queries only a sublinear number of input bits and soundness only means that, with high probability, the input is close to an accepting input. We show that using a small amount of interaction, our ULVC for CSP can be, in a sense, ``emulated" by an IPP, yielding a 3-round IPP for CSP with sublinear communication and query complexity.

Joint work with Oded Goldreich

April 20th, 2016: Periodicity in Rectangular Arrays

A&C Seminar

Speaker: Taylor Jonathan Smith

Abstract: We say that a nonempty word is primitive if it cannot be written as the power of some other word; that is, a nonempty word z is primitive if it cannot be written in the form z = w^e for some word w and some integer e >= 2. In this talk, we extend the notion of primitivity to two-dimensional words. We discuss a two-dimensional generalization of the familiar Lyndon-Schutzenberger theorem for words, present a method to enumerate the number of primitive words of size m x n, and present a linear time algorithm to test primitivity and to compute the primitive root of a two-dimensional word. This is a joint work with Jeffrey Shallit.

February 10th, 2016: Title not recorded

A&C Seminar

Speaker: Hamideh Vosoughpour Yazdchi

Abstract: “Morphing” is the standard term used since the 80s to address the continuous transformation between shapes while some properties are maintained all the time during this transformation. Morphing is used in computer graphics and animation, and also includes reconfiguration problems such as linkage unfolding, and morphing between two different graph drawings or triangulations. In this talk I will review some of these morphing problems and will focus on “straight-line morphing”. Straight-line morphing is a continuous transformation of a planar graph drawing from an initial straight-line drawing to a final straight-line drawing in which planarity is preserved, and each vertex moves on the straight-line segment connecting its initial position to its final position without backtracking. There is freedom to move the vertices at different speeds along their straight-line segments. We will give some algorithms and some NP-hardness results on straight-line morphing of disjoint segments.

This is a joint work with Anna Lubiw.

February 3rd, 2016: Title not recorded

A&C Seminar

Speaker: Amit Levi

Abstract: We consider the problem of estimating the number of triangles in a graph. This problem has been extensively studied in both theory and practice, but all existing algorithms read the entire graph. In this work we design a {\em sublinear-time\/} algorithm for approximating the number of triangles in a graph, where the algorithm is given query access to the graph. The allowed queries are degree queries, vertex-pair queries and neighbor queries. We show that for any given approximation parameter $0<\epsilon<1$, the algorithm provides an estimate $\widehat{\tr}$ such that with high constant probability, $(1-\epsilon)\cdot \tr< \widehat{\tr}<(1+\epsilon)\cdot \tr$, where $t$ is the number of triangles in the graph $G$. The expected query complexity of the algorithm is $\!\left(\frac{n}{\tr^{1/3}} + \min\left\{m, \frac{m^{3/2}}{\tr}\right\}\right)\cdot {\rm poly}(\log n, 1/\epsilon)$, where $n$ is the number of vertices in the graph and $m$ is the number of edges, and the expected running time is $\!\left(\frac{n}{\tr^{1/3}} + \frac{m^{3/2}}{\tr}\right)\cdot {\rm poly}(\log n, 1/\epsilon)$. We also prove that $\Omega\!\left(\frac{n}{\tr^{1/3}} + \min\left\{m, \frac{m^{3/2}}{\tr}\right\}\right)$ queries are necessary, thus establishing that the query complexity of this algorithm is optimal up to polylogarithmic factors in $n$ (and the dependence on $1/\epsilon$).

Joint work with Talya Eden, Dana Ron and C. Seshadhri.

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2015

December 9th, 2015: Title not recorded

A&C Seminar

Speaker: Martin Derka

Abstract: List Colouring and Partial List Colouring of Graphs On-Line

We investigate the problem of graph list colouring in the on-line setting. We provide several results on paintability of graphs in the model introduced by Schauz and Zhu in 2009. We prove that the on-line version of Ohba’s conjecture is true for the class of planar graphs. We show that the conjecture for partial list colouring brought to the on-line setting holds for several graph classes, namely claw-free graphs, maximal planar graphs, series-parallel graphs, and chordal graphs. We propose and analyse other possible on-line models for the list colouring problem.

December 2nd, 2015: Title not recorded

A&C Seminar

Speaker: Hicham El-Zein

Abstract: We investigate the problem of succinctly representing an arbitrary unlabeled permutation, so that power queries can be computed quickly. We consider the problem in several scenarios:

- Labeling schemes where we assign labels to elements and the query is to be answered by just examining the labels of the queried elements: we show that a label space of: sum (i=1 to n) of (floor(n/i).i) is necessary and sufficient. In other words, 2lg(n) bits of space are necessary and sufficient for representing each of the labels.

- Succinct data structures for the problem where we assign labels to the n elements from the label set {1,…,cn} where c >= 1: we show that O(n^(1/2)) bits are necessary and sufficient to represent the permutation. Moreover, we support queries in such a structure in O(1) time in the standard word-RAM model.

- Succinct data structures for the problem where we assign labels to the n elements from the label set {1,…,cn^1+epsilon} where c is a constant and 0 < epsilon < 1: we show that O(n^(1-epsilon)/2) bits are necessary and sufficient to represent the permutation. We can also support queries in such a structure in O(1) time in the standard word-RAM model.

October 21st, 2015: Title not recorded

A&C Seminar

Speaker: Kevin Chen, Rutgers

Abstract: I will present novel spectral algorithms for learning tree-structured hidden Markov models that have provable PAC-style guarantees on the sample complexity. Our algorithm fits into a general tensor decomposition framework for learning graphical models that was recently developed in the theoretical machine learning community. This approach has been applied to other probabilistic models such as mixtures of Gaussians, latent tree graphical models, etc. and is very different from traditional local search techniques such as the Expectation-Maximization algorithm for maximum likelihood estimation. Our algorithm development is inspired by the emergence of several large international consortia in biology that have recently released massive amounts of epigenetics data - chemical modifications of the human genome that specify when and where genes are switched on and off. I will describe experiments on these biological data sets and show that our theoretical work improves on previous methods from the bioinformatics community. Joint work with Kamalika Chaudhuri, Chicheng Zhang and Jimin Song

October 13th, 2015: Automatic Sequences and Curves over Finite Fields

A&C Seminar

Speaker: Andrew Bridy, University of Rochester

Abstract: An amazing theorem of Christol states that a power series y with coefficients in a finite field is an algebraic function if and only if its coefficient sequence can be produced by a finite automaton. It is then possible to represent the automaton by a family of differential operators acting on a curve whose function field contains y. I study this connection in detail and show how it can be used to draw a precise link between the complexity of the automaton and algebraic invariants of the function y, such as its degree and height.

September 30th, 2015: Three Geometric Optimization Problems on Orthogonal Polygons

A&C Seminar

Speaker: Saeed Mehrabi

Abstract: In this talk, we present some recent results on a variant of the art gallery problem in which sliding cameras are used to guard the polygon. We consider two versions of this problem on an orthogonal polygon P: the Minimum-Cardinality Sliding Cameras (MCSC) problem in which we want to guard P with the minimum number of sliding cameras, and the Minimum-Length Sliding Cameras (MLSC) problem in which the goal is to compute a set S of sliding cameras for guarding P so as to minimize the total length of trajectories along which the cameras in S travel. We answer questions posed by Katz and Morgenstern (2011) by presenting the following results: we first show that the MLSC problem is solvable in polynomial time even for orthogonal polygons with holes. Next, we show that the MCSC problem is NP-complete when P is allowed to have holes. Finally, an O(n)-time exact algorithm for the MCSC problem on monotone polygons is discussed. We conclude the talk by a discussion on open problems.

September 22nd, 2015: Intersection Non-Emptiness for Tree Shaped Finite Automata

A&C Seminar

Speaker: Michael Wehar, SUNY Buffalo

Abstract: We consider the problem of deciding whether a finite list of regular languages has a non-empty intersection. That is, given a finite list of DFA’s (deterministic finite automata), does there exist a string that satisfies all of the DFA’s? This problem is in PSPACE rather than NP because the shortest satisfying string may have exponential length. In fact, this problem is PSPACE-complete. Classical time and space complexity lower bounds for natural problems are hard to come by. Surprisingly, the intersection non-emptiness problem has both classical time and space complexity lower bounds. When restricted to k-many DFA’s, the intersection nonemptiness problem cannot be solved using less than klog(n) bits of memory. If, in addition to the k-many DFA’s, we allow for one pushdown automaton, then the problem cannot be solved in less than n^k time. Recent developments suggest that the intersection non-emptiness problem for tree shaped DFA’s may fundamentally be connected with limited non-determinism. In particular, for k-many tree shaped DFA’s, the problem is solvable efficiently with limited nondeterminism. However, if there exists an efficient deterministic algorithm, then the exponential time hypothesis is false.

September 21st, 2015: Global Information in Distributed Computing

A&C Seminar

Speaker: Avery Miller, TAU

Abstract: In distributed computing, many nodes/processors/agents need to cooperate in order to solve some task. In order to do so efficiently, it is often the case that each node is initially provided some “local” information about itself, such as its identifier or degree, and provided some “global” information about the entire system, such as network diameter or size. Ideally, the amount of global information needed by an algorithm should be minimized, since it might be costly or impossible to provide accurate information about the entire system each time that the algorithm is executed. In this talk, I will discuss how the “algorithm with advice” framework is used to prove upper and lower bounds on the amount of global information sufficient to solve distributed tasks, and present some recent results about tasks such as leader election and rendezvous.

July 15th, 2015: A unified approach to linear probing hashing

A&C Seminar

Speaker: Alfredo Viola

Abstract: We give a unified analysis of linear probing hashing with a general bucket size. We use both a combinatorial approach, giving exact formulas for generating functions, and a probabilistic approach, giving simple derivations of asymptotic results. Both approaches complement nicely, and give a good insight in the relation between linear probing and random walks. The Poisson Transform links in a natural way both approaches. A key methodological contribution, at the core of Analytic Combinatorics, is the use of the symbolic method (based on q-calculus) to directly derive the generating functions to analyze.

This is a joint work with Svante Janson.

June 17th, 2015: Towards the solution of the Cerný conjecture

A&C Seminar

Speaker: Marek Szykula, University of Wroclaw

Abstract: he talk is an overview of some recent results concerning synchronizing automata and the Cerný conjecture. This is based on my joint work with Mikhail Berlinkov and Andrzej Kisielewicz. The Cerný conjecture is one of most longstanding open problems in the combinatorial theory of finite automata. It states that every n-state synchronizing automaton has a reset word of length at most (n - 1)^2. We discuss two commonly used approaches for finding upper bounds on the length of the shortest reset words (reset thresholds). These are the so-called compression and extension methods. We exhibit a certain series of strongly connected synchronizing automata with subsets whose shortest extending words are of length \theta(n^2), where n is the number of states of the automaton. This disproves a conjecture that every subset in such an automaton is cn-extendable, for some constant c, and in particular, shows that the cubic upper bound on the length of the shortest reset words cannot be improved generally by means of the extension method. Some relaxation of this conjecture is proposed. By joining the two approaches, and using Markow chains and some linear algebra, we obtain new upper bounds for the reset thresholds of automata with a short word of a small ranks. The bounds are applied to make several improvements in the area. In particular, we improve the best general upper bound for reset thresholds of finite prefix codes (Huffman codes): we show that an n-state synchronizing decoder has a reset word of length at most O(n log^3n). Also we prove the Cerný conjecture for n-state automata with a letter of rank at most sqrt3(6n - 6). In another corollary, based on the recent results of Cyril Nicaud, we show that the probability that the Cerný conjecture does not hold for a random synchronizing binary automaton is exponentially small in terms of the number of states, and the expected value of the reset threshold of an n-state random synchronizing binary automaton is at most n^(7/4+o(1)). Moreover, reset words of the lengths within our bounds are computable in polynomial time. We present suitable algorithms for this task for various classes of automata for which our results can be applied.

May 13th, 2015: 1-String VPG Representations of Planar Graphs and their Subclasses

A&C Seminar

Speaker: Martin Derka

April 22nd, 2015: Title not recorded

A&C Seminar

Speaker: Alex Lopez-Ortiz

Abstract: We consider the problem of discovering overlapping communities in social networks. This can be modeled as generalizations of the Set and Graph Packing problems with overlap. More formally we seek a sub-collection S consisting of at least k sets subject to certain disjointness restrictions. In the r-Set Packing with t-Membership, each element belongs to at most t sets of S while in r-Set Packing with t-Overlap each pair of sets in S overlaps in at most t elements.

Similarly, both of our graph packing problems seek a sub-collection of at least k subgraphs in a graph G each isomorphic to one of several graphs H of at most r vertices. In H-Packing with t-Membership, each vertex of G belongs to at most t subgraphs in the sub-collection while in H-Packing with t-Overlap each pair of subgraphs overlaps in at most t vertices.

Here, we show NP-Completeness results for all of our packing problems. Furthermore, we give a dichotomy result for H-Packing with t-Membership. Given this intractability, we reduce r-Set Packing with t-Membership and t-Overlap to problem kernels with O((r+1)^r k^{r}) and O(r^r k^{r-t-1}) elements, respectively. Similarly, we reduce H-Packing with t-Membership and t-Overlap to instances with O((r+1)^r k^{r}) and O(r^r k^{r-t-1}) vertices, respectively. In all cases, k is the input parameter while t and r are constants.

This is joint work with Jazmin Romero (Waterloo) and Henning Fernau (Trier, Germany).

February 12th, 2015: Title not recorded

A&C Seminar

Speaker: Yakov Nekrich

Abstract: In this talk we give an overview of problems related to storage and analysis of biological sequence data. Due to the arrival of the next-generation sequencing hardware, the amount of biological data to be stored and processed has increased dramatically. Therefore we need methods to process and analyze very large volumes of data that is kept in compressed form.

We focus on the problem of maintaining a sequence or a collection of sequences so that pattern matching queries and its variants can be answered efficiently. The most basic problem of this kind is to report all occurrences of a string P in a sequence S. We also consider more advanced problems, such as identifying sequences that contain a string and identifying locations in a sequence that approximately match the query pattern.

February 4th, 2015: Reconfiguration on sparse graphs

A&C Seminar

Speaker: Amer Mouawad

Abstract: A vertex-subset graph problem Q defines which subsets of the vertices of an input graph G are feasible solutions. The reconfiguration version of a vertex-subset problem asks, given two feasible solutions S and T of size k, whether it is possible to transform S into T by a sequence of vertex additions and deletions such that each intermediate set is also a feasible solution of size bounded by k. In this talk, we consider reconfiguration variants of two classical vertex-subset problems, namely Independent Set Reconfiguration (ISR) and Dominating Set Reconfiguration (DSR). ISR is one of the most studied reconfiguration problems and is known to be PSPACE-complete on planar graphs and graphs of bounded degree. On the positive side, the problem is known to be solvable in O(f(k) n^{O(1)}) time on planar graphs and in O(g(k,d) n^{O(1)}) on graphs of degree at most d, for some computable functions f and g and integer constant d. In other words, the problem is fixed-parameter tractable parameterized by k. We push this boundary even further and show that ISR remains fixed-parameter tractable parameterized by k when the input graph is of bounded degeneracy or nowhere-dense, answering positively an open question concerning the parameterized complexity of the problem on graphs of bounded treewidth (i.e we prove the existence of an algorithm solving the problem in O(h(k,tw) n^{O(1)}), where tw denotes the treewidth of the input graph G and h is some computable function). For DSR, we show the problem is fixed-parameter tractable parameterized by k when the input graph does not contain large bicliques, a class of graphs which includes graphs of bounded degeneracy and nowhere-dense graphs.

Joint work with Daniel Lokshtanov, Fahad Panolan, M.S. Ramanujan, and Saket Saurabh.

January 28th, 2015: Vertex cover reconfiguration and beyond

A&C Seminar

Speaker: Amer Mouawad

Abstract: Under the reconfiguration framework, instead of finding a feasible solution to some instance I of a search problem Q, we are interested in structural and algorithmic questions related to the solution space of Q. Naturally, given some adjacency relation A defined over feasible solutions of Q, the solution space can be represented using a graph R(I). R(I) contains one vertex for each feasible solution of Q on instance I and two vertices share an edge whenever their corresponding solutions are adjacent under A. An edge in R(I) corresponds to a reconfiguration step, a walk in R(I) is a sequence of such steps, a reconfiguration sequence, and R(I) is a reconfiguration graph. Studying problems related to reconfiguration graphs has received considerable attention in recent literature, the most popular problem being to determine whether there exists a reconfiguration sequence between two given feasible solution. In this talk, we consider the Vertex Cover Reconfiguration (VCR) problem, where given graph G, positive integers k and ell, and two vertex covers S and T of G of size at most k, the goal is to determine whether S can be transformed into T by a sequence of at most ell vertex additions or removals such that each operation results in a vertex cover of size at most k. Motivated by recent results establishing the hardness of VCR under both classical and parameterized complexity assumptions, we delineate the complexity of the problem restricted to various graph classes. In particular, we give an overview of some of the tools and techniques used to prove negative (i.e. hardness) and positive results (i.e. polynomial-time or fixed-parameter tractable algorithms) for VCR and other closely related reconfiguration problems.

Joint work with Naomi Nishimura, Venkatesh Raman, and Marcin Wrochna.

January 21st, 2015: Efficient Online Strategies for Renting Servers in the Cloud

A&C Seminar

Speaker: Shahin Kamali

Abstract: This talk is about a recent application of the online bin packing problem in the Cloud systems. In certain applications of these systems, e.g., in online gaming, we often deal with jobs that arrive and depart in an online manner. Upon its arrival, a job should be assigned to a server. Each job has a size which defines the amount of resources that it needs. Servers have uniform capacity and, at all times, the total size of jobs assigned to a server should not exceed the capacity. This setting is closely related to the classic bin packing problem. The difference is that, in bin packing, the objective is to minimize the total number of used servers. In the Cloud, however, the charge for each server is proportional to the length of the time interval it is rented for, and the goal is to minimize the cost involved in renting all used servers.

Recently, certain bin packing strategies were considered for renting servers in the Cloud [Li et al. SPAA'14]. There, it is proved that all Any-Fit bin packing strategy has a competitive ratio of at least μ, where μ is the max/min interval length ratio of jobs. It is also shown that First Fit has a competitive ratio of 2μ+13 while Best Fit is not competitive at all. We observe that the lower bound of μ extends to all online algorithms. We also prove that, surprisingly, Next Fit algorithm has competitive ratio of at most 2μ+1. We also show that a variant of Next Fit achieves a competitive ratio of K × max{1,μ/(K−1)} +1 where K is a parameter of the algorithm. In particular, if the value of μ is known, the algorithm has a competitive ratio of μ+2; this improves upon the existing upper bound of μ+8. Finally, we introduce a simple algorithm called Move To Front (MTF) which has a competitive ratio of at most 6μ+7 and also promising average-case performance. We experimentally study the average-case performance of different algorithms and observe that the typical behavior of MTF is distinctively better than other algorithms.

The talk starts with an overview of the classic bin packing problem and continues with a slow review of the above-mentioned results.

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2014

December 9th, 2014: On balanced holes in bicolored point sets

A&C Seminar

Speaker: Jorge Urrutia, Universidad Nacional Autónoma de México

Abstract: Let P be a bicolored set of points in the plane, that is each element of P is coloured red or blue. A k-hole is a simple polygon (not necessarily convex) with k vertices such that its vertices are elements of P, and it contains no points of P in its interior. A 2k-hole is balanced if it contains the same number of red and blue vertices. In this talk we prove that if P has n blue, and n red points, then it contains a quadratic number of balanced 4-holes. We will review other problems on holes of point sets.

December 8th, 2014: Reconfiguring Triangulations

A&C Seminar

Speaker: Vinayak Pathak

Abstract: The results in this thesis lie at the confluence of triangulations and reconfiguration. We make the observation that certain solved and unsolved problems about triangulations can be cast as reconfiguration problems. We then solve some reconfiguration problems that provide us new insights about triangulations. Following are the main contributions of this thesis:

1. We show that computing the flip distance between two triangulations of a point set is NP-complete. A flip is an operation that changes one triangulation into another by replacing one diagonal of a convex quadrilateral by the other diagonal. The flip distance, then, is the smallest number of flips needed to transform one triangulation into another. For the special case when the points are in convex position, the problem of computing the flip distance is a long-standing open problem.

2. Inspired by the problem of computing the flip distance, we start an investigation into computing shortest reconfiguration paths in reconfiguration graphs. We consider the reconfiguration graph of satisfying assignments of Boolean formulas where there is a node for each satisfying assignment of a formula and an edge whenever one assignment can be changed to another by changing the value of exactly one variable from 0 to 1 or from 1 to 0. We show that computing the shortest path between two satisfying assignments in the reconfiguration graph is either in P, NP-complete, or PSPACE-complete depending on the class the Boolean formula lies in.

3. We initiate the study of labelled reconfiguration. For the case of triangulations, we assign a unique label to each edge of the triangulation and a flip of an edge from e to e’ assigns the same label to e’ as e. We show that adding labels may make the reconfiguration graph disconnected. We also show that the worst-case reconfiguration distance changes when we assign labels. We show tight bounds on the worst case reconfiguration distance for edge-labelled triangulations of a convex polygon and of a spiral polygon, and edge-labelled spanning trees of a graph. We generalize the result on spanning trees to labelled bases of a matroid and show non-trivial upper bounds on the reconfiguration distance.

December 3rd, 2014: On the Succinct Representation of Equivalence Classes

A&C Seminar

Speaker: Hicham El-Zein

Abstract: Given a set of n elements that are partitioned into equivalence classes, we study the problem of assigning unique labels to these elements in order to support the query that asks whether the elements corresponding to two given labels belong to the same equivalence class. This problem has been studied by Katz et al, Alstrup et al, and Lewenstein et al. Lewenstein et al showed that if the labels were to be assigned from the set {1,…,n}, a data structure of size $\Theta(\sqrt n)$ bits is necessary and sufficient to represent the equivalence classes. They also showed that with no auxiliary data structure, a label space of size $\sum_{i=1}^{n}\lfloor\frac{n}{i}\rfloor$ is necessary and sufficient. Our main result is that if we allow a label space of size cn for any constant c > 1, a data structure of size $\Theta(\log n)$ bits is necessary and sufficient. We also show that the equivalence query in such a data structure can be answered in $\Theta(1)$ time. We believe that our work can trigger further work on tradeoffs between label space and auxiliary data structure space for other labeling problems.

November 20th, 2014: Title not recorded

A&C Seminar

Speaker: Or Sheffet, Harvard

Abstract: The problem of analyzing the effect of privacy concerns on the behavior of selfish utility-maximizing agents has received much attention lately. Privacy concerns are often modeled by altering the utility functions of agents to consider also their privacy loss. Such privacy aware agents prefer to take a randomized strategy even in very simple games in which non-privacy aware agents play pure strategies. In some cases, the behavior of privacy aware agents follows the framework of Randomized Response, a well-known mechanism that preserves differential privacy.

Our work is aimed at better understanding the behavior of agents in settings where their privacy concerns are explicitly given. We consider a toy setting where agent A, in an attempt to discover the secret type of agent B, offers B a gift that one type of B agent likes and the other type dislikes. As opposed to previous works, B’s incentive to keep her type a secret isn’t the result of “hardwiring” B’s utility function to consider privacy, but rather takes the form of a payment between B and A. We investigate three different types of payment functions and analyze B’s behavior in each of the resulting games. As we show, under some payments, B’s behavior is very different than the behavior of agents with hardwired privacy concerns and might even be deterministic. Under a different payment we show that B’s BNE strategy does fall into the framework of Randomized Response.

Joint work with Yiling Chen and Salil Vadhan. We assume no prior knowledge from the audience.

November 19th, 2014: Title not recorded

A&C Seminar

Speaker: Guangyu Feng

Abstract: A central task of computational linguistics is to decide if two pieces of texts have similar meanings. Ideally, this depends on an intuitive notion of semantic distance. While this semantic distance is most likely undefinable and uncomputable, in practice it is approximated heuristically, consciously or unconsciously. In this talk, we present a theory, and its implementation, to approximate the elusive semantic distance by the well-defined information distance. It is mathematically proven that any computable approximation of the intuitive concept of semantic distance is ``covered’’ by our theory. We have implemented our theory to question answering (QA) and performed large scale experiments on 35 million question-answer pairs. Experiments demonstrate that our initial implementation of the theory produces convincingly fewer errors in classification compared to other academic models and commercial systems.

November 12th, 2014: Title not recorded

A&C Seminar

Speaker: Zahed Rahmati, U. Victoria

Abstract: This is a talk on simple approaches for obtaining faster kinetic solutions for some fundamental problems in computational geometry and graph drawing (from my own research): the closest pair, all k-nearest neighbors, reverse k-nearest neighbors, Euclidean minimum spanning tree, and point set embedding problems.

Here is problem statement: Given a set of moving points, one wants to answer the queries (e.g. what is the closest pair?) at any time quickly without solving problems from scratch. We use the standard kinetic data structures framework to continuously maintain/track some properties of moving points, which in fact will able us to answer the queries efficiently.

October 22nd, 2014: Title not recorded

A&C Seminar

Speaker: Therese Biedl

Abstract: One of the first algorithmic results in graph drawing was how to find a planar straight-line drawing such that vertices are at grid-points with polynomial coordinates. But not until 2007 was it proved that finding such a grid-drawing with optimal area is NP-hard, and the result was only for disconnected graphs.

In this talk, I will show that for graphs with bounded treewidth one can find area-optimal planar straight-line drawings in one of the following two scenarios: (1) when faces have bounded degree and the planar embedding is fixed, or (2) when we want all faces to be drawn convex. I’ll also give NP-hardness results to show that none of these restrictions can be dropped. In particular, finding area-minimal drawings is NP-hard for triangulated graphs minus one edge.

October 8th, 2014: Speeding Up Graph Algorithms via Switching Classes

A&C Seminar

Speaker: Nathan Lindzey

Abstract: Given a graph G, a vertex switch of v results in a new graph where neighbors of v become nonneighbors and vice versa. This operation gives rise to an equivalence relation over the set of labeled digraphs on n vertices. The equivalence class of G with respect to the switching operation is commonly referred to as G’s switching class. The algebraic and combinatorial properties of switching classes have been studied in depth; however, they have not been studied as thoroughly from an algorithmic point of view. The intent of this work is to further investigate the algorithmic properties of switching classes. In particular, we show that switching classes can be used to asymptotically speed up several super-linear unweighted graph algorithms. The current techniques for speeding up graph algorithms are all somewhat involved insofar that they employ sophisticated pre-processing, data-structures, or use “word tricks” on the RAM model to achieve at most a O(log(n)) speed up for sufficiently dense graphs. Our methods are much simpler and can result in super-polylogarithmic speedups. In particular, we achieve better bounds for diameter, transitive closure, bipartite maximum matching, and general maximum matching.

Link to the paper: http://arxiv.org/abs/1408.4900

September 17th, 2014: Title not recorded

A&C Seminar

Speaker: Vinayak Pathak

Abstract: The problem of computing the minimum number of flips to transform one triangulation of a convex polygon to another is not known to be in P or NP-complete. A flip sequence determines a one-to-one correspondence between the edges of the two triangulations. As a step towards understanding the source of difficulty, we investigate the case when this edge correspondence is given, i.e., we want the flip distance between two edge-labelled triangulations of a convex polygon. We provide bounds on the number of flips required in the worst case and show some connections with sorting permutations.

Joint work with: Prosenjit Bose, Anna Lubiw, Sander Verdonschot

September 10th, 2014: Title not recorded

A&C Seminar

Speaker: Shahin Kamali

Abstract: In this thesis we introduce and evaluate new algorithms and models for the analysis of online bin packing and list update problems. These are two classical online problems which are extensively studied in the literature and have many applications in the real world. Similar to other online problems, the framework of competitive analysis is often used to study the list update and bin packing algorithms. Under this framework, the behaviour of online algorithms is compared to an optimal offline algorithm on the worst possible input. This is aligned with the traditional algorithm theory built around the concept of worst-case analysis. However, the pessimistic nature of the competitive analysis along with unrealistic assumptions behind the proposed models for the problems often result in situations where the existing theory is not quite useful in practice. The main goal of this thesis is to develop new approaches for studying online problems, and in particular bin packing and list update, to guide development of practical algorithms performing quite well on real-world inputs. In doing so, we introduce new algorithms with good performance (not only under the competitive analysis) as well as new models which are more realistic for certain applications of the studied problems.

For many online problems, competitive analysis fails to provide a theoretical justification for observations made in practice. This is partially because, as a worst-case analysis method, competitive analysis does not necessarily reflect the typical behaviour of algorithms. In the case of bin packing problem, the classical Best Fit and First Fit algorithms are widely used in practice. There are, however, other algorithms with better competitive ratios which are rarely used in practice since they perform poorly on average. We show that it is possible to optimize for both cases. In doing so, we introduce online bin packing algorithms which outperform Best Fit and First Fit in terms of competitive ratio while maintaining their good average-case performance.

An alternative for analysis of online problems is the advice model which has received significant attention in the past few years. Under the advice model, an online algorithm receives a number of bits of advice about the unrevealed parts of the sequence. Generally, there is a trade-off between the size of the advice and the performance of online algorithms. The advice model generalizes the existing frameworks in which an online algorithm has partial knowledge about the input sequence, e.g., the access graph model for the paging problem. We study list update and bin packing problems under the advice model and answer several relevant questions about the advice complexity of these problems.

Classical online problems are usually studied under specific settings which are not necessarily valid for all applications of the problem. As an example, online bin packing algorithms are widely used for server consolidation to minimize the number of active servers in a data center. In some applications, e.g., tenant placement in the Cloud, often a ‘fault-tolerant’ solution for server consolidation is required. In this setting, the problem becomes different and the classical algorithms cannot be used any further. We study a fault-tolerant model for the bin packing problem and analyse algorithms which fit this particular application of the problem. Similarly, the list update problem was initially proposed for maintaining self-adjusting linked lists. However, presently, the main application of the problem is in the data compression realm. We show that the standard cost model is not suitable for compression purposes and study a compression cost model for the list update problem. Our analysis justifies the advantage of the compression schemes which are based on Move-To-Front algorithm and might lead to improved compression algorithms.

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2013

February 27th, 2013: I/O-Efficient Shared-Constrained Range Reporting

A&C Seminar

Speaker: Sharma Thankachan

Abstract: Many database applications benefit from the indexes which answer range queries in two or more dimensions. We consider the cases of range queries which seem to have 4 constraints but at least two of them are shared. It is known that if an orthogonal range query has 4 or more constraints, then one cannot hope for I/O efficient index which achieves linear or near linear space. However, we consider cases where 2 of these 4 constraints are shared. We take the following as our central problem: given a set of n weighted points in 2D, we consider the Q(a, b, k ), where the task is to report all those points with weight >= k within the box [a, b]x(-\infty, a). Even though there are four constraints to the range query problem, the number of independent constraints is only three. In other words, one constraint is shared. We call this as Shared-Constrained Range Reporting (SCRR) problem. We derive an O(n log log n) word space data structure with optimal query I/Os for solving SCRR. We show that SCRR solution can be used for other orthogonal range searching and interval problems where query has only 3 independent constraints. Our approach leads to direct application for top-k color queries and document retrieval problems.

February 20th, 2013: Space Efficient Construction of Wavelet Trees and Matrices

A&C Seminar

Speaker: Francisco Claude

Abstract: Wavelet trees [Grossi, Gupta and Vitter, SODA 2003] are one of the main building blocks in many space-efficient data structures.

In this talk, we present new techniques for constructing this structure in little space. This represents a step forward into practical space-efficient data structures, allowing construction for massive data sets.

The techniques presented rely on the machinery for in-place sorting. By making use of the specific properties of wavelet trees, our techniques allow us to build the structure in place, and also revert this building process to reconstruct the original input. This last point is of great value for libraries implementing such structures.

We extend our results to a resent proposal that has similar properties as the wavelet tree, called the wavelet matrix [Claude and Navarro, SPIRE 2012]. We prove that our techniques allow building the wavelet matrix more efficiently than the original wavelet tree.

This is joint work with Patrick Nicholson and Diego Seco.

January 23rd, 2013: Shortest path problem in CAT(0) rectangular complexes

A&C Seminar

Speaker: Daniela Maftuleac

Abstract: Introduced in full generality by M. Gromov in 1987, CAT(0) metric spaces, i.e., metric spaces B of B global non-positive curvature, constitute B a far-reaching generalization of Euclidean spaces, hyperbolic spaces and simple polygons: any two points $x$ and $y$ of a CAT(0) metric space are connected by a unique shortest path $\gamma(x,y).$

This talk will focus on the shortest path problem in some particular CAT(0) metric spaces, namely CAT(0) rectangular complexes.

I will describe an algorithm for efficiently solving two-point shortest path queries in CAT(0) rectangular complexes and two of theirs subclasses, ramified rectilinear polygons and squaregraphs.

This work is joint with Victor CHEPOI and will appear in CGTA (January 2013).

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2012

December 14th, 2012: A Space Efficient Framework for Dynamic Point Location

A&C Seminar

Speaker: Patrick Nicholson

Abstract: Let G be a planar subdivision with n vertices and bounded face size. A succinct geometric index for G is a data structure that occupies o(n) bits beyond the space required to store the coordinates of the vertices of G, while supporting efficient queries. We describe a general framework for converting dynamic data structures for planar point location into succinct geometric indexes. Using this framework, we obtain several succinct geometric indexes for dynamic planar point location on G with query times matching the currently best (non-succinct) data structures and polylogarithmic update times.

This is joint work with Meng He and Norbert Zeh and will appear in ISAAC 2012

December 13th, 2012: A Framework for Succinct Labeled Ordinal Trees over Large Alphabets

A&C Seminar

Speaker: Gelin Zhou

Abstract: We consider succinct representations of labeled ordinal trees that support a rich set of operations. Our new representations support a much broader collection of operations than previous work. In our approach, labels of nodes are stored in a preorder label sequence, which can be compressed using any succinct index for strings that supports rank_alpha and select_alpha operations. In other words, we present a framework for succinct representations of labeled ordinal trees that allows alphabets to be large. This answers an open problem presented by [Geary et al., TALG 2006]. We further extend our work and present the first succinct representation of dynamic labeled ordinal trees that supports several label-based operations including finding the level ancestor with a given label.

This is joint work with Meng He and J. Ian Munro, and will appear in ISAAC 2012.

December 12th, 2012: Fast, precise and dynamic distance queries

A&C Seminar

Speaker: Moshe Lewenstein

Abstract: We present an approximate distance oracle for a point set S with n points and doubling dimension \lambda. For every epsilon > 0, the oracle supports (1 + epsilon) - approximate distance queries in (universal) constant time, occupies space [\epsilon^{-O(lambda)} + 2^{O(\lambda \log \lambda)}]n, and can be constructed in [2^{O(\lambda)}\log^3 n + \eps^{-O(\lambda)} + 2^{O(\lambda \log \lambda)}]n expected time. This improves upon the best previously known constructions, presented by Har-Peled and Mendel. Furthermore, the oracle can be made fully dynamic with expected O(1) query time and only 2^{O(\lambda)} \log n + \eps^{-O(\lambda)} + 2^{O(\lambda \log \lambda)} update time. This is the first fully dynamic (1+\epsilon)-distance oracle.

Joint work with: Yair Bartal, Lee-Ad Gottlieb, Tsvi Kopelowitz, Liam Roditty

November 28th, 2012: Bend-Bounded Path Intersection Graphs: Sausages, Noodles, and Waffles on a Grill

A&C Seminar

Speaker: Steven Chaplick

Abstract: In this paper we study properties of intersection graphs of k-bend paths in the rectangular grid. A k-bend path is a path with at most k 90 degree turns. The class of graphs representable by intersections of k-bend paths is denoted by B_k-VPG. We show here that for every fixed k, B_k-VPG \subsetneq B_{k+1}-VPG and that recognition of graphs from B_k-VPG is NP-complete even when the input graph is given by a B_{k+1}-VPG representation. We also show that the class B_k-VPG (for k >= 1) is in no inclusion relation with the class of intersection graphs of straight line segments in the plane.

This is joint work with V. Jelinek, J. Kratochvil and T. Vyskocil and was presented at WG 2012.

November 23rd, 2012: Computing with Uncertainty

A&C Seminar

Speaker: Thomas Erlebach

Abstract: We consider problems where the input data is initially uncertain but the exact value of an input item can be obtained at a certain cost. For example, a typical setting is that instead of an exact value, only an interval containing the exact value is given. An update of an input item then reveals its exact value. An algorithm performs a number of updates until it has gathered sufficient information to output a correct solution to the problem. The goal is to minimise the number of updates.

We discuss several problems in the setting of computing with uncertainty, including the minimum spanning tree problem and the minimum multicut problem in trees. Algorithms are evaluated using competitive analysis, comparing the number of updates that the algorithm makes on an instance of the problem to the best possible number of updates that suffices to solve that instance.

(The talk is based on joint work with Michael Hoffmann, Frank Kammer, Danny Krizanc, Matus Mihalak, and Rajeev Raman.)

November 21st, 2012: Efficient Fetch-and-Increment

A&C Seminar

Speaker: Faith Ellen

Abstract: A Fetch-and-Increment object stores a non-negative integer and supports a single operation, FI, that returns the value of the object and increments it. Such objects are used in asynchronous shared memory algorithms for many distributed computing problems, such as renaming, mutual exclusion, and barrier synchronization. This talk will present an efficient implementation of a wait-free Fetch-and-Increment object from read-write registers and load-linked/store conditional (LL/SC) objects, based on a new data structure. In a system with p processes, every FI operation finishes in O(log^2 p) steps, and only a polynomial number of registers (large enough to hold a value) and O(log p)-bit LL/SC objects are needed.

This work is joint with Philipp Woelfel and Vijaya Ramachandran and appeared at DISC 2012.

November 14th, 2012: Reconfiguration of List L(2,1)-Labelings in a Graph

A&C Seminar

Speaker: Kazuto Kawamura

Abstract: For an integer k >= 0, suppose that each vertex v of a graph G has a set C(v) \subseteq {0,1, …, k} of labels, called a list of v. A list L(2,1)-labeling of G is an assignment of a label in C(v) to each vertex v of G such that every two adjacent vertices receive labels which differ by at least 2 and every two vertices of distance two receive labels which differ by at least 1. In this paper, we study the problem of reconfiguring one list L(2,1)- labeling of a graph into another list L(2,1)-labeling of the same graph by changing only one label assignment at a time, while at all times maintaining a list L(2,1)-labeling. First we show that this decision problem is PSPACE-complete, even for bipartite planar graphs and k >= 6. In contrast, we then show that the problem can be solved in linear time for general graphs if k <= 4. We finally consider the problem restricted to trees, and give a sufficient condition for which any two list L(2,1)- labelings of a tree can be transformed into each other.

This appears in ISAAC 2012.

November 7th, 2012: Online Unit Clustering

A&C Seminar

Speaker: Kim S. Larsen

Abstract: Unit Clustering is the problem of dividing a set of points from a metric space into a minimal number of subsets such that the points in each subset are enclosable by a unit ball. We continue the work, initiated by Chan and Zarrabi-Zadeh, on determining the competitive ratio of the online version of this problem. For the one-dimensional case, we develop a deterministic algorithm, improving the best known upper bound of 7/4 by Epstein and van Stee to 5/3. This narrows the gap to the best known lower bound of 8/5 to only 1/15. Our algorithm automatically leads to improvements in all higher dimensions as well. Finally, we strengthen the deterministic lower bound in two dimensions and higher from 2 to 13/6.

This is joint work with Martin Ehmsen.

October 24th, 2012: Succinct Data Structures for Path Queries

A&C Seminar

Speaker: Gelin Zhou

Abstract: Consider a tree T on n nodes, each having a weight drawn from [1..\sigma]. In this paper, we design succinct data structures to encode T using nH(W_T) + o(n\lg\sigma) bits of space, such that we can support path counting queries in O(lg\sigma / lglgn + 1) time, path reporting queries in O((occ+1)(lg\sigma / lglgn + 1)) time, and path median and path selection queries in O(lg\sigma / lglg\sigma) time, where H(W_T) is the entropy of the multiset of the weights of the nodes in T. Our results not only improve the best known linear space data structures, but also match the lower bounds for these path queries when \sigma = \Omega(n / polylog(n)).

This is joint work with Meng He and J. Ian Munro, and appears in ESA 2012.

October 10th, 2012: Odd Cycle Transversal and Vertex Cover above matching

A&C Seminar

Speaker: Venkatesh Raman

Abstract: The odd cycle transversal in a graph is a subset of vertices whose removal results in a bipartite graph; i.e. it is a subset of vertices that hits all odd cycles in the graph. We address the question of determining whether a given undirected graph on n vertices has an odd cycle transversal of size at most k. Reed, Smith and Vetta gave a 3^k n^{O(1)} algorithm for the problem in 2004, and this bound has been improved to (2.32)^k n^{O(1)} recently (STACS 2012). This improvement is obtained by a reduction to the vertex cover problem, that asks whether there is a vertex cover in the graph whose size is at most k more than the size of the maximum matching. The talk will describe (a simpler version of) this improved algorithm, particularly for the `vertex cover above matching’ problem. The talk will start with an introduction to parameterized complexity including recent research issues in the area.

October 3rd, 2012: A Small Depth-16 Circuit for the AES S-Box

A&C Seminar

Speaker: Joan Boyar

Abstract: New techniques for reducing the depth of circuits for cryptographic applications are described. These techniques also keep the number of gates quite small. The result, when applied to the AES S-Box, is a circuit with depth 16 and only 128 gates. For the inverse, it is also depth 16 and has only 127 gates. There is a shared middle part, common to both the S-Box and its inverse, consisting of 63 gates. The best previous comparable design for the AES S-Box has depth 22 and size 148 [Nogami, Nekado, Toyota, Hongo, Morikawa, 2010]. This is joint work with René Peralta.

September 26th, 2012: On the complexity of reconfiguration problems

A&C Seminar

Speaker: Takehiro Ito

Abstract: Reconfiguration problems arise when we wish to find a step-by-step transformation between two feasible solutions of a problem such that all intermediate results are also feasible. For example, in the SAT reconfiguration problem, we are given a CNF formula and two satisfying truth assignments, and we are asked whether they are connected in the subgraph of the hypercube induced only by the satisfying truth assignments; this problem is known to be PSPACE-complete. We demonstrate that a host of reconfiguration problems derived from NP-complete problems are PSPACE-complete. In contrast, several reconfiguration versions of problems in P (e.g., matching) are solvable in polynomial time.

September 5th, 2012: Succinct Posets

A&C Seminar

Speaker: Patrick Nicholson

Abstract: We describe an algorithm for compressing a partially ordered set, or poset, so that it occupies space matching the information theory lower bound (to within lower order terms), in the worst case. Using this algorithm, we design a succinct data structure for representing a poset that, given two elements, can report whether one precedes the other in constant time. This is equivalent to succinctly representing the transitive closure graph of the poset, and we note that the same method can also be used to succinctly represent the transitive reduction graph. For an n element poset, the data structure occupies n^2/4 + o(n^2) bits, in the worst case, which is roughly half the space occupied by an upper triangular matrix. Furthermore, a slight extension to this data structure yields a succinct oracle for reachability in arbitrary directed graphs. Thus, using roughly a quarter of the space required to represent an arbitrary directed graph, reachability queries can be supported in constant time.

This is joint work with Ian Munro, and will appear in ESA 2012.

August 29th, 2012: Minimizing Cache Usage in Paging

A&C Seminar

Speaker: Alejandro Salinger

Abstract: Traditional paging models seek algorithms that maximize their performance while using the maximum amount of cache resources available. However, in many applications this resource is shared or its usage involves a cost. In this work we introduce the Minimum Cache Usage problem, which is an extension to the classic paging problem that accounts for the efficient use of cache resources by paging algorithms. In this problem, the cost of a paging algorithm is a combination of both its number of faults and the amount of cache it uses, where the relative cost of faults and cache usage can vary with the application. We present a simple family of online paging algorithms that adapt to the ratio alpha between cache and fault costs, achieving competitive ratios that vary with alpha, and that are between 2 and the cache size k. Furthermore, for sequences with high locality of reference, we show that the competitive ratio is at most 2, and provide evidence of the competitiveness of our algorithms on real world traces. Finally, we show that the offline problem admits a polynomial time algorithm. In doing so we define a reduction of paging with cache usage to weighted interval scheduling on identical machines.

This is joint work with Alex Lopez-Ortiz, and will appear in WAOA 2012.

August 27th, 2012: Streaming Cryptography

A&C Seminar

Speaker: Periklis Papakonstantinou

Abstract: Three years ago we introduced the concept of Streaming Cryptography; i.e. the possibility of computing cryptographic primitives using a device severely restricted in memory and in its ability to scan its external tape(s). Is it possible to perform cryptography in such a setting, or do the limitations of these devices rule-out such a possibility? For several, natural, black-box settings, and pretty much all popular cryptographic assumptions used today we can show the impossibility of realizing cryptography in a streaming setting.

Recently we showed that one can circumvent lower bounds for specific kinds of constructions by devising a non-black-box technique, showing that Streaming Cryptography is possible if one uses two external streams, and in total 3 passes over them. Roughly speaking, this development encrypts in a streaming fashion not the output itself but enough information regarding the computation of the primitive. This idea is in the work of Applebaum-Ishai-Kushilevitch on Crypto in NC^0 (although their so-called “randomized encodings” cannot be computed in a streaming fashion). In Streaming Crypto we are able to make somewhat counter-intuitive statements, such as: computing the product of two numbers (and permuted variants) is unconditionally impossible in a streaming fashion, however we are still able to base streaming cryptography on the hardness of factoring a composite integer.

To the best of our knowledge, this is the first practical, non-black-box cryptographic construction (note that the vast majority of cryptographic constructions are black-box).

August 22nd, 2012: Dual-Sorted Inverted Lists in Practice

A&C Seminar

Speaker: Roberto Konow

Abstract: We implement a recent theoretical proposal to represent inverted lists in memory, in a way that docid-sorted and weight-sorted lists are simultaneously represented in a single wavelet tree data structure.We compare our implementation with classical representations, where the ordering favors either bag-of-word queries or Boolean and weighted conjunctive queries, and demonstrate that the new data structure is faster than the state of the art for conjunctive queries, while it offers an attractive space/time tradeoff, when both kinds of queries are of interest.

August 15th, 2012: Path Queries in Weighted Trees

A&C Seminar

Speaker: Gelin Zhou

Abstract: Trees are fundamental structures in computer science, being widely used in modeling and representing different types of data in numerous computer applications. In many cases, properties of objects being modeled are stored as weights or labels on the nodes of trees. Thus researchers have studied the preprocessing of weighted trees in which each node is assigned a weight, in order to support various path queries, for which a certain function over the weights of the nodes along a given query path in the tree is computed [3, 14, 22, 26].

In this thesis, we consider the problem of supporting several various path queries over a tree on n weighted nodes, where the weights are drawn from a set of sigma distinct values. One query we support is the path median query, which asks for the median weight on a path between two given nodes. For this and the more general path selection query, we present a linear space data structure that answers queries in O(lg sigma) time under the word RAM model. This greatly improves previous results on the same problem, as previous data structures achieving O(lg n) query time use O(n lg^2 n) space, and previous linear space data structures require O(n ^eps) time to answer a query for any positive constant eps [26].

We also consider the path counting query and the path reporting query, where a path counting query asks for the number of nodes on a query path whose weights are in a query range, and a path reporting query requires to report these nodes. Our linear space data structure supports path counting queries with O(lg sigma) query time. This matches the result of Chazelle [14] when sigma is close to n, and has better performance when sigma is significantly smaller than n. The same data structure can also support path reporting queries in O(lg sigma + occ lg sigma) time, where occ is the size of output. In addition, we present a data structure that answers path reporting queries in O(lg sigma + occ lg lg sigma) time, using O(n lg lg sigma) words of space. These are the first data structures that answer path reporting queries.

August 15th, 2012: Prefix Free Codes in Linear Time

A&C Seminar

Speaker: Jeremy Barbay

Abstract: Considering the computation of an *optimal prefix-free code* over $n$ symbols from their frequencies $\{f_1,\ldots,f_n\}$ given in arbitrary order, we give a *linear time algorithm* in the *word-RAM memory model* and an (input order oblivious) *instance optimal* algorithm in the *algebraic model*. These complexities improve over both the traditional $O(n\lg n)$ algorithm from Huffman and the $O(nk)$ adaptive algorithm from Belal and Elmasry, where $k$ is the number of distinct code lengths in any optimal code.

July 18th, 2012: The k-Server Problem with Advice

A&C Seminar

Speaker: Marc Renault

Abstract: Traditionally, online algorithms receive their input piece by piece and all actions for a given piece of the input must be performed before the next piece is received. That is, that an online algorithm must make a decision for the current piece of the input without knowledge of the future. Competitive analysis, the ratio between the performance of an online algorithm and an optimal algorithm with full knowledge of the future, is used to analyze online algorithms.

One of the most studied online problems is the k-Server Problem which consists of a metric space, k mobile servers and a request sequence, where each request is for a node of the metric space. At the time of the request, if a node is not covered, one of the k servers must be moved to the node. The goal is to minimize the distance traveled by the k servers over the entire request sequence.

Emek, Fraigniaud, Korman and Rosen (ICALP 2009) presented a model of online computation with advice, where, along with each request, an algorithm has access to a quantified amount of information about the future. We present an algorithm that is \lceil(log k)\rceil/(b-1)\rceil, where 3 <= b <= log k, which improves on previous results. More importantly, we present an algorithm and analysis that is more intuitive and simpler than previous ones. Further, we give a 1-competitive algorithm with 2+2\lceil(log p + 1)\rceil bits of advice, where p is the caterpillar dimension of the tree. Lastly, we present an optimal algorithm for the line with 1 bit of advice.

June 6th, 2012: Robust Non-Parametric Data Approximation of Pointsets via Data Reduction

A&C Seminar

Speaker: Jason Morrison

Abstract: This talk will discuss a novel non-parametric method of simplifying piecewise linear curves and we apply this method as a statistical approximation of structure within sequential data in the plane. We consider the problem of minimizing the average length of sequences of consecutive input points that lie on any one side of the simplified curve. Specifically, given a sequence $P$ of $n$ points in the plane that determine a simple polygonal chain consisting of $n-1$ segments, we describe algorithms for selecting an ordered subset $Q \subset P$ (including the first and last points of $P$) that determines a second polygonal chain to estimate $P$, such that the number of crossings between the two polygonal chains is maximized, and the cardinality of $Q$ is minimized among all such maximizing subsets of $P$. Our algorithms have respective running times $O(n^2\log n)$ when $P$ is monotonic and $O(n^2\log^2 n)$ when $P$ is an arbitrary simple polyline. Finally, we examine the application of our algorithms iteratively in a bootstrapping technique to define a smooth robust non-parametric approximation of the original monotonic sequence.

April 25th, 2012: Satisfiability Allows No Nontrivial Sparsification Unless The Polynomial-Time Hierarchy Collapses

A&C Seminar

Speaker: Dieter van Melkebeek

Abstract: Abstract: Consider the following two-player communication process to decide a language L: The first player holds the entire input x but is polynomially bounded; the second player is computationally unbounded but does not know any part of x; their goal is to cooperatively decide whether x belongs to L at small cost, where the cost measure is the number of bits of communication from the first player to the second player.

For any integer d>=3 and positive real epsilon we show that if satisfiability for n-variable d-CNF formulas has a protocol of cost O(n^{d-epsilon}) then the polynomial-time hierarchy collapses. The result is tight as there exists a trivial protocol for epsilon = 0. Under the standard complexity-theoretic hypothesis that the polynomial-time hierarchy does not collapse, the result implies tight lower bounds for parameters of interest in several areas, including sparsification, probabilistically checkable proofs, instance compression, and kernelization in parameterized complexity.

By reduction similar results hold for other NP-complete problems. For the vertex cover problem on n-vertex d-uniform hypergraphs, the above statement holds for any integer d >= 2. The case d=2 implies that no NP-hard vertex deletion problem based on a graph property that is inherited by subgraphs can have kernels consisting of O(k^{2-epsilon}) edges unless coNP is in NP/poly, where k denotes the size of the deletion set. Kernels consisting of O(k^2) edges are known for several problems in the class, including vertex cover, feedback vertex set, and bounded-degree deletion.

The proofs hinge on the existence of high-density subsets of the integers without nontrivial arithmetic progressions of length three.

Joint work with Holger Dell.

April 11th, 2012: The Incremental Constraint of k-Server

A&C Seminar

Speaker: Caelyn McAulay

Abstract: Abstract: Online algorithms are characterized by operating on an input sequence revealed over time versus a single static input. Instead of generating a single solution, they produce a sequence of incremental solutions corresponding to the input seen so far. An online algorithm’s ignorance of future inputs limits its ability to produce optimal solutions. The incremental nature of its solutions is also an obstacle. The two factors can be differentiated by examining the corresponding incremental algorithm, which has knowledge of future inputs, but must still provide a competitive solution at each step We talk both about this method in general and its application to the k-server problem. We prove that the incremental constraint of the k-server problem is bounded below by 2.

April 4th, 2012: Incorporating all inputs in the search for closest strings

A&C Seminar

Speaker: Narges Simjour

Abstract: The closest string problem asks for a ``centre’’ string within Hamming distance $d$ of a set of equal-length $m$ input strings. In this presentation, we briefly describe the previous parameterized algorithms for the problem for the parameter $d$. We then show that a slight modification to these algorithms will make them enumerative for minimal $d$. We also give a new algorithm that improves the previous best asymptotic time bound when the inputs are binary and $m\in O(2^{o(d)})$.

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2011

October 26th, 2011: Variations on the Erdos Discrepancy Problem

A&C Seminar

Speaker: Alex Leong

Abstract: In this talk I will introduce the notion of D-discrepancy of infinite sequences and how it relates to the Erdos Discrepancy Problem. I will discuss the conjecture that there are finitely many infinite sequences with ODD-discrepancy 1. Finally, I’ll define discrepancy maximal sets, give some examples, and conclude with some open problems.

May 18th, 2011: Finite Orbits of Language Operations

A&C Seminar

Speaker: Emilie Charlier

Abstract: We consider a set of natural operations on languages, and prove that the orbit of any language L under the monoid generated by this set is finite and bounded, independently of L. This generalizes previous results about complement, Kleene closure, and positive closure.

This is a 20-25 minute practice talk for LATA, and is joint work with Mike Domaratzki, Tero Harju, and Jeffrey Shallit.

March 9th, 2011: Alphabet Partitioning for Compressed Rank/Select and Applications

A&C Seminar

Speaker: Travis Gagie

Abstract: We present a data structure that stores a string $s[1..n]$ over the alphabet $[1..\sigma]$ in $nH_0(s) + o(n)(H_0(s){+}1)$ bits, where $H_0(s)$ is the zero-order entropy of $s$. This data structure supports the queries \access\ and \rank\ in time $\Oh{\lg\lg\sigma}$, and the \select\ query in constant time. This result improves on previously known data structures using $nH_0(s)+o(n\lg\sigma)$ bits, where on highly compressible instances the redundancy $o(n\lg\sigma)$ cease to be negligible compared to the $nH_0(s)$ bits that encode the data. The technique is based on combining previous results through an ingenious partitioning of the alphabet, and practical enough to be implementable. It applies not only to strings, but also to several other compact data structures. For example, we achieve $(i)$ faster search times and lower redundancy for the smallest existing full-text self-index; $(ii)$ compressed permutations $\pi$ with times for $\pi()$ and $\pi^{-1}()$ improved to log-logarithmic; and $(iii)$ the first compressed representation of dynamic collections of disjoint sets. This is joint work with Jeremy Barbay, Gonzalo Navarro, and Yakov Nekrich and appeared in ISAAC 2010.

March 2nd, 2011: The Streaming Complexity of Validating XML Documents

A&C Seminar

Speaker: Christian Konrad

Abstract: This talk discusses the problem of checking validity of XML files w.r.t. arbitrary DTDs in the streaming model. While prior works restrict on the set of DTDs for which there are sub-linear space streaming algorithms, we do not make any restrictions and consider the general setting. Checking validity of arbitrary XML files can not be established in the usual streaming setting with sub-linear space and a constant number of passes. Therefore we consider a more powerful streaming model, that is a streaming model with auxiliary streams onto which the algorithm is allowed to write and read. In this model we develop an algorithm using 3 auxiliary read/write streams which performs O(log N) passes over these streams and which uses O(polylog N) random access memory.

February 9th, 2011: Decomposing the Competitive Ratio

A&C Seminar

Speaker: Alexa Sharp

Abstract: Online algorithms are given an input sequence revealed over time and produce a sequence of incremental solutions corresponding to the input seen so far. The changing nature of the input can make it difficult to achieve optimality, but the incremental condition can be a fundamental obstacle as well. The quality of an online algorithm is often measured by its competitive ratio, which accounts for its two limitations: not knowing the input sequence in advance, and making irreversible (incremental) decisions. Both factors can contribute to an algorithm’s performance, but the competitive ratio fails to discern which is more significant.

In this talk, we will differentiate the two factors by examining incremental optimization. As with online algorithms, incremental algorithms are given an input sequence, and produce a sequence of solutions that build incrementally while adapting to the additional input. Unlike their online counterparts, however, they are given the input sequence in advance, even though they must still make irreversible decisions. In this way, the study of incremental problems allows us to isolate the effects of the two factors which contribute to the competitive ratio.

February 8th, 2011: Edge Intersection Graphs of Single Bend Paths in a Grid

A&C Seminar

Speaker: Martin Charles Golumbic

Abstract: Let $\cal{P}$ be a collection of nontrivial simple paths on a host graph $H$. The edge intersection graph of $\cal{P}$, denoted by $EP_H(\cal{P})$, has vertex set that corresponds to the members of $\cal{P}$, where two vertices are joined by an edge if and only if the corresponding members of $\cal{P}$ share at least one common edge in $H$. An undirected graph $\Gamma$ is called an edge intersection graph of paths in a tree (EPT) if $\Gamma$ = $EP_T(\cal{P})$ for some $\cal{P}$ and tree $T$. Similarly, $\Gamma$ is called an edge intersection graph of paths in a grid (EPG) if $\Gamma$ = $EP_G(\cal{P})$ for some $\cal{P}$ and grid $G$. The EPT and EPG graphs can be useful in network and circuit applications, where scheduling and layout problems are often equivalent to coloring an EPT or EPG graph.

In this lecture, we will survey the mathematical and algorithmic results on various types of EPT and EPG graphs and some of their generalizations, together with several restrictions on the representations. The class of EPT graphs was first investigated by Golumbic and Jamison in two papers appearing in 1985, and subsequently, further research has been carried out by a number of algorithmic graph theorists. In a series of papers during the past 5 years, Golumbic, Lipshteyn and Stern studied the hierarchy of related EPT graph classes giving some structure theorems.

Very recently, Golumbic, Lipshteyn and Stern introduced EPG graphs, proving that every graph is an EPG graph, and then turning their attention to the subclass of graphs that admit an EPG representation in which every path has at most a single bend, called B$_1$-EPG graphs. They proved that any tree is a B$_1$-EPG graph and gave a structural property that enables generating non B$_1$-EPG graphs. A characterization of the representation of cliques and chordless 4-cycles in B$_1$-EPG graphs was given, and also prove that single bend paths on a grid have Strong Helly number 3 when the paths satisfy the usual Helly property, and otherwise have Strong Helly number 4.

We will also survey recent results by our colleagues Andrei Asinowski, Andrew Suk and Bernard Ries on edge intersection graphs of systems of paths on a grid with a bounded number of bends and some further research by a team in Germany. We conclude with some open problems and future work.

February 2nd, 2011: Testing Simultaneous Planarity when the Common Graph is 2-connected

A&C Seminar

Speaker: Krishnam Raju Jampani

Abstract: Two planar graphs G1 and G2 sharing some vertices and edges are said to be simultaneously planar if they have planar drawings such that a shared vertex [edge] is represented by the same point [curve] in both drawings. It is an open problem to determine whether simultaneous planarity can be tested efficiently. In this talk, I present a linear-time algorithm to test simultaneous planarity when the two graphs share a 2-connected subgraph.

This is joint work with Bernhard Haeupler and Anna Lubiw and appeared in ISAAC 2010.

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2010

December 1st, 2010: Simultaneous Interval Graphs

A&C Seminar

Speaker: Krishnam Raju Jampani, University of Waterloo

Abstract: In a recent paper, we introduced the simultaneous representation problem and studied it for chordal, comparability and permutation graphs. For interval graphs, the problem is defined as follows. Two interval graphs G1 and G2, sharing some vertices I, and the edges induced by I, are said to be `simultaneous interval graphs’ if there exist interval representations of G1 and G2, such that any vertex of I is mapped to the same interval in both representations. The simultaneous representation problem for interval graphs asks whether G1 and G2 are simultaneous interval graphs.

Simultaneous representation problems are related to several well-studied problems in the literature including simultaneous planar embeddings, graph sandwich problems and probe graphs, and have application in any situation where two related graphs should be represented consistently. In this talk we present an O(n^2*logn) time algorithm for recognizing simultaneous interval graphs.

This is joint work with Anna Lubiw and will be presented at ISAAC 2010.

November 12th, 2010: Linear-Space Data Structures for Range Mode Query in Arrays

A&C Seminar

Speaker: Stephane Durocher

Abstract: A mode of a multiset S is an element x in S of maximum multiplicity; that is, x occurs at least as frequently as any other element in S. Given a list A[1 : n] of n items, we consider the problem of constructing a data structure that efficiently answers range mode queries on A. Each query consists of an input pair of indices (i, j) for which a mode of A[i : j] must be returned. We present an O(n)-space static data structure that supports range mode queries in O(min{sqrt(n), k, |j-i|+1}) time in the worst case, where k denotes the number of distinct elements in A. This is the first linear-space data structure to guarantee O(sqrt(n)) query time.

This work is joint with Jason Morrison.

November 10th, 2010: Minimum Enclosing Area Triangle with a Fixed Angle

A&C Seminar

Speaker: Jean-Lou De Carufel

Abstract: Given a set $S$ of $n$ points in the plane and a fixed angle $0 < \omega < \pi$, I will explain how to find all triangles of minimum area with angle $\omega$ that enclose $S$ in $O(n log n)$ time. I will also demonstrate that in general, the solution cannot be written without using cubic root.

This is joint work with Prosenjit Bose.

October 14th, 2010: Towards Universal Semantic Communication

Joint A&C Seminar and Distinguished Lecture Series [video]

Speaker: Madhu Sudan, MIT

Abstract: Is it possible for two intelligent players to communicate meaningfully with each other, without any prior common background? What does it even mean for the two players to understand each other? We claim that this question, in addition to being of philosophical/linguistic interest, goes to the essence of modern communication/computation. Modern communicating devices are extremely diverse, constantly evolving, and misunderstandings (mismatches in protocols) between communicating devices are a major source of errors. We believe that questions like this need to be answered to set the foundations for a robust theory of (meaningful) communication.

In this talk, I will describe our approach towards this problem. We argue that in order to solve such problems, we must first articulate why intelligent players (and/or powerful computers) communicate. This leads us to a formal theory of “goals” of communication. We then show that when progress towards the goal is “verifiable” then players can detect misunderstandings. Under a complexity-theoretic lens, we show roughly that verifiability is really the essence of resolving misunderstandings: Verifiable goals are more powerful than goals that can be achieved without communication (so communication is good), but very restricted compared to “unverifiable” goals (so there is need for moderation).

Most of the talk will focus on the definitions of various concepts such as “goals,” “(mis)understanding” and resort to theorems based on computational complexity to support these definitions. Based on joint works with Oded Goldreich (Weizmann) and Brendan Juba (MIT).

September 8th, 2010: Geometric On-line Ray Searching Under Probability of Placement Scenarios

A&C Seminar

Speaker: Ying (Catherine) Liu

Abstract: Competitive analysis is a standard measure for analysis of online algorithms. It has been applied to many online problems in diverse areas ranging from robot navigation, to network routing, to scheduling, to onine graph coloring. In this thesis, we first study three classic online problems, namely the Cow-Path problem, the Processor-Allocation problem and the Robot-Search-Ray problem and highlight connections between them. Second, the main result is for the One-Robot-Two-Rays problem for which we consider the weighted scenario, in which the robot locates on a ray with preferential probability p, which we term as 1-STRAW (and in general k-STRAW for k searchers). We propose a search strategy which is optimal among weighted geometric states. Additionally, we prove a tight lower bound of the worst case competitive ratio and conjecture a lower bound of the average case competitive ratio for 1-STRAW.

July 28th, 2010: A Grouped Hamming Network

A&C Seminar

Speaker: Bryan Logan, School of Computer Science, University of Waterloo

Abstract: A distributed hash table (DHT) is a type of peer-to-peer (P2P) network that, like traditional hash tables, maps keys to values. Unlike traditional hash tables, however, the data is distributed across a network with each node being responsible for a particular range of keys. Numerous other DHTs have been presented and have become the cornerstone of wildly popular P2P file-sharing applications, such as BitTorrent. Each of these DHTs trades-off the number of pointers maintained per node with the overhead and lookup time; storing more pointers decreases the lookup time at the expense of increased overhead.

A Grouped Hamming Network (GHN), the overlay network presented in this thesis, allows for the number of pointers per node to be any increasing function of the number of nodes, n, where P(n) = Ω(lg n). The system presented assumes that nodes fail independently and uniformly at random with some probability q=1-p. Three different schemes for routing in a GHN are presented. For each routing scheme a theoretical estimate on the probability of failure is given and optimal configurations in terms of n and P(n) are given. Simulations of GHNs with various configurations indicate that the given estimates are indeed accurate for reasonable values of q and that the optimal configurations are accurate.

July 14th, 2010: Neighborship Voronoi Games

A&C Seminar

Speaker: Masud Hasan

Abstract: We study the following two general problems: Given n Player1 sites in the plane and their Delaunay triangulation D, place minimum possible Player2 sites such that in the resulting Delaunay triangulation D’ of the sites of both players, (1) Player2 gets all sites of Player1 as neighbors, and (2) neighborship among Player1 sites is as small as possible. We study several variations of these two problems and formulate them as one round Vornoi games. We give upper bounds and winning strategies for Player2, if they exist, or show how the games end in ties.

The talk presents the results of the following two papers: - Md. Muhibur Rasheed, Masud Hasan, and M. Sohel Rahman, Maximum neighbor Voronoi games, in Proc. WALCOM 2009 - Syed Ishtiaque Ahmed, Masud Hasan, and Awalin Sopan, Vindictive Voronoi games and stabbing Delaunay circles, to be presented at ISVD 2010

June 23rd, 2010: Algorithms for computing NE/SNE repeats in a string and NE multirepeats in multiple sequences

A&C Seminar

Speaker: Munina Yusufu

Abstract: In this talk, we first describe four variants of a new fast algorithm RPT1 that compute the complete nonextendible (NE) repeats in a given string. The variants RPT1-3 and RPT1-4 execute in O(n) time independent of alphabet size and are faster than the two other algorithms previously proposed for this problem. Furthermore, we describe new algorithm RPT2 for computing complete supernonextendible (SNE) repeats. We conclude with a brief discussion of applications to bioinformatics and data compression. This is joint work with W. F. Smyth and Simon J. Puglisi. The initial result of the work was presented in PSC 2008, the full version was published in Mathematics in Computer Science 3-4 (2010).

Then we present the new algorithms for finding the NE multirepeats in a set of N strings of average length n under various constraints. We show that RPT1 can be extended to locate the multirepeats based on the investigation of the properties of the multirepeats and various strategies. We formulate two problems - first with no restriction on gap length, then with bounded gaps. For the first problem, we propose two algorithms with worst-case time complexities O(Nn+ alpha*logN) and O(Nn+ alpha) with different space complexities, where alpha is the alphabet size. For the second problem, we describe an algorithm with worst-case time complexity O(RNn), where R is the number of multirepeats output. We demonstrate that our algorithms are faster, more flexible and much more space efficient than algorithms previously proposed for this problem. This is joint work with W. F. Smyth and C. S. Iliopoulos. The result of the work was published in Fundamenta Informaticae 97-3 (2009).

June 2nd, 2010: Recognizing well-parenthesized expressions in the streaming model

A&C Seminar

Speaker: Frederic Magniez

Abstract: Motivated by a concrete problem and with the goal of understanding the sense in which the complexity of streaming algorithms is related to the complexity of formal languages, we investigate the problem Dyck(s) of checking matching parentheses, with $s$ different types of parenthesis.

We present a one-pass randomized streaming algorithm for Dyck(2) with space $\Order(\sqrt{n}\log n)$, time per letter $\polylog (n)$, and one-sided error. We prove that this one-pass algorithm is optimal, up to a $\polylog n$ factor, even when two-sided error is allowed. For the lower bound, we prove a direct sum result on hard instances by following the “information cost” approach, but with a few twists. Indeed, we play a subtle game between public and private coins. This mixture between public and private coins results from a balancing act between the direct sum result and a combinatorial lower bound for the base case. Surprisingly, the space requirement shrinks drastically if we have access to the input stream in reverse. We present a two-pass randomized streaming algorithm for Dyck(2) with space $\Order((\log n)^2)$, time $\polylog (n)$ and one-sided error, where the second pass is in the reverse direction. Both algorithms can be extended to Dyck(s) since this problem is reducible to Dyck(2) for a suitable notion of reduction in the streaming model.

May 26th, 2010: Dynamic Planar Orthogonal 3-sided Range Reporting in Expected Doubly Logarithmic Time

A&C Seminar

Speaker: Konstantinos Tsakalidis

Abstract: We consider the problem of maintaining dynamically a set of points in the plane and supporting 3-sided orthogonal range queries (namely, of the type [a,b]×(-∞,c]). All previous results in the Pointer Machine, the RAM and the I/O model, achieve nearly logarithmic worst case query and update time using linear space. By assuming that the input coordinates are drawn from various probabilistic distributions, we show how to decrease these complexities to expected doubly logarithmic.

Let $n$ be the number of points currently stored in the data structure, let $t$ be the number of reported points and $B$ the size of a block in external memory. We first reduce the amortized update time to $O(\log\log n)$ in the RAM, when both x and y-coordinates are drawn from an unknown continuous \emph{$\mu$-random distribution}. Respectively, in the I/O model we achieve $O(\log_B \log n)$ amortized update I/Os. Next, we additionally reduce the query time to $O(\log\log n)$, when the x-coordinates are drawn from the \emph{Zipfian} distribution. In the I/O model we achieve $O(\log_B \log n + t/B)$ query I/Os, by moreover imposing the y-coordinates to be \emph{smoothly} distributed. So far all improved complexities are expected with high probability. [K. Tsakalidis et al. ICDT'10]

Independently, we show how to achieve $O(\log \log n + t)$ expected with high probability query time and $O(\log \log n)$ expected amortized update time, in the RAM, by only assuming that the x-coordinates are smoothly distributed. In the I/O model, the same assumption leads to $O(\log \log_B n + t/B)$ expected with high probability query I/Os and $O(\log_B \log n)$ expected amortized I/Os. [K. Tsakalidis, et al. ISAAC'09]

May 5th, 2010: Simultaneously Embedding Planar Graphs at Fixed Vertex Locations

A&C Seminar

Speaker: Taylor Gordon

Abstract: We discuss the problem of embedding planar graphs onto the plane with pre-specified vertex locations. In particular, we introduce a method for constructing such an embedding for both the case where the mapping from the vertices onto the vertex locations is fixed and the case where this mapping can be chosen. Moreover, the technique we present is sufficiently abstract to generalize to a method for constructing simultaneous planar embeddings with fixed vertex locations. In all cases, we are concerned with minimizing the number of bends per edge in the embeddings we produce.

In the case where the mapping is fixed, our technique guarantees embeddings with at most 8n bends per edge in the worst case and, on average, at most Bn bends per edge, where B < 4.1318402. This result improves previously known techniques by a significant constant factor.

When the mapping is not pre-specified, our technique guarantees embeddings with at most O(n^(1 - 2^(1-k))) bends per edge in the worst case and, on average, at most O(n^(1 - 1/k)) bends per edge, where k is the number of graphs in the simultaneous embedding. This improves upon the previously known O(n) bound on the number of bends per edge for k at least 2. Moreover, we give an average-case lower bound on the number of bends that has similar asymptotic behaviour to our upper bound.

April 28th, 2010: Problems Related to Shortest Strings in Formal Languages

A&C Seminar

Speaker: Thomas Ang

Abstract: In formal language theory, studying shortest strings in languages, and variations thereof, can be useful since these strings can serve as small witnesses for properties of the languages, and can also provide bounds for other problems involving languages. For example, the length of the shortest string accepted by a regular language provides a lower bound on the state complexity of the language. We present new bounds for length of shortest strings in the intersection of regular languages, and for length of shortest epsilon-reducible strings in regular languages defined over free groups.

April 7th, 2010: RootChord

A&C Seminar

Speaker: Lukasz Cwik

Abstract: We present a distributed hash table which is able to adapt to changes in the size of the network and answer lookup queries within a guaranteed 2 hops while maintaining a routing table of size $\Theta(N^{\frac{1}{2}})$. We will cover all aspects of the protocol including routing, joining, maintaining, and departing the network. In addition we discuss the practical implementation issues of parallelization, data replication, dead node discovery, and network messaging convergence.

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2009

December 9th, 2009: I/O and Space-Efficient Path Traversal in Planar Graphs

A&C Seminar

Speaker: Meng He

Abstract: We present a technique for representing bounded-degree planar graphs succinctly while permitting I/O-efficient path traversal. To represent a graph G on N vertices, each with an associated key of q = O(lg N) bits, we use Nq + O(N) + o(Nq) bits. Using this representation, a path of length K can be traversed with O(K/ lg B) I/Os, where B is the disk block size. Our structure may be adapted to represent, with similar space bounds, a terrain modeled as a triangular-irregular network to support traversal of a path that visits K triangles using O(K/lg B) I/Os. This structure can be used to answer a number of useful queries efficiently, such as reporting terrain profiles, trickle paths and connected components.

November 25th, 2009: On a Problem Posed by Steve Smale

A&C Seminar

Speaker: Felipe Cucker, City University of Hong Kong

Abstract: The 17th of the problems proposed by Steve Smale for the 21st century asks for the existence of a deterministic algorithm computing an approximate solution of a system of $n$ complex polynomials in $n$ unknowns in time polynomial, on the average, on the size $N$ of the input system. A partial solution to this problem was given by Carlos Beltran and Luis Miguel Pardo who exhibited a randomized algorithm doing so. In this paper we further extend this result in several directions. Firstly, we exhibit a linear homotopy algorithm that efficiently implements a non-constructive idea of Mike Shub. This algorithm is then used in a randomized algorithm, call it LV, a la Beltran-Pardo. Secondly, we perform a smoothed analysis (in the sense of Spielman and Teng) of algorithm LV and prove that its smoothed complexity is polynomial in the input size and $\sigma^{-1}$, where $\sigma$ controls the size of of the random perturbation of the input systems. Thirdly, we perform a condition-based analysis of LV. That is, we give a bound, for each system $f$, of the expected running time of LV with input $f$. In addition to its dependence on $N$ this bound also depends on the condition of $f$. Fourthly, and to conclude, we return to Smale’s 17th problem as originally formulated for deterministic algorithms. We exhibit such an algorithm and show that its complexity is polynomial in $N$ for systems with moderate degree.

Felipe Cucker is a Chair Professor of Mathematics at City University of Hong Kong. His main research interest is on the theoretical aspects of numeric computations. He sits on the editorial board of several journals and is the present chairman of the Society for the Foundations of Computational Mathematics.

November 18th, 2009: Practical Discrete Unit Disk Cover Using an Exact Line-Separable Algorithm

A&C Seminar

Speaker: Robert Fraser

Abstract: Given m unit disks and n points in the plane, the discrete unit disk cover problem is to select a minimum subset of the disks to cover the points. This problem is NP-hard and the best previous practical solution is a 38-approximation algorithm by Carmi et al. We first consider the line-separable discrete unit disk cover problem (the set of disk centres can be separated from the set of points by a line) for which we present an O(m^2n)-time algorithm that finds an exact solution. Combining our line-separable algorithm with techniques from the algorithm of Carmi et al. results in an O(m^2n^4) time 22-approximate solution to the discrete unit disk cover problem.

This describes joint work with Francisco Claude, Reza Dorrigiv, Stephane Durocher, Alejandro Lopez-Ortiz and Alejandro Salinger. It has been accepted for publication at ISAAC 2009, and has been invited to a special issue of DMAA.

November 4th, 2009: Vertex Intersection Graphs of Paths on a Grid

A&C Seminar

Speaker: Elad Cohen

Abstract: Let P be a set of simple paths on a grid. The Vertex Intersection Graph of Paths on a Grid r(P) has a vertex set V, where every vertex v in V correspond to a path p in P, and an edge set E, where (u,v) is in E if and only if the corresponding paths Pu,Pv intersect. We call the graph G the Vertex Intersection Graph of Paths on a Grid (VPG) if G=r(P), for some P. If Pk is a set of simple paths on a grid, where each path has at most k bends, then the graph G=r(Pk) is called Bk-VPG. The Intersection Graph of Curves on a Plane, also known as String graph, is defined in a similar way to the VPG graph, here the vertex set corresponds to a set of curves on a plane and two vertices are adjacent if and only if the corresponding curves intersect. We show that the class of VPG graphs is equivalent to the class of string graphs. It follows that the recognition of VPG graphs is NP-complete since recognizing string graphs is NP-complete. Moreover, the recognition of Bk-VPG graphs has been shown to be NP-complete, for k>=0. The coloring problem on string graphs is also NP-complete ,we prove that it is also NP-complete for Bk-VPG graphs, for k>=0. We study the relations between Bk-VPG graphs and different known graphs classes, and we give several properties of B0-VPG graphs and examples for graphs which are not B0-VPG graphs.

This is joint work with Andrei Asinowski, Martin Charles Golumbic, Vincent Limouzy, Marina Lipshteyn and Michal Stern.

October 16th, 2009: Left-leaning Red-Black Trees

A&C Seminar

Speaker: Robert Sedgewick

Abstract: The red-black tree model for implementing balanced search trees, introduced by Guibas and Sedgewick thirty years ago, is now found throughout our computational infrastructure. Red-black trees are described in standard textbooks and are the underlying data structure for symbol-table implementations within C++, Java, Python, BSD Unix, and many other modern systems. However, many of these implementations have sacrificed some of the original design goals (primarily in order to develop an effective implementation of the delete operation, which was incompletely specified in the original paper), so a new look is worthwhile. In this paper, we describe a new variant of redblack trees that meets many of the original design goals and leads to substantially simpler code for insert/delete, less than one-fourth as much code as in implementations in common use. All red-black trees are based on implementing 2-3 or 2-3-4 trees within a binary tree, using red links to bind together internal nodes into 3-nodes or 4-nodes. The new code is based on combining three ideas:

* Use a recursive implementation. * Require that all 3-nodes lean left. * Perform rotations on the way up the tree (after the recursive calls).

Not only do these ideas lead to simple code, but they also unify the algorithms: for example, the leftleaning versions of 2-3 trees and top-down 2-3-4 trees differ in the position of one line of code. All of the red-black tree algorithms that have been proposed are characterized by a worst-case search time bounded by a small constant multiple of lg N in a tree of N keys, and the behavior observed in practice is typically that same multiple faster than the worst-case bound, close the to optimal lg N nodes examined that would be observed in a perfectly balanced tree. This performance is also conjectured (but not yet proved) for trees built from random keys, for all the major variants of red-black trees. Can we analyze average-case performance with random keys for this new, simpler version? This paper describes experimental results that shed light on the fascinating dynamic behavior of the growth of these trees. Specifically, in a left-leaning red-black 2-3 tree built from N random keys:

* A random successful search examines lg N – 0.5 nodes. * The average tree height is about 2 ln N (!) * The average size of left subtree exhibits log-oscillating behavior.

The development of a mathematical model explaining this behavior for random keys remains one of the outstanding problems in the analysis of algorithms. From a practical standpoint, left-leaning red-black trees (LLRB trees) have a number of attractive characteristics:

* Experimental studies have not been able to distinguish these algorithms from optimal. * They can be implemented by adding just a few lines of code to standard BST algorithms. * Unlike hashing, they support ordered operations such as select, rank, and range search.

Thus, LLRB trees are useful for a broad variety of symbol-table applications and are prime candidates to serve as the basis for symbol tables in software libraries in the future.

October 14th, 2009: Finding Empty Cubes in Any Dimension

A&C Seminar

Speaker: Mark Keil

Abstract: Given a set of n points, inside a bounding box, in a d-dimensional Euclidean space, we seek to find the largest volume empty axis parallel cube. We also consider a bichromatic variant of this problem. Given a set of n red points and a set of n blue points in a d-dimensional Euclidean space, find an axis parallel cube that contains as many blue points as possible, but no red points. Combining our results with previous work we now have near linear time algorithms for both problems in two dimensions, polynomial time sweep-plane algorithms in any fixed dimensions, and we know that both problems are NP-complete if the dimension d is part of the input.

October 7th, 2009: The Quasi-G-Packing Problem

A&C Seminar

Speaker: Jazmin Romero

Abstract: We introduce the k-quasi-G-packing(t) problem as a formalization of the problem of finding communities in a network. In the k-quasi-G-packing(t) problem, we search for at least k communities with possible overlap, and we regulate this overlap through a parameter t, in contrast with previous work where communities are disjoint. In this paper, our focus is to study the parameterized complexity of the k-quasi-G-packing(t) problem. In particular, we provide a new technique for the k-quasi-G-packing(t) problem generalizing the crown decomposition technique . Using our global rule, we achieve a kernel with size bounded by 2(rk-r) for the k-quasi-G-packing(t) problem when t=r-2 and G=Kr.

September 30th, 2009: Improved Parameterized Algorithms for the Kemeny Aggregation Problem

A&C Seminar

Speaker: Narges Simjour

Abstract: We give improved fixed parameter tractable (FPT) algorithms to solve the Kemeny aggregation problem, where the task is to summarize a multi-set of preference lists, called votes, over a set of alternatives, called candidates, into a single preference list that has the minimum total tau-distance from the votes. The tau-distance between two preference lists is the number of pairs of candidates that are ordered differently in the two lists. We study the problem for preference lists that are total orders.

We develop algorithms of running times O*(1.403^kt ), O*(5.823^{kt/m}) \leq O*(5.823^kavg) and O*(4.829^kmax) for the problem, ignoring the polynomial factors in the O* notation, where kt is the optimum total tau-distance, m is the number of votes, and kavg (resp, kmax ) is the average (resp, maximum) over pairwise tau-distances of votes. Our algorithms improve the best previously known running times of O*(1.53^kt) and O*(16^kavg)\leq O*(16^kmax), which also implies an O*(16^{4kt/m}) running time. We also show how to enumerate all optimal solutions in O*(36^{kt/m})\leq O*(36^kavg) time.

This work was presented at IWPEC 2009.

September 16th, 2009: Small drawings of series-parallel graphs and other planar graphs

A&C Seminar

Speaker: Therese Biedl

Abstract: In this paper, we study small planar drawings of planar graphs. For arbitrary planar graphs, Theta(n^2) is the established upper and lower bound on the worst-case area. It is a long-standing open problem for what graphs smaller area can be achieved, with results known only for trees and outer-planar graphs. We show here that series-parallel can be drawn in O(n^{3/2}) area, but 2-outer-planar graphs and planar graphs of proper pathwidth 3 require Omega(n^2) area.

To be presented at Graph Drawing 2009.

September 16th, 2009: Drawing planar 3-trees with fixed face areas

A&C Seminar

Speaker: Elena Lesvia Velasquez Ruiz

Abstract: We study straight-line drawings of planar graphs such that each interior face has a prescribed area. It was known that such drawings exist for all planar graphs with maximum degree 3. We show here that such drawings exist for all planar partial 3-trees. Moreover, vertices have rational coordinates if the face-areas are rational, and we can bound the resolution. We also give some negative results for other graph classes.

Joint work with T. Biedl. To be presented at Graph Drawing 2009.

September 2nd, 2009: Five topics from CCCG 2009

A&C Seminar

Speaker: Robert Fraser

Abstract: In this talk I will give a high level overview of five results that were presented this year at CCCG 2009. The topics are diverse, but each is interesting. The papers:

Gary Miller, Todd Phillips and Don Sheehy. The Centervertex Theorem for Wedge Depth

Muriel Dulieu, Dania El-Khechen, John Iacono and Nikolaj van Omme. Packing 2x2 unit squares into grid polygons is NP-complete

Yoshio Okamoto and Ryuhei Uehara. How to make a picturesque maze

Prosenjit Bose, Luc Devroye, Maarten Loffler, Jack Snoeyink and Vishal Verma. The spanning ratio of the Delaunay triangulation is greater than pi/2

Tetsuo Asano and Gunter Rote. Constant-Working-Space Algorithms for Geometric Problems

August 12th, 2009: The Simultaneous Representation Problem for Chordal, Compatability and Permutation Graphs

A&C Seminar

Speaker: Krishnam Raju Jampani

Abstract: We introduce the simultaneous representation problem, defined for any graph class C characterized in terms of representations, e.g. any class of intersection graphs. Two graphs G1 and G2, sharing some vertices X (and the corresponding induced edges), are said to have a simultaneous representation with respect to a graph class C, if there exist representations R1 and R2 of G1 and G2 that are “consistent” on X. Equivalently (for the classes C that we consider) there exist edges E’ between G1-X and G2-X such that G1 U G2 U E’ belongs to class C.

Simultaneous representation problems are related to graph sandwich problems, probe graph recognition problems and simultaneous planar embeddings and have applications in any situation where it is desirable to consistently represent two related graphs.

We give efficient algorithms for the simultaneous representation problem on chordal, comparability and permutation graphs. These results complement the recent poly-time algorithms for recognizing probe graphs for the above classes and imply that the graph sandwich problem for these classes is solvable for an interesting special case: when the set of optional edges induce a complete bipartite graph. Moreover for comparability and permutation graphs, our results can be extended to solve a generalized version of the simultaneous representation problem when there are k graphs any two of which share a common vertex set X. This generalized version is equivalent to the graph sandwich problem when the set of optional edges induce a k-partite graph.

This is a joint work with Anna Lubiw.

August 5th, 2009: Self-Indexed Text Compression using Straight-Line Programs

A&C Seminar

Speaker: Francisco Claude

Abstract: Straight-line programs (SLPs) offer powerful text compression by representing a text T[1,u] in terms of a restricted context-free grammar of n rules, so that T can be recovered in O(u) time. However, the problem of operating the grammar in compressed form has not been studied much. We present a grammar representation whose size is of the same order of that of a plain SLP representation, and can answer other queries apart from expanding nonterminals. This can be of independent interest. We then extend it to achieve the first grammar representation able of extracting text substrings, and of searching the text for patterns, in time o(n). We also give byproducts on representing binary relations.

This is joint work with Gonzalo Navarro.

August 5th, 2009: Succinct Orthogonal Range Search Structures on a Grid with Applications to Text Indexing

A&C Seminar

Speaker: Meng He

Abstract: We present a succinct representation of a set of n points on an $n\times n$ grid using $n\lg n + o(n\lg n)$ bits to support orthogonal range counting in $O(\lg n /\lg\lg n)$ time, and range reporting in $O(k\lg n/\lg\lg n)$ time, where k is the size of the output. This achieves an improvement on query time by a factor of $\lg\lg n$ upon the previous result of M{\“a}kinen and Navarro, while using essentially the information-theoretic minimum space. Our data structure not only can be used as a key component in solutions to the general orthogonal range search problem to save storage cost, but also has applications in text indexing. In particular, we apply it to improve two previous space-efficient text indexes that support substring search and position-restricted substring search. We also use it to extend previous results on succinct representations of sequences of small integers, and to design succinct data structures supporting certain types of orthogonal range query in the plane.

July 29th, 2009: Finding a Hausdorff Core of a Polygon: On Convex Polygon Containment with Bounded Hausdorff Distance

A&C Seminar

Speaker: Robert Fraser

Abstract: Given a simple polygon P, we consider the problem of finding a convex polygon Q contained in P that minimizes H(P,Q), where H denotes the Hausdorff distance. We call such a polygon Q a Hausdorff core of P. We describe polynomial-time approximations for both the minimization and decision versions of the Hausdorff core problem.

July 22nd, 2009: Distributed Policing with Full Utilization and Rate Guarantees

Thesis Presentation

Speaker: Albert Choi

Abstract: A network service provider typically sells service at a fixed traffic rate to customers. This rate is enforced by allowing or dropping packets that pass through, in a process called policing. Distributed policing is a version of the problem where a number of policers must limit their combined traffic allowance to the specified rate. The policers must coordinate their behaviour such that customers are fully allowed the rate they pay for, without receiving too much more, while maintaining some semblance of fairness between packets arriving at one policer versus another.

We present a new global ``leaky bucket’’ approach that provably prevents starvation, guarantees full utilization, and provides a simple upper bound on the rate allowed under any incoming traffic pattern. We find that the algorithm guarantees a minimum 1/n share of the rate for each policer, and achieves close to max-min fairness in many, but not all cases. We also suggest some experimental modifications that could improve the fairness in practice.

July 8th, 2009: Reconstructing hv-convex polyominoes with multiple colours

Thesis Presentation

Speaker: Adam Bains

Abstract: This thesis examines the problem of reconstructing multiple discrete 2D objects, represented by a set of cells arranged in an m by n grid, from their projections. The objects being constructed are disjoint, hv-convex polyominoes, each of which consists of a separate colour. The main results presented here are two algorithms for unordered c-colour reconstruction that have time complexities of O(c^2n^{2c + 1}m^{2c + 1}) and O(c^2\min(n^{2c}, m^{2c})nm), an ordered c-colour reconstruction algorithm that is also O(c^2\min(n^{2c}, m^{2c})nm), and an NP-completeness proof when the number of colours is unbounded.

June 24th, 2009: Three New Algorithms for Regular Language Enumeration

A&C Seminar

Speaker: Margareta Ackerman

Abstract: We present new, efficient and elegant algorithms for the following problems: (1) The min-word problem: find the lexicographically minimal word of length n accepted by a given NFA (2) The cross-section problem: list all words of length n accepted by an NFA in lexicographical order (3) The enumeration problem: list the first m words accepted by an NFA according to length-lexicographic order For the min-word and cross-section problems, we present algorithms with better asymptotic running times than previously known algorithms. Additionally, for each problem, we present algorithms with better practical running times than previously known algorithms.

This is joint work with Erkki Makinen

June 17th, 2009: Optimal trees with clusters of stars constraints: characterizations and algorithms

A&C Seminar

Speaker: Michal Stern

Abstract: We consider the following optimal stars clustering tree problem, denoted by OS: Given a complete graph G=(V,E) with a weight on every edge and a collection S of subsets of V, the aim is to find a spanning tree T with minimum weight where each subset in S induces a star in T. We consider another version of the OS problem where the objective is to find in G a tree T such that each subset of the collection induces a star in T and the sum of all weights of the stars is minimum. We call this problem the median optimal stars clustering tree problem and denote it by MOS. We present, characterize and solve three incomparable cases of OS and MOS: (i) The complete stars case, for which we prove a structure theorem that lead to a polynomial time algorithm. (ii) The disjoint centers case, i.e. a node can be a center of at most one star, we solve by a polynomial time dynamic programming algorithm. (iii) The case in which there is no restrictions on the stars to be complete and on the centers to be disjoint, for which we give a polynomial time algorithm. For each case we solve where the intersection graph of the subsets is connected and also where the intersection graph is not necessarily connected. Applications to the problem come from the area of communication networks, from the area of database systems with synchronous replications and from the area of key management for secure group communication.

This work is a joint work with Ephraim Korach, Ben-Gurion University of the Negev, Israel.

June 11th, 2009: Communication Complexity and Its Applications

Joint A&C Seminar and Distinguished Lecture Series

Speaker: Andrew Yao, Tsinghua University

Abstract: For any function f(x, y), its communication complexity is the minimum number of bits needed to be exchanged between two parties holding integers x and y respectively. Invented thirty years ago, communication complexity has been a central research area in theoretical computer science with rich applications to algorithmic problems in data structure, circuit complexity, and cryptography. In this talk, we give an overview of communication complexity, high-lighting notable recent results and the diverse mathematical techniques needed to obtain these results.

May 21st, 2009: Hyperbolic dovetailing

A&C Seminar

Speaker: David Kirkpatrick

Abstract: A familiar quandary, in many settings (computational and otherwise), arises when there are several {\em possible} alternatives for the solution of a problem, but no way of knowing which, if any, are viable for a particular problem instance. Faced with this uncertainty, we are forced to simulate the parallel exploration of alternatives through some kind of co-ordinated interleaving ({\em dovetailing}) process. The goal, as usual, is to find a solution with low total cost. Much of the existing work on such problems has assumed, implicitly or explicitly, that at most one of the alternatives is viable. This assumption provides support for a competitive analysis of algorithms (using the cost of the unique viable alternative as a benchmark).

However, just as it is unrealistic to analyse algorithms in terms of the worst case cost of the alternative solutions or their worst-case ordering (giving rise to competitive analysis), it is also unrealistic in many scenarios to make the worst-case assumption that at most one of the alternatives is viable. In this paper, we relax this assumption in revisiting several familiar dovetailing problems.

Our main contribution is the introduction of a novel process interleaving technique, called {\em hyperbolic dovetailing} that achieves a competitive ratio that is within a logarithmic factor of optimal on {\em all} inputs in the worst, average and expected cases, over all possible deterministic (and randomized) dovetailing schemes. We also show that no other dovetailing strategy can guarantee an asymptotically smaller competitive ratio for all inputs.

An interesting application of hyperbolic dovetailing arises in the design of what we call {\em input-thrifty} algorithms, algorithms that are designed to minimize the total precision of the input requested in order to evaluate some given predicate. We show that for some very basic predicates involving real numbers (such as certifying that the numbers are not all identical) we can use hyperbolic dovetailing to provide input-thrifty algorithms that are competitive, in this novel cost measure, with the best algorithms that solve these problems.

May 6th, 2009: Random Hyperplane Search Trees

A&C Seminar

Speaker: James King

Abstract: A random hyperplane search tree is a binary search tree structure for storing d-dimensional Euclidean point sets; I define their construction below. After giving an overview of random binary search trees and geometric split trees, I will discuss techniques used to analyze the height and depth of such random trees, then apply these techniques to random hyperplane search trees.

A random hyperplane search tree is constructed as follows. From our input set S of n points in general position, d are chosen uniformly at random to be stored in the root. These d points define a hyperplane that partitions the remaining n-d points into two disjoint sets S_1 and S_2. The left and right subtrees of the root are defined recursively as random hyperplane search trees constructed from S_1 and S_2 respectively. A set of fewer than d points is stored in a single external node.

April 15th, 2009: The application of the in-tree knapsack problem to routing prefix caches

A&C Seminar

Speaker: Patrick Nicholson

Abstract: Modern routers use specialized hardware, such as Ternary Content Addressable Memory (TCAM), to solve the Longest Matching Prefix Problem (LMPP) quickly. Due to the fact that TCAM is a non-standard type of memory and inherently parallel, there are concerns about its cost and power consumption. This problem is exacerbated by the growth in routing tables, which demands ever larger TCAMs.

To reduce the size of the TCAMs in a distributed forwarding environment, a batch caching model is proposed and analysed. The problem of determining which routing prefixes to store in the TCAMs reduces to the In-Tree Knapsack Problem (ITKP) for unit weight vertices in this model.

Several algorithms are analysed for solving the ITKP, both in the general case and when the problem is restricted to unit weight vertices. Additionally, a variant of the ITKP is proposed and analysed, which exploits the caching model to provide better solutions. This thesis concludes with discussion of open problems and future experimental work.

April 1st, 2009: A hierarchy between P and RP

A&C Seminar

Speaker: Periklis Papakonstantinou

Abstract: How does a randomized algorithm use its randomness? Instead of the common approach which focuses on the number of random bits an algorithm uses, we propose an orthogonal concept by defining the “essential use of randomness”, i.e. we quantify on the number of times a randomized algorithm accesses its random bits. Building on (Cook'71) we observe that logarithmic space-bounded Turing Machines, equipped with an unbounded stack and an external poly-long random tape characterizes RP, BPP etc, depending on the error condition. Parameterizing on the number of head-reversals on the random stream (tape) we define a hierarchy of classes between P and probabilistic polytime. This definition provides a possible framework for understanding how algorithms make use of their randomness.

In this talk I’ll focus on two contributions of [1].

We introduce the model of Randomness Compilers, and we show that it is equivalent to SMs with limited access to randomness. In this model a polytime algorithm on a given input x, it compiles all the randomness it needs in a polysize, swallow circuit. Parametrizing on the depth of this circuit we characterize the levels of our hierarchy. A perhaps surprising consequence of the above characterization is that derandomizing RNC (a class believed to be strictly inside P) implies derandomization along our hierarchy (where even the first level contains P).

Another main contribution is an unconditional lower bound on the use of randomness by a large family of algorithms for Polynomial Identity Testing, or PIT. The majority of PIT algorithms proceed by evaluating the given polynomial at the provided random points (Schwartz-Zippel-like algorithms). We show a streaming lower-bound using a NIH communication complexity reduction, which implies that every such algorithm makes polynomially many reversals on the provided randomness.

This is joint work with Matei David, and Anastasios Sidiropoulos.

[1] Polynomial Time with Restricted Use of Randomness Matei David, Periklis Papakonstantinou, Anastasios Sidiropoulos

March 25th, 2009: Optimal Scheduling of Contract Algorithms with Soft Deadlines

A&C Seminar

Speaker: Angele Hamel

Abstract: A contract algorithm is an algorithm which is given, as part of its input, a specified amount of allowable computation time. In contrast, interruptible algorithms may be interrupted throughout their execution, at which point they must report their current solution. Simulating interruptible algorithms by means of schedules of executions of contract algorithms in parallel processors is a well-studied problem with significant applications in AI.

In the classical case, the interruptions are hard deadlines in which a solution must be reported immediately at the time the interruption occurs. In this paper we study the more general setting of scheduling contract algorithms at the presence of soft deadlines. This is motivated by the observation of practitioners that soft deadlines are as common an occurrence as hard deadlines, if not more common. In our setting, at the time t of interruption the algorithm is given an additional window of time w(t) <= ct to continue the contract or, indeed, start a new contract (for some fixed constant c). We explore this variation using the acceleration ratio, which is the canonical measure of performance for these schedules, and derive schedules of optimal acceleration ratio for all functions w.

This is joint work with Spyros Angelopoulos and Alex Lopez-Ortiz.

March 18th, 2009: Agnostic Online Learning

A&C Seminar

Speaker: David Pal

Abstract: We generalize the Littlestone’s online learning model to the agnostic setting where no hypothesis makes zero error. Littlestone defined a combinatorial parameter of the hypothesis class, which we call Littlestone’s dimension and which determines the worst-case number of prediction mistakes made by an online learning algorithm in the realizable setting. Point of our paper is that Littlestone’s dimension characterizes learnability in the agnostic case as well. Namely, we give upper and lower bounds on the so called regret (i.e. #mistakes of the learner - #mistakes of the best hypothesis) in terms of the Littlestone’s dimension.

This is a similar story to what happened to the Valiant’s PAC model. The key quantity there is Vapnik-Chervonekis dimension. Valiant’s PAC model is the ``realizable case’’. Our work can be paralleled to what Haussler and others did in 1992 when they generalized the PAC model to the agnostic setting and showed that Vapnik-Chervonekis dimension remains the key quantity characterizing learnability there as well.

joint work with Shai Ben-David and Shai Shalev-Shwartz from TTI-Chicago

March 11th, 2009: Towards Universal Succinct Representations of Trees

A&C Seminar

Speaker: Arash Farzan

Abstract: A succinct representation of a combinatorial object is a highly space-efficient representation on the RAM with logarithmic word size that supports dynamic queries in constant time. In this talk, we mainly focus on succinct representation of trees which have found an increasing number of applications in indexing massive collections of textual and semi-structured data.

In the first part of the talk, we present a new approach in succinct representation of trees which unifies and enhances the power of pre-existing succinct representations of trees for various families of trees (such as ordered and k-ary trees).

In the second part of the talk, we discuss some related topics in succinct representations. We first sketch a dynamic version of our succinct representation of trees, where the encoded tree is changing by insertion and deletion of nodes. We conclude by exploring the topic of space lower bounds for succinct encodings.

This is joint work with J.Ian Munro, Rajeev Raman, and S.Srinivasa Rao.

February 25th, 2009: Decision Problems For Convex Languages

A&C Seminar

Speaker: Zhi Xu

Abstract: We examine decision problems for various classes of convex languages, previously studied by Ang and Brzozowski under the name ``continuous languages’’. We can decide whether a language $L$ is prefix-, suffix-, factor-, or subword-convex in polynomial time if $L$ is represented by a DFA, but the problem is PSPACE-hard if $L$ is represented by an NFA. If a regular language is not convex, we prove tight upper bounds on the length of the shortest words demonstrating this fact, in terms of the number of states of an accepting DFA. Similar results are proved for some subclasses of convex languages: the prefix-, suffix-, factor-, and subword-closed languages, and the prefix-, suffix-, factor-, and subword-free languages.

This paper is to be in LATA09 and a joint work with Janusz Brzozowski and Jeffrey Shallit.

February 11th, 2009: Adaptive (Analysis of) Algorithms for Convex Hulls and Related Problems

A&C Seminar

Speaker: Jeremy Barbay

Abstract: Adaptive analysis is a well known technique in computational geometry, which refines the traditional worst case analysis over all instances of fixed input size by taking into account some other parameters, such as the size of the output in the case of output sensitive analysis. We present two adaptive techniques for the computation of the convex hull in two and three dimensions and of related problems. The first analysis technique is based on the *input order* and yields results on the computation of convex hulls in two and three dimensions, and the first adaptive algorithms for Voronoi and Delaunay diagrams, through the entropy of a partition of the input in easier instances. The second analysis technique is based on the *structural entropy* of the instance, and yields results on the computational complexity of planar convex hull and of multiset sorting, through a generalization of output sensitivity and a more precise analysis of the complexity of Kirkpatrick and Seidel’s algorithm. Our approach yields adaptive algorithms which perform faster on many classes of instances, while performing asymptotically no worse in the worst case over all instances of fixed size.

January 14th, 2009: Market Equilibrium and Convex Programs (with two agents)

A&C Seminar

Speaker: Deeparnab Chakrabarty

Abstract: In the Fisher model of markets, a single seller owns all the divisible items and buyers come to the market with moneys with utility functions for the items. A pricing and allocation of the items is said to be in equilibrium if three conditions are satisfied: 1) All items are sold completely (no excess supply) 2) All buyers spend all their money (no excess demand) 3) Each buyer gets the “best bundle” at the specified prices (agent rationality). In 1959, Eisenberg and Gale showed that in the case when the utility functions are linear the equilibrium can be captured as a solution to a convex program thus converting the equilibrium question into a optimization question.

Recently, Jain and Vazirani (STOC ‘07) proposed a study of all markets whose equilibrium is captured via similar convex programs. They called these markets EG markets and showed a lot of markets studied in the Fisher model fall under this classification. Such an abstract way of investigating markets leads to a general approach to study many properties of the market in a unified fashion.

In this talk, after introducing EG markets in some more detail, I would present the result that any EG market with two agents is rational, that is the equilibrium prices and allocations are rational numbers as long as the parameters of the market are rational. Such a property is not true when one takes 3 agents. One could draw a parallel with the theory of Nash equilibrium in games where $2$ player games always have rational equilibria while there exist $3$ player games with irrational ones. (That said, the result on games doesn’t imply our result in any way)

Moreover, we will also describe a polynomial time algorithm to compute the equilibrium prices and allocations. Furthermore in the case of many interesting markets (which we will talk about in the talk), the algorithm would run in strongly polynomial time - that is, it would not depend on, say, the number of bits required to encode the moneys of buyers. We should remark that EG markets by definition have a polynomial (and not strongly) time approximation for the equilibrium – a convex program captures the equilibria; thus our algorithms can be thought of as strongly polynomial time solvers of a certain class of convex programs.

This is joint work with Nikhil R. Devanur (Microsoft Research, Redmond) and Vijay V. Vazirani (Georgia Tech). A draft can be found at http://www.math.uwaterloo.ca/~deepc/RESEARCH/REPORTS/EG2.pdf

January 9th, 2009: Spanning trees with O(1) average stretch factor

A&C Seminar

Speaker: Michiel Smid

Abstract: Let G be a connected graph in which each edge has a weight, and let T be a spanning tree of G. The stretch of two vertices p and q is the ratio of the the distance between p and q in T and the distance between p and q in G. In SODA 2007, Abraham, Bartal and Neiman showed that a spanning tree T exists such that the average stretch (over all (n choose 2) vertex pairs) is bounded by a constant.

I will present a simple proof of this result for the case when G is the complete graph on a set of points in the plane, and the edge weights represent Euclidean distances.

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2008

December 17th, 2008: Showing Relevant Ads via Context Multi-Armed Bandits

A&C Seminar

Speaker: David Pal

Abstract: We study context multi-armed bandit problems where the context comes from a metric space and the payoff satisfies a Lipschitz condition with respect to the metric. Abstractly, a context multi-armed bandit problem models a situation where, in a sequence of independent trials, an online algorithm chooses an action based on a given context (side information) from a set of possible actions so as to maximize the total payoff of the chosen actions. The payoff depends on both the action chosen and the context. In contrast, context-free multi-armed bandit problems, a focus of much previous research, model situations where no side information is available and the payoff depends only on the action chosen.

Our problem is motivated by sponsored web search, where the task is to display ads to a user of an Internet search engine based on her search query (context) so as to maximize the click-through rate of the ads displayed. We cast this problem as a context multi-armed bandit problem where queries and ads form metric spaces and the payoff function is Lipschitz with respect to both the metrics. For any $\epsilon > 0$ we present an algorithm with regret $O(T^{\frac{a+b+1}{a+b+2} + \epsilon})$ where $a,b$ are the covering dimensions of the query space and the ad space respectively. We prove a lower bound $\Omega(T^{\frac{\tilde{a}+\tilde{b}+1}{\tilde{a}+\tilde{b}+2} - \epsilon})$ for the regret of any algorithm where $\tilde{a}, \tilde{b}$ are packing dimensions of the query spaces and the ad space respectively. For finite spaces or bounded subsets of Euclidean spaces, this gives an (almost) matching upper and lower bound.

joint work with Tyler Lu and Martin Pal

December 10th, 2008: Shortest Paths Avoiding Forbidden Subpaths

A&C Seminar

Speaker: Mustaq Ahmed

Abstract: In this talk we study a variant of the shortest path problem in graphs: given a weighted graph G and vertices s and t, and given a set X of forbidden paths in G, find a shortest s-t path P such that no path in X is a subpath of P. Path P is allowed to repeat vertices and edges. We call each path in X an exception, and our desired path a shortest exception-avoiding path. We formulate a new version of the problem where the algorithm has no a priori knowledge of X, and finds out about an exception x in X only when a path containing x fails. This situation arises in computing shortest paths in optical networks. We give an algorithm that finds a shortest exception avoiding path in time polynomial in |G| and |X|. The main idea is to run Dijkstra’s algorithm incrementally after replicating vertices when an exception is discovered.

This is a joint work with Anna Lubiw.

December 3rd, 2008: Bounding the Locality of Distributed Routing Algorithms

A&C Seminar

Speaker: Stephane Durocher

Abstract: We examine theoretical bounds on the locality of routing. A local routing algorithm A at a network node v receives a packet P and selects one of its neighbours to which to forward P using only local information. Specifically, in addition to knowing the node for which P is destined, algorithm A may also know the node from which P originated, the neighbour from which node v received P, and the graph corresponding to all nodes within k hops of node v. Our objective is to determine which of these parameters are necessary and/or sufficient to permit local routing as k varies, where the network is modelled by an undirected graph. In particular, we establish tight bounds on k for the feasibility of deterministic k-local routing for various combinations of these parameters. Although motivated by applications in networks, our results involve combinatorial and graph-theoretic arguments.

This is joint work with Prosenjit Bose and Paz Carmi.

November 27th, 2008: Bit-Optimal Lempel-Ziv Compression

A&C Seminar

Speaker: Igor Nitto

Abstract: This talk will present results about the Lempel and Ziv compression scheme. In particular,we propose an efficient solution to the problem of computing a bit-optimal Lempel-Ziv parsing of a string. In the Lempel and Ziv compression scheme, an input text is compressed by replacing repeated substrings with “codewords”, where each codeword encodes a reference to a previous occurence of a repeated substring. In order to locate repetitions, the input text has to be partitioned into phrases, which can be either repeated substrings or single chars. This task, called “parsing”, is fundamental since it determines the size of the compressed text. The problem of bit-optimally parsing a text asks to parse a text minimizing the total size in bits of the encoded codewords, thus achieving the maximum compression. We will show that, for a general class of variable-length codeword encodings typically used in data compression, the problem above admits a time and space efficient solution.

Joint work with Paolo Ferragina and Rossano Venturini (University of Pisa)

November 26th, 2008: Multi-pass geometric algorithms

A&C Seminar

Speaker: Eric Chen

Abstract: We initiate the study of exact geometric algorithms that require limited storage and make only a small number of passes over the input. Fundamental problems such as low-dimensional linear programming and convex hulls are considered. This is a joint work with Timothy Chan.

November 19th, 2008: Computational challenges and opportunities in RNA secondary structure prediction

A&C Seminar

Speaker: Anne Condon

Abstract: DNA and RNA molecules have proven to be very versatile materials. Structures formed by RNA molecules play important regulatory and catalytic roles in the cell, and show promise in therapeutics. Molecular engineers can now design and realize nano-scale structures and sensors, and even simple machines with moving parts, built from DNA. Function follows form in the molecular world, and so our ability to understand nucleic acid function in the cell, as well as to design novel structures, is enhanced by reliable means for structure prediction.

Computational work on RNA structure prediction has focused on secondary structure - the set of base pairs that form when the molecule folds on itself. Algorithms which predict RNA and DNA secondary structure from the base sequence typically rely on models of the physical (thermodynamic) aspects of molecular folding. In this talk, we will describe some algorithmic and inference problems that arise in DNA/RNA secondary structure prediction from the base sequence, progress in solving these problems, and directions for future research. No prior biological background will be assumed for the talk. Biography: Anne Condon is a Professor of Computer Science at U. British Columbia and Associate Dean in the Faculty of Science. Much of her current research focuses on computational prediction of nucleic acid structure, with applications to design of novel structures and gene synthesis. Her research contributions span computational complexity theory, hardware verification, biomolecular computation, and combinatorial auctions. She has received an ACM Distinguished Dissertation Award, NSF National Young Investigator Award, and University College Cork Distinguished Alumna Award for her work, and currently holds the NSERC/GM Canada Chair. Anne received her B.Sc. degree (1982) from University College Cork, Ireland and Ph.D. (1987) from U. Washington. Prior to her position at UBC, she was a faculty member at U. Wisconsin from 1987-1999.

November 19th, 2008: Practical Approximation Improvements for Segment Minimization in Intensity-Modulated Radiation Therapy

A&C Seminar

Speaker: Maxwell Young

Abstract: We devise three algorithms for segment minimization that yield constant factor approximation improvements over previous results. Central to these algorithms is our proof that the single row segmentation problem is fixed parameter tractable in the largest value in the row. Additionally, we demonstrate a construction for translating any solution to a $1\times{}n$ intensity matrix into an improved solution for a full $m\times{}n$ intensity matrix. Therefore, any future improvements in approximating the $1\times{}n$ intensity matrix lead immediately to an improved approximation algorithm for the $m\times{}n$ problem. Finally, a performance evaluation of several approximation algorithms is presented using real world and synthetic data sets. Our experiments indicate that the improved approximation results translate into significant performance gains in practice.

This is joint work with Therese Biedl, Stephane Durocher, Holger Hoos, Shuang Luan, and Jared Saia.

November 5th, 2008: Coloring perfect graphs by contraction

A&C Seminar

Speaker: Benjamin Leveque

Abstract: Many applied problems can be modelised by the vertex coloring problem of a graph, which is NP-complete in general but polynomial on the class of perfect graphs introduced by Berge. The coloring algorithm of perfect graphs, due to Grotschel, Lovasz and Schrijver, is not really efficient in practice and it is still interesting to find a purely combinatorial algorithm to color perfect graphs in polynomial time. We give several simple and fast algorithms that enable to color some subclasses of perfect graphs. These algorithms use in particular the notion of even pair contraction, introduced by Fonlupt and Uhry, about which several conjectures are still open. We also use search algorithms like LexBFS, due to Rose, Tarjan and Lueker, to prove some structural results on the considered graphs.

October 29th, 2008: Shortest Anisotropic Paths with Few Bends is NP-complete

A&C Seminar

Speaker: Mustaq Ahmed

Abstract: In the Shortest Anisotropic Path problem, the goal is to minimize the weighted length of a path on a triangulated terrain, where the weight of a path segment ab depends both on the face containing ab and the direction of ab. The problem is a generalization of the Weighted Region Problem. It is known that a shortest anisotropic path can have many bends in the interior of a face, and there can be infinitely many shortest anisotropic paths even between two points in a common face. In practice, however, not all of these shortest paths are equally good. For example, for robot motion planning, a shortest anisotropic path with fewer bends is preferable because a robot needs extra time and energy to change its direction at a bend. It is therefore natural to look for a shortest anisotropic path with a limited number of bends. We show using a reduction from 3-SAT that the problem is NP-complete.

This is a joint work with Anna Lubiw, and will be presented in FWCG 2008.

October 24th, 2008: Well-quasi-ordering tournaments and Rao’s degree-sequence conjecture

Joint A&C Seminar and Tutte Colloquium

Speaker: Paul Seymour

Abstract: Rao conjectured about 1980 that in every infinite set of degree sequences (of graphs), there are two degree sequences with graphs one of which is an induced subgraph of the other. In the last month or so we seem to have found a proof, and we sketch the main ideas. The problem turns out to be related to ordering digraphs by immersion (vertices are mapped to vertices, and edges to edge-disjoint directed paths). Immersion is not a well-quasi-order for the set of all digraphs, but for certain restricted sets (for instance, the set of tournaments) we prove it is a well-quasi-order. The connection between Rao’s conjecture and tournament immersion is as follows. One key lemma reduces Rao’s conjecture to proving the same assertion for degree sequences of split graphs (a split graph is a graph whose vertex set is the union of a clique and a stable set); and to handle split graphs it helps to encode the split graph as a directed complete bipartite graph, and to replace Rao’s containment relation with immersion.

Joint work with Maria Chudnovsky (Columbia University).

October 17th, 2008: How easily can we find our way without a map?

A&C Seminar

Speaker: Prosenjit Bose

Abstract: Computing a path between two vertices in a graph is a fundamental problem that has been widely studied in the literature within various contexts and settings. There are many different algorithms for computing such paths depending on the type of graph as well as the type of path one is searching for. In this talk, we review some results on what can be said about computing paths when the underlying graph is a geometric graph (i.e. vertices are points and edges are segments) and one only has local information (such as knowledge of the vertices adjacent to the current position and location of the destination). We highlight some of the things that can and cannot be done based on the properties known about the underlying graph.

October 8th, 2008: Cliques in Edge Intersection Graphs of Subtrees of a Tree

A&C Seminar

Speaker: Elad Cohen

Abstract: Let $\phi$ be a set of subtrees of a tree T . We call $G(\phi) = (V, E)$ the edge in- tersection graph of $\phi$ if for each member of $\phi$ a vertex $v\in V$ is assigned and two vertices are adjacent if and only if the corresponding subtrees intersect in at least one edge. The set $\phi$ is called the representation of $G(\phi)$ . It is known, by the Helly property, that any clique in an edge intersection graph of subtrees of a tree is representable on a substar of the hosting tree. We ask what is the maximum degree of a star which represents a clique in an edge intersection graph of subtrees of a tree, where the maximum degree of the subtrees is bounded by some interger s. This problem is related to the following definition: A Semi Projective Plane of order k (SPP(k)), is a pair, B = (P, L), of a set of Points P and a set of subsets of points, called Lines, which satisfy three axioms:

1. The cardinality of a largest line is k + 1. 2. Every two lines intersect in at least one point. 3. For every $x\in P$ , there exists $A_x,B_x\in L$, such that $A_x\cap B_x = {x}$.

We show that the problem mentioned above is equivalent to the problem of finding the maximum number of points in an SPP(k). We prove that the maximum number of points in an SPP(2) is seven and we give a lower bound for the problem. In addition, we show an upper bound for |P| based on a result of Bollobas (1965).

October 3rd, 2008: On the Complexity of Game and Market Equilibria

Joint A&C Seminar and Tutte Colloquium

Speaker: Shang-Hua Teng

Abstract: I will present some recent results in Algorithmic Game Theory and particularly in computing and approximating game and market equilibria. As you may have already known, the notion of the Nash equilibrium has captured the imagination of much of the computer science theory community, both for its many applications in the growing domain of online interactions and for its deep and fundamental mathematical structures. As the scale of typical internet applications increases, the problems of efficiently analyzing their game-theoretic properties become more pointed. I will discuss the recent results in settling several open questions about Nash equilibria. I will focus on the approximation and smoothed complexity of equilibrium computation in noncooperative two-player games. I will also address the extensions of these results to other equilibrium problems such as in trading and market economies.

Joint work conducted with Xi Chen (IAS, Princeton), Xiaotie Deng (The City University of Hong Kong); also with Li-Sha Huang (Google China), Paul Valiant (MIT), and Kyle Burke (Boston University).

October 1st, 2008: Succinct Representations of Arbitrary Graphs

A&C Seminar

Speaker: Arash Farzan

Abstract: We consider the problem of encoding a graph with $n$ vertices and $m$ edges compactly supporting adjacency, neighborhood and degree queries in constant time in the log(n)-bit word RAM model. The adjacency query asks whether there is an edge between two vertices, the neighborhood query reports the neighbors of a given vertex in constant time per neighbor, and the degree query reports the number of incident edges to a given vertex.

We study the problem in the context of succinctness, where the goal is to achieve the optimal space requirement as a function of $n$ and $m$, to within lower order terms. We prove a lower bound in the cell probe model that it is impossible to achieve the information-theory lower bound within lower order terms unless the graph is too sparse (namely $m=o(n^\delta)$ for any constant $\delta > 0$) or too dense (namely $m = o(n^{2-\delta})$ for any constant $\delta > 0$).

Furthermore, we present a succinct encoding for graphs for all values of $n,m$ supporting queries in constant time. The space requirement of the representation is always within a multiplicative $1+\epsilon$ factor of the information-theory lower bound for any arbitrarily small constant $\epsilon > 0$. This is the best achievable space bound according to our lower bound where it applies. The space requirement of the representation achieves the information-theory lower bound tightly to within lower order terms when the graph is sparse ($m=o(n^\delta)$ for any constant $\delta > 0$).

This is a joint work with Ian Munro.

September 24th, 2008: Approximating Geometric Steiner Forest

A&C Seminar

Speaker: Glencora Borradaile

Abstract: Given a set of pairs of points in space, what is the minimum-length network that connects each pair of points? This is the geometric Steiner forest problem, also known as the generalized Steiner tree problem. In graphs and high-dimension Euclidean space, the problem admits no polynomial-time approximation scheme (PTAS) and the best known approximation is 2. For a fixed epsilon, we give an O(n^2 log n)-time algorithm that finds a (1+epsilon)-approximation for instances in the Euclidean plane. In this talk, I will review a PTAS due to Arora for the geometric Steiner tree problem. I will illustrate the major pitfalls that prevent direct application of this PTAS to the Steiner forest problem and present the techniques we use to overcome these hurdles.

September 23rd, 2008: A Ramble of Recent Results

Joint A&C Seminar and C&O Seminar

Speaker: David Pritchard

Abstract: This will be an extremely informal talk in which I describe some of results from the recent ALGO 2008 and the upcoming SODA 2009 that seem interesting. I will try to present a couple of open problems. Most, but not all of the results, will have to do with approximation, network design, or combinatorial optimization. Interval scheduling played a large role at ALGO and I will briefly show the elegant 2-approximation by Fung, Poon and Zheng for online unit-length interval scheduling.

September 22nd, 2008: Optimal Halfspace Range Reporting in Three Dimensions

A&C Seminar

Speaker: Peyman Afshani

Abstract: We give the first optimal solution to a standard problem in computational geometry: three-dimensional halfspace range reporting. We show that $n$ points in 3-d can be stored in a linear-space data structure so that all $k$ points inside a query halfspace can be reported in $O(\log n+k)$ time. The data structure can be built in $O(n\log n)$ expected time. The previous methods with optimal query time requires superlinear ($O(n\log\log n)$) space.

We also mention consequences, for example, to higher dimensions and to external-memory data structures. As an aside, we partially answer another open question concerning the crossing number in Matou\v sek’s {\em shallow partition theorem\/} in the 3-d case (a tool used in many known halfspace range reporting methods).

This is a joint work with Timothy Chan and will be presented in SODA 2009 conference.

September 18th, 2008: The Steiner ratio for the obstacle-avoiding Steiner tree problem

Thesis Presentation

Speaker: Mena Razaghpour

Abstract: The Steiner ratio measures the performance of several approximation algorithms for the Steiner tree problem. It is defined as the minimum possible ratio of the length of a minimum Steiner tree to the length of a minimum spanning tree for the same set of terminals. In this talk, we survey the background literature on the Steiner ratio, and discuss how to generalize it to the case of obstacles. We intoduce the obstacle-avoiding Steiner ratio and prove that it is equal to the classical Steiner ratio for the rectilinear metric.

September 17th, 2008: The Bad Santa Problem and Energy-Efficient Reliable Broadcast for Radio Networks

A&C Seminar

Speaker: Maxwell Young

Abstract: We address the problem of minimizing power consumption when broadcasting a message from one node to all the other nodes in a radio network. To enable power savings for such a problem, we introduce a compelling new data streaming problem that we call the {\it Bad Santa problem}. Our results on this problem apply for any situation where: 1) a node can listen to a set of $n$ nodes, out of which at least half are non-faulty and know the correct message; and 2) each of these $n$ nodes sends according to some previously known predetermined schedule which assigns each of them its own unique time slot. In this situation, we show that in order to receive the correct message with probability 1, it is necessary and sufficient for the listening node to listen to an $O(\sqrt{n})$ expected number of time slots. Moreover, if we allow for repetitions of transmissions so that each sending node sends the message $O(\log^{*} n)$ times (i.e. in $O(\log^* n)$ rounds each consisting of the $n$ time slots), then listening to $O(\log^{*} n)$ expected number of time slots suffices. We show that this is near optimal.

We describe an application of our result to the popular grid model for a radio network. Each node in the network is located on a point in a two dimensional grid, and whenever a node sends a message, all awake nodes within $L_{\infty}$ distance $r$ receive the message. In this model, up to $t<\frac{r}{2}(2r+1)$ nodes within any $2r+1$ by $2r+1$ square in the grid can suffer Byzantine faults. Moreover, we assume that the nodes thatsuffer Byzantine faults are chosen and controlled by an adaptive adversary that knows everything except for the random bits of each non-faulty node. This type of adaptive adversary models worst-case behavior due to malicious attacks on the network; mobile nodes moving around in the network; or static nodes losing power or ceasing to function. Let $n=r(2r+1)$. We show how to solve the broadcast problem in this model with each node awake only an expected $O(1/\sqrt{n})$ fraction of the time. Moreover, if we allow each node to send $O(\log^{*} n)$ times, we can increase the energy savings so that each node is awake only an expected $O((\log^{*}n)/ n)$ fraction of the time. This compares favorably with previous protocols that required each node to be awake for every time step.

This is joint work with Valerie King, Cynthia Phillips and Jared Saia. Our paper will be presented at the Twenty-Seventh Annual ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing (PODC 2008).

September 12th, 2008: Combinatorial reasoning in Information Theory

Joint A&C Seminar and C&O Seminar

Speaker: Noga Alon

Abstract: Combinatorial arguments have played a crucial role in the investigation of several surprising phenomena in Information Theory. After a a brief discussion of some of these results I will describe a recent example, based on joint work with Lubetzky and Stav, and with Weinstein, in which properties of graph powers, colorings of Cayley graphs, and the chromatic numbers of Kneser graphs are applied in the study of a broadcasting problem with side information.

September 12th, 2008: Integral Multicommodity Flow in Trees

Joint A&C Seminar and C&O Seminar

Speaker: David Pritchard

Abstract: The maximum flow problem is a well-known combinatorial optimization problem that can be solved efficiently. However, generalizing to multiple noninterchangeable source-sink pairs makes the problem NP-complete, even in tree graphs; hence the best we can hope for is an efficient approximation algorithm. We will show how a recent technique, iterated rounding of linear programs, leads to an improved approximation guarantee for the problem.

The talk is based on joint work with Jochen Konemann and Ojas Parekh.

September 10th, 2008: On Dominance Reporting in 3D

A&C Seminar

Speaker: Peyman Afshani

Abstract: In this paper, we study the 3D dominance reporting problem in different models of computations and offer optimal results in the pointer machine and the external memory models and a near optimal result in the RAM model; all our results consume linear space.

These improve the results of various papers, such as Makris and Tsakalidis (IPL'98), Vengroff and Vitter (STOC'96) and Nekrich (SOCG'07). With a $\log^3 n$ fold increase in the space complexity these can be turned into orthogonal range reporting algorithms with matching query times, improving the previous orthogonal range searching results in the external memory and RAM models. Using our 3D results as base cases, we can provide improved orthogonal range reporting algorithms in higher dimensions as well.

This work will be presented in ESA 2008 conference.

September 3rd, 2008: On set cover in geometric settings

A&C Seminar

Speaker: Sariel Har-Peled

Abstract: Speaker: Sariel Har-Peled

In this talk, we will survey some of the known results on weighted set cover in geometric settings, and how the known approximation algorithms work. In the second part of the talk, we will discuss some recent results, on the weighted set cover for weighted halfplanes, and fat shapes in the plane.

Sariel Har-Peled is a faculty in the CS department in UIUC working on geometric approximation algorithms.

August 21st, 2008: Algorithms for Optimizing Search Schedules in a Polygon

Thesis Presentation

Speaker: Stephen Bahun

Abstract: In the area of motion planning, considerable work has been done on guarding problems, where guards, modelled as points, must guard a polygonal space from intruders. Different variants of this problem involve varying a number of factors. The guards performing the search may vary in terms of their number, their mobility, and their range of vision. The model of intruders may or may not allow them to move. The polygon being searched may have a specified starting point, a specified ending point, or neither of these. The typical question asked about one of these problems is whether or not certain polygons can be searched under a particular guarding paradigm defined by the types of guards and intruders.

In this presentation, we focus on two cases of a chain of guards searching a room (polygon with a specific starting point) for mobile intruders. The intruders must never be allowed to escape through the door undetected. In the case of the two guard problem, the guards must start at the door point and move in opposite directions along the boundary of the polygon, never crossing the door point. At all times, the guards must be able to see each other. The search is complete once both guards occupy the same spot elsewhere on the polygon. In the case of a chain of three guards, consecutive guards in the chain must always be visible. Again, the search starts at the door point, and the outer guards of the chain must move from the door in opposite directions. These outer guards must always remain on the boundary of the polygon. The search is complete once the chain lies entirely on a portion of the polygon boundary not containing the door point.

Determining whether a polygon can be searched is a problem in the area of visibility in polygons; further to that, our work is related to the area of planning algorithms. We look for ways to find optimal schedules that minimize the distance or time required to complete the search. This is done by finding shortest paths in visibility diagrams that indicate valid positions for the guards. In the case of the two-guard room search, we are able to find the shortest distance schedule and the quickest schedule. The shortest distance schedule is found in O(n2) time by solving an L_1 shortest path problem among curved obstacles in two dimensions. The quickest search schedule is found in O(n4) time by solving an L_1 shortest path problem among curved obstacles in two dimensions. For the chain of three guards, a search schedule minimizing the total distance travelled by the outer guards is found in O(n6) time by solving an L_1 shortest path problem among curved obstacles in two dimensions.

August 20th, 2008: Generating Variant Sudoku Puzzles

A&C Seminar

Speaker: Matthew Skala

Abstract: Sudoku is a case of the exact-cover problem set as a puzzle for human solution. Scientific work on it has focused on traditional algorithmic questions like the complexity class of solving an instance; or combinatorial questions like how many solution grids exist. We introduce novel variant rule sets for Sudoku-like puzzles, and describe the implementation of a puzzle generator for multiple rule sets, based on a general exact-cover solver in the CHR constraint programming system. Along the way we consider some issues specific to the human solvers, such as the difference between easy and hard or boring and fun instances.

August 6th, 2008: Adaptive Searching in One and Two Dimensions

A&C Seminar

Speaker: Reza Dorrigiv

Abstract: Searching in a geometric space is an active area of research, predating computer technology. Online geometric search algorithms are usually analyzed using competitive analysis. In this paper we apply ideas from adaptive analysis to study two elementary problems in online search: search on the real line and looking around a corner. We show that small changes in the model lead to results that are more consistent with real life applications.

This is a joint work with Alejandro Lopez-Ortiz and will be presented in CCCG 2008.

August 6th, 2008: Maintaining Coresets in Data Streams

A&C Seminar

Speaker: Hamid Zarrabi-Zadeh

Abstract: In this talk, I will define a class of algorithms for constructing coresets of (geometric) data sets, and show that algorithms in this class can be dynamized efficiently in the insertion-only (data stream) model.

As an application of this framework, I will show that epsilon-coresets for the k-center problem in fixed dimensions (in both additive and multiplicative forms) can be maintained using a data structure whose size is independent of the size of the input. This independency to the input size is of particular interest as the input size in data streams is typically huge. Our framework also leads to faster streaming algorithms for some basic geometric problems such as diameter and minimum enclosing ball in fixed dimensions.

This work will be presented at CCCG 2008.

August 5th, 2008: Lower Bounds and Derandomization

Thesis Presentation

Speaker: Jaffer Gardezi

Abstract: A major open problem in complexity is to determine whether randomized complexity classes such as BPP, MA, and AM have any nontrivial derandomization. This talk will discuss derandomization of two randomized versions of the polynomial hierarchy.

August 5th, 2008: Pattern Matching with Springs: an Adaptive Analysis

A&C Seminar

Speaker: Jeremy Barbay

Abstract: We propose an adaptive algorithm for context queries (queries expressed as preorder and ancestor- descendant relations on labeled nodes), which can be used to find patterns in XML documents. Our algorithm takes advantage of the correlation between terms of the query without any preprocessed information, and it runs in time (kd(lg lg min(n,s)+lg lg(r))) in the RAM model, where k is the number of terms in the query, d is the non-deterministic complexity of the query on the multi-labeled tree (i.e. the minimum number of operations required to check the answer to the query), n is the number of nodes in the tree, s is the number of relations between nodes and labels, and r is the maximal number of nodes matching a label on any rooted path in the tree.

July 30th, 2008: The Monoid Frobenius Problem and the De Bruijn Graph

A&C Seminar

Speaker: Zhi Xu

Abstract: The de Bruijn graph is a special graph that has various applications in combinatorics. Eulerian and Hamiltonian circuits always exist in a de Bruijn graph, and each represents a de Bruijn sequence, which is the circular sequence of length $|\Sigma|^n$ such that every word of length $n$ occurs exactly once.

The monoid Frobenius problem is a generalization of the Frobenius problem. Given $k$ words over an alphabet $\Sigma$, the monoid Frobenius problem is to find the length of the longest word(s) that cannot be written as a concatenation of the given words, if it is finite.

The de Bruijn graph and the monoid Frobenius problem are strongly related in the sense that the word graph $G(m,n)$ for a subproblem of the latter is a generalization of the former. I will present an upper bound for the number of edges of the maximum acyclic subgraph of $G(m,n)$, namely, $|\Sigma|^n-\Omega(|\Sigma|^n/n)$, providing a linear upper bound in terms of the input length for the subproblem and proving that the subproblem is in P. In $G(m,n)$, both Eulerian circuits and Hamiltonian circuits exist, which leads to examples that achieve a bound of $\Omega(|Sigma|^n)$ for the monoid Frobenius problem.

July 23rd, 2008: Coresets and Geometric Approximation

A&C Seminar

Speaker: Hamid Zarrabi-Zadeh

Abstract: The coreset framework has recently attracted considerable attention as a powerful tool for approximating various measures of a geometric data set. In this framework, a small subset of the input, called a coreset, is extracted in such a way that solving the optimization problem on the coreset yields an approximate solution to the entire set.

In this talk, I will give a brief survey on the coreset technique and its application in geometric approximation. In particular, I will consider the problem of constructing geometric coresets in the data stream model (where points arrive one at a time), and present a new streaming algorithm that maintains an epsilon-coreset under the extent measure in near optimal space. This immediately improves the best previous streaming algorithms for a number of fundamental geometric optimization problems in fixed dimensions, including width, minimum enclosing cylinder, minimum-width enclosing annulus, etc.

This work will be presented at ESA 2008.

July 18th, 2008: On Approximating Four Covering and Packing Problems with Applications to Bioinformatics

Joint A&C Seminar and Bioinformatics Seminar

Speaker: Bhaskar DasGupta, University of Illinois at Chicago

July 16th, 2008: Morphing Planar Graph Drawings with Bent Edges

A&C Seminar

Speaker: Mark Petrick

Abstract: A morph is the continuous transformation of one object or configuration to another, often with the requirement of preserving some geometric structure. This seminar is about morphs of graph drawings. More specifically, we will look at morphs on planar graph drawings that preserve planarity throughout the morph.

We give an algorithm to morph between two planar drawings of a graph, preserving planarity, but allowing edges to bend during the course of the morph. The morph uses a polynomial number of elementary steps, where each elementary step is a linear morph that moves each vertex in a straight line at uniform speed. Although there are planarity-preserving morphs that do not require edge bends, it is an open problem to find polynomial-size morphs. We achieve polynomial size at the expense of edge bends.

This is joint work with Anna Lubiw.

July 16th, 2008: Error Detection in Number-Theoretic and Algebraic Algorithms

A&C Seminar

Speaker: Troy Vasiga

Abstract: Computers currently are, and always have been, unreliable. Whether “computers” were humans calculating arithmetic formulae or modern-day electronic devices, mistakes occur during the execution of algorithms. In this talk, we analyze various number-theoretic and algebraic algorithms under a specific error model and quantify the robustness of these algorithms.

July 9th, 2008: Online Steiner Trees in Directed Graphs

A&C Seminar

Speaker: Spyros Angelopoulos

Abstract: The Steiner Tree problem in graphs is one of the fundamental problems in combinatorial optimization: given an edge-weighted graph and a specified subset of its vertices, we seek the minimum-cost subgraph that spans the subset in question. Most of the existing work on Steiner trees is focused on undirected graphs; however, in several applications (e.g., algorithms over the internet) the underlying graph is in fact directed.

In this talk I will present some recent work on the online version of the problem. Namely, requests appear in a sequence, and we want to maintain a Steiner arborescence of small cost. The analysis appeals to the concept of the edge-asymmetry of a graph, defined as the maximum ratio of the weights of antiparallel edges in the graph. I will also outline some related variations of the Steiner tree problem which, likewise, apply in the on-line setting.

June 27th, 2008: Scarf’s Lemma and the Stable Paths Problem

Joint A&C Seminar and Tutte Colloquium

Speaker: Penny Haxell

Abstract: We address a question in graphs called the stable paths problem, which is an abstraction of a network routing problem concerning the Border Gateway Protocol (BGP). The main tool we use is Scarf’s Lemma. This talk will describe Scarf’s Lemma and how it is related to other results more familiar to combinatorialists, and then will explain its implications for the stable paths problem.

June 25th, 2008: Design and Analysis of Algorithms for Multicore Architectures

A&C Seminar

Speaker: Alejandro Salinger

Abstract: Over the last five years, major microprocessor manufacturers have released plans for a rapidly increasing number of cores per microprossesor, with upwards of 64 cores by 2015. In this setting, a sequential RAM computer will no longer accurately reflect the architecture on which algorithms are being executed. In this work we propose a model of low degree parallelism (LoPRAM) which builds upon the RAM and PRAM models yet better reflects recent advances in parallel (multi-core) architectures. This model supports a high level of abstraction that simplifies the design and analysis of parallel programs. More importantly we show that in many instances it naturally leads to work-optimal parallel algorithms via simple modifications to sequential algorithms.

This is joint work with Alex Lopez-Ortiz and Reza Dorrigiv.

June 11th, 2008: Geometric k-Centres of Unit Disc Graphs

A&C Seminar

Speaker: Stephane Durocher

Abstract: Motivated by the gateway placement problem in wireless networks, we consider the geometric k-centre problem on unit disc graphs: given a set of points P in the plane, find a set F of k points in the plane that minimizes the maximum graph distance from any vertex in P to the nearest vertex in F in the unit disc graph induced by P union F. We describe exact and approximate polynomial-time solutions to this problem for any fixed k and show that the problem is NP-hard when k is an arbitrary input parameter.

This work is joint with Raju Krishnam Jampani, Anna Lubiw, and Lata Narayanan.

June 4th, 2008: A Subproblem of the Monoid Frobenius Problem

A&C Seminar

Speaker: Zhi Xu

Abstract: The monoid Frobenius problem is a generalization of the classical Frobenius Problem on integers. Given $k$ words on alphabet $\Sigma$, if there are only finitely many words that cannot be written as a concatenation of given words, the monoid Frobenius problem is to find the longest such word(s) or the length of the longest such word(s).

An interesting subproblem of the monoid Frobenius problem comes from the restriction that the given words are only of two different lengths. One (tight) upper bound of this subproblem will be discussed. I will present two equivalent problems and a polynomial-time algorithm, which is based on a word graph for given words.

May 29th, 2008: Repetitions in Strings

A&C Seminar

Speaker: Lucian Ilie

Abstract: Repetitions in strings constitute a basic area of string combinatorics with important applications to text algorithms and data compression. While 100 years ago Axel Thue was initiating stringology by constructing infinite strings without squares (here is square: hotshots = (hots)^2), in the past decades the effort was focused on finding efficient algorithms for computing all repetitions in a string. This was finally achieved by Kolpakov and Kucherov (FOCS'99) who used previous techniques to construct an algorithm that computes all maximal repetitions (that is, repetitions that cannot be extended – they encode all repetitions) in time proportional to the size of the output. The major advance was a proof that the number of runs in a string of length n is at most cn, c a constant. Unexpectedly, no bound on the value of c followed from the proof and significant subsequent work concentrated on this problem. Based on numerical computations, it has been conjectured that c<=1.

Besides the obvious mathematical importance of the question, good bounds on c have direct implications to the complexity analysis of algorithms computing all repetitions in strings.

I will give an overview of the history and latest results on this problem and related ones, concerning both enumeration and computation of various types of repetitions. In particular, the constant c above has been recently shown to be included in the interval [0.948 .. 1.048], which essentially settles the practical aspect of the conjecture.

May 28th, 2008: Aspects of Metric Spaces in Computation

A&C Seminar

Speaker: Matthew Skala

May 28th, 2008: Locality and Location Awareness

A&C Seminar

Speaker: Evangelos Kranakis

Abstract: The nature of ad hoc networking imposes an additional requirement that algorithms should be local in the sense that messages propagate only a constant number of hops and each host makes decisions based solely on information obtained from hosts located a constant (independent of the size of the network) number of steps away from it. We will discuss the interaction between local computation and global communication in such systems and give an overview of methods that are more robust for dynamically changing ad hoc networks, including routing, topology control, channel allocation and clustering.

May 7th, 2008: Continuous Languages

A&C Seminar

Speaker: John Brzozowski

Abstract: A language is prefix-continuous if it satisfies the condition that, if a word w and its prefix u are in the language, then so is every prefix of w that has u as a prefix. Prefix-continuous languages include prefix-closed languages at one end of the spectrum, and prefix-free languages, which include prefix codes, at the other. In a similar way, we define suffix-, bifix-, factor-, and subword-continuous languages and their closed and free counterparts. We generalize these notions to arbitrary binary relations on the set of all words over a finite alphabet. This provides a common framework for diverse languages such as codes, factorial languages and ideals. We examine the relationships among these languages and their closure properties.

April 25th, 2008: Unsolved Problems in Graph Theory

Joint A&C Seminar and Tutte Colloquium

Speaker: U. S. R. Murty

Abstract: A book entitled “Graph Theory”, co-authored by myself and Dr. J.A. Bondy, has recently been published by Springer. It has an Appendix on unsolved problems. The aim of this talk is to present a selection of problems from that appendix.

April 25th, 2008: The Algorithmic Lens: How the Computational Perspective is Transforming the Sciences

Joint A&C Seminar and Distinguished Lecture Series

Speaker: Christos Papadimitriou, University of California, Berkeley

Abstract: Computational research transforms the sciences (physical, mathematical, life or social) not just by empowering them analytically, but mainly by providing a novel and powerful perspective which often leads to unforeseen insights. Examples abound: quantum computation provides the right forum for questioning and testing some of the most basic tenets of quantum physics, while statistical mechanics has found in the efficiency of randomized algorithms a powerful metaphor for phase transitions. In mathematics, the P vs. NP problem has joined the list of the most profound and consequential problems, and in economics considerations of computational complexity revise predictions of economic behavior and affect the design of economic mechanisms such as auctions. Finally, in biology some of the most fundamental problems, such as understanding the brain and evolution, can be productively recast in computational terms. My talk is structured around eight vignettes exemplifying this pattern.

Biography: Christos H. Papadimitriou is C. Lester Hogan Professor of Computer Science at UC Berkeley. Before joining Berkeley in 1996 he taught at Harvard, MIT, Athens Polytechnic, Stanford, and UCSD. He has written five textbooks and many research articles on algorithms and complexity, and their applications to optimization, databases, AI, economics, and the Internet. He holds a PhD from Princeton, and honorary doctorates from ETH (Zurich), the University of Macedonia, and the University of Athens. He is a member of the American Academy of Arts and Sciences and of the National Academy of Engineering, and a fellow of the ACM. His novel .“Turing”. was published by MIT Press in 2003, and his graphic novel “Logicomix” (with Apostolos Doxiadis) will be published by Bloomsbury in 2008.

April 16th, 2008: General Auction Mechanism for Search Advertising

A&C Seminar

Speaker: David Pal

Abstract: speaker: David Pal co-authors: Gagan Aggarwal, S. Muthukrishnan, Martin Pal

(This is the second part of the talk given by David Pal on March 12, 2008).

Internet advertising is often sold by the way of an online automated auction. Variants of the Generalized Second Price (GSP) mechanism are heavily used by major search engines. An adaptation of the Vickerey-Clarke-Groves (VCG) framework has been studied as a truth-revealing alternative to GSP. Extensions to these mechanisms have been proposed to account for things like position constraints and various bidder objectives.

Our central thesis is that each of the above auction mechanisms can be viewed as computing a “bidder-optimal stable matching” with suitably defined preferences of the auction participants. The theory of stable matchings has been pioneered by Gale and Shapley and has became an indispensable tool in studying two-sided matching markets. We define a simple model of preference relations of the bidders (and the auctioneer) in which each of the existing auctions can be expressed. In our model, we prove the existence and uniqueness (under mild assumptions) of bidder-optimal stable matchings, and show how to compute them in polynomial $O(nk^3)$ time.

Our machinery allows us to define a wide variety of generalizations of the existing auction mechanisms. In particular, it allows us to design combined mechanisms for diverse pools of bidders, some of whom prefer to specify (say) a maximum bid per impression (like in GSP), while others want to specify their value of a click. By adopting existing results on stable matchings, we are able to show that the combined mechanisms are truth revealing as well.

April 9th, 2008: I/O-Efficient Algorithms for Computing Contour Lines on a Terrain

A&C Seminar

Speaker: Bardia Sadri

Abstract: A terrain Mˆ’ is the graph of a bivariate function. We assume that Mˆ’ is represented as a triangulated surface with n vertices. A “contour” of Mˆ’ is a connected component of a level set of Mˆ’. Generically, each contour is a closed polygonal curve; at €œcritical levels these curves may touch each other. We present I/O-efficient algorithms for the following two problems related to computing contours of Mˆ’:

Given two real parameters h and d > 0, we present an I/O-optimal algorithm that report all contours of Mˆ’ at heights h + kd, for every positive integer k, using O(Sort(N)+T/B) I/Os, where T is the total number edges in the output contours, B is the “€œblock size”,€ and Sort(N) is the number of I/Os needed to sort N elements. The algorithm uses O(N/B) disk blocks. Each contour is generated individually with its composing segments sorted in clockwise order.

We can preprocess Mˆ’, using O(Sort(N)) I/Os, into a linear-size data structure so that all contours at a given height can be reported using O(log_B N + T/B) I/Os, where T is the output size. Each contour is generated individually with its composing segments sorted in clockwise order.

This is joint work with Lars Arge, Pankaj K. Agarwal, and Thomas Moelhave.

April 4th, 2008: Approximation algorithms for (two) discrete stochastic optimization problems

Joint A&C Seminar and Tutte Colloquium

Speaker: David Shmoys

Abstract: One of the most active areas of research in the design of approximation algorithms is for discrete {\it stochastic} optimization problems, with a particular focus on 2-stage problems with recourse. Here, we are given a probability distribution over inputs, and the aim is to find a feasible solution that minimizes the expected cost of the solution found (with respect to the input distribution); an approximation algorithm finds a solution that is guaranteed to be nearly optimal. Techniques initially developed in the context of deterministic approximation, including rounding approaches, primal-dual algorithms, and randomization, have proved to be important in this context as well. We will focus on two specific examples, the {\it a priori} traveling salesman problem and a 2-stage stochastic single-machine scheduling problem, and for each, we will give an algorithm that is guaranteed to find a solution within a constant factor of optimal.

This is joint work with Mauro Sozio and Kunal Talwar.

April 2nd, 2008: Noisy sorting without resampling

A&C Seminar

Speaker: Mark Braverman

Abstract: Sorting using pairwise comparisons is one of the most basic algorithmic problems. An important variant of the problem is sorting in the presence of noise. We consider a setting where one is allowed to query every pair of elements (a,b), and receive the correct order relationship with probability p>0.5, and the incorrect relationship with probability 1-p, and where resampling of the same pair is not allowed.

Such a situation arises, for example, when ranking items using an expert algorithm. The expert algorithm is more likely than not to be correct on each given pair, but running the same algorithm again and again on the same pair provides no new information. The goal is to find the ordering that is as consistent with the noisy queries as possible (in other words, the maximum likelihood order). We prove a structure theorem relating the optimal solution to the original ordering, and use it to give an efficient algorithm for the Noisy Sorting Without Resampling problem.

Joint work with Elchanan Mossel

April 1st, 2008: A Combinatorial Characterization of Decoy Sets

Joint A&C Seminar and C&O Seminar

Speaker: Christina Boucher

Abstract: Joint work with Dan Brown and Stephane Durocher

The motif recognition problem is the task of discovering similar subsequences in a given set of sequences without knowledge of the shared pattern or its position within a sequence. Central to motif recognition is the Consensus String problem; for a given set of sequences $S$ and a degeneracy parameter $d$, the Consensus String problem asks whether there exists a sequence that has Hamming distance at most $d$ from each sequence in $S$. A valid motif set is a set of sequences for which such a consensus sequence exits, while a decoy set is a set of sequences which does not have a consensus sequence but whose pairwise Hamming distances are all at most $2d$. At present, no efficient solution is known to the Consensus String problem; current solutions involve exhaustive enumeration of candidate consensus sequences to distinguish between decoy sets and valid motif sets. In this talk, I will present a combinatorial decomposition theorem of decoy sets over binary alphabets. This is the first such characterization, and will be undoubtedly be useful in developing new efficient motif-finding algorithms for the Consensus String problem and its numerous variants.

March 26th, 2008: State-Complexity Hierarchies of Uniform Languages of Alphabet-Size Length

A&C Seminar

Speaker: John Brzozowski

Abstract: We study the state complexity of a special class of simple languages. If A is an alphabet of k letters, then a k-language is a nonempty set of words of length k, that is, a uniform language of length k. We show that every k-language of maximal state complexity is also a uniform language of length k of maximal state complexity. Moreover, we prove that, for every i between the minimal and the maximal state complex- ities, there is a language of complexity i. The proof is constructive: for each i we exhibit a language of complexity i. We introduce a family of “pi automata” accepting languages whose words are permutations of the alphabet. The complexities of these languages form a complete hierarchy between k^2 − k + 3 and 2^k +1. We construct another family of automata, based on k-ary trees, whose languages define a complete hierarchy of complexities between 2^k +1 and the maximal complexity.

March 19th, 2008: Fast Distributed Computation of Cuts via Random Circulations

A&C Seminar

Speaker: David Pritchard

Abstract: In the model of distributed computing in a graph, a network wants to compute properties of its own unknown topology. Nodes initially know only their own ID and their neighbours’ IDs. In this talk, we study the problem of distributively identifying all cut vertices, cut edges, and cut-edge-pairs of the network. Following the standard model, every message has to be O(log V) bits long; this limits the power of the model but makes things much more interesting and the algorithms more practical.

We give a unifying approach to compute these “small cuts” using the notion of a random circulation. Roughly speaking, a circulation is an assignment of oriented values (modulo k) to each edge so that at each node, the sum in equals the sum out. Our first observation is that it is easy to uniformly sample from the universe of all modulo k circulations. This leads to simple and efficient sequential and distributed algorithms for cut vertices, cut edges, and cut-edge-pairs; in particular, we eliminate the need for a depth-first search and obtain the first O(Diameter)-time distributed algorithms for cut edges and cut-edge-pairs. Under certain natural conditions, we show that O(Diameter) time is optimal for these problems on every graph.

March 12th, 2008: General Auction Mechanism for Search Advertising

A&C Seminar

Speaker: David Pal

Abstract: speaker: David Pal co-authors: Gagan Aggarwal, S. Muthukrishnan, Martin Pal

Internet advertising is often sold by the way of an online automated auction. Variants of the Generalized Second Price (GSP) mechanism are heavily used by major search engines. An adaptation of the Vickerey-Clarke-Groves (VCG) framework has been studied as a truth-revealing alternative to GSP. Extensions to these mechanisms have been proposed to account for things like position constraints and various bidder objectives.

Our central thesis is that each of the above auction mechanisms can be viewed as computing a “bidder-optimal stable matching” with suitably defined preferences of the auction participants. The theory of stable matchings has been pioneered by Gale and Shapley and has became an indispensable tool in studying two-sided matching markets. We define a simple model of preference relations of the bidders (and the auctioneer) in which each of the existing auctions can be expressed. In our model, we prove the existence and uniqueness (under mild assumptions) of bidder-optimal stable matchings, and show how to compute them in polynomial $O(nk^3)$ time.

Our machinery allows us to define a wide variety of generalizations of the existing auction mechanisms. In particular, it allows us to design combined mechanisms for diverse pools of bidders, some of whom prefer to specify (say) a maximum bid per impression (like in GSP), while others want to specify their value of a click. By adopting existing results on stable matchings, we are able to show that the combined mechanisms are truth revealing as well.

March 4th, 2008: Disjoint paths. II. Easy disjoint path problems

Joint A&C Seminar and C&O Seminar

Speaker: Jim Geelen

Abstract: This is one in a sequence of four talks on some algorithmic aspects of the graph minors project of Robertson and Seymour. This talk is mostly self-contained, so it does not matter if you missed the first one.

These talks were initially given at the ADONET-CIRM Summer School on Graphs and Algorithms, Trento, Italy, 2007.

February 26th, 2008: Disjoint paths. I. Structural results

Joint A&C Seminar and C&O Seminar

Speaker: Jim Geelen

Abstract: This is one in a sequence of four talks on some algorithmic aspects of the graph minors project of Robertson and Seymour. In particular, we give an overview of the $k$-disjoint paths problem. In this talk we discuss the graph minors structure theorem.

These talks were initially given at the ADONET-CIRM Summer School on Graphs and Algorithms, Trento, Italy, 2007.

February 20th, 2008: Title not recorded

A&C Seminar

Speaker: Glencora Borradaile

Abstract: We present a framework for designing polynomial-time approximation schemes for network design problems such as Steiner tree and 2-edge connectivity in planar graphs. For a fixed epsilon, a polynomial-time approximation scheme finds, in polynomial time, a solution whose value is within 1+epsilon of the optimal solution.

In this talk I will overview the framework and discuss how to use it to solve a variety of connectivity problems.

February 6th, 2008: On the Complexity of Reverse Similarity Search

A&C Seminar

Speaker: Matthew Skala

Abstract: Two decision problems are presented that arise from reversing the operation of a distance-based indexing tree. Whereas similarity search finds a leaf in the tree given a query point, reverse similarity search begins with a leaf and finds a point that would be stored in that leaf. The problems are analysed for spaces of strings and vectors with a variety of metrics: strings with Hamming distance; the usual (Levenshtein) edit distance; an edit distance we introduce called Superghost distance; arbitrary weighted tree metrics; and real vectors with Minkowski $L_p$ metrics (of which the Euclidean distance is a special case). They are found to inhabit different complexity classes depending on the metric. In particular, the reverse similarity search problem derived from a VP- or GH-tree is NP-complete for any $L_p$ metric except that it is in P for a GH-tree with the Euclidean metric.

January 30th, 2008: Title not recorded

A&C Seminar

Speaker: Guy Blelloch

Abstract: In this talk I’ll describe a data-structure that supports dynamic orthogonal range reporting, orthogonal intersection reporting and point-location for horizontal lines. For all problems the structure requires O(n) space, O(log n) time for the first result of a query and O(log n) amortized time for updates. The approach we use is based on “short” pointers to represent the links between layers in a fractional-cascading structure.

January 29th, 2008: A Six-Pack of SODA Nuggets

A&C Seminar

Speaker: David Pritchard, SODA 2008

Abstract: The Symposium on Discrete Algorithms (SODA) is an annual conference whose 2008 instantiation recently took place. In this informal talk, I’ll give a flavour of six results from SODA ‘08. The combo includes theorems, open problems, true things needing better proofs, and true things appearing to be false. The tentative topics are:

- how far you can overhang n books off the side of a table - greedy routing in planar graphs - counting hamilton cycles in 3-regular graphs* - why shuffling cards is the same as adding numbers** - a max-min generalization of Edmonds’ disjoint branchings theorem - using LPs to improve a geometric bound from sqrt(2) to sqrt(2)-0.0001

January 21st, 2008: Title not recorded

A&C Seminar

Speaker: Gwenael Richomme

Abstract: TITLE Quasiperiodic Sturmian and Episturmian Words

SPEAKER Gwenael Richomme, University of Picardie Jules Verne, France

January 10th, 2008: Meshing in Fixed Dimension in Near Optimal Work and Time

A&C Seminar

Speaker: Gary L. Miller

Abstract: A new meshing algorithm will be presented for meshing with boundaries in any fixed dimension, Sparse Voronoi Refinement (SVR). The meshing problem in 3D, for example, takes as input a a domain and a collection of features(points, edges, and faces) and decomposes the domain into tetrahedra. There are four important properties that a meshing algorithm should have: 1) The tetrahedra should have good aspect ratio, no small angles. 2) The mesh should conform to the features. 3) The size should be competitive to an optimal-size mesh. 4) The algorithm should be work and time competitive with a optimal algorithm. SVR is the first algorithm known to have all four properties even in 3D for a reasonable assumption about the input.

Over the last 17 years computer scientists have been in the forefront in designing algorithms with guarantees for all four conditions, beginning with the pioneering work of Bern, Eppstein, and Gilbert on quadtree meshing in 1990. Their algorithm has all 4 guarantees for 2D points where the work is O(n log L/s + m). Here L/s is the ratio of the size of the domain over the smallest input feature. In 1993 Ruppert proposed a method called Delaunay Refinement which included guarantees for the first three conditions in 2D.

The 3D octtree algorithms starts by insuring 1) always and finishes by insuring 2), while Delaunay Refinement algorithms first insure 2) then refine until 1) is satisfied. SVR can be viewed as a compromise by alternately insuring 1) and then 2). SVR has sequential-time/work bounds of O(n log L/s + m) for inputs in any fixed dimension with piecewise-linear constraining (PLC) features. The parallel time is O(log L/s log m)$ on an EREW PRAM, with the same work. SVR is straightforward enough that it is likely to be extremely fast in practice.

This represent joint work with Benoit Hudson and Todd Phillips

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2007

December 12th, 2007: On Routing with Guaranteed Delivery in Three-Dimensional Ad Hoc Wireless Networks

A&C Seminar

Speaker: Stephane Durocher

Abstract: We study routing algorithms for three-dimensional ad hoc networks that guarantee delivery and are $k$-local, i.e., each intermediate node $v$’s routing decision only depends on knowledge of the labels of the source and destination nodes, of the subgraph induced by nodes within distance $k$ of $v$, and of the neighbour of $v$ from which the message was received. We model a three-dimensional ad hoc network by a unit ball graph, where nodes are points in $R^3$, and nodes $u$ and $v$ are joined by an edge if and only if the distance between $u$ and $v$ is at most one.

The question of whether there is a simple local routing algorithm that guarantees delivery in unit ball graphs has been open for some time. In this paper, we answer this question in the negative: we show that for any fixed $k$, there can be no $k$-local routing algorithm that guarantees delivery on all unit ball graphs. This result is in contrast with the two-dimensional case, where $1$-local routing algorithms that guarantee delivery are known. Specifically, we show that guaranteed delivery is possible if the nodes of the unit ball graph are contained in a slab of thickness $1/\sqrt{2}$. However, there is no $k$-local routing algorithm that guarantees delivery for the class of unit ball graphs contained in thicker slabs, i.e., slabs of thickness $1/\sqrt{2} + \epsilon$ for some $\epsilon > 0$. The algorithm for routing in thin slabs derives from a transformation of unit ball graphs contained in thin slabs into quasi unit disc graphs, which yields a $2$-local routing algorithm. We also show several results that further elaborate on the relationship between these two classes of graphs.

This work is joint with David Kirkpatrick and Lata Narayanan.

December 5th, 2007: Geometric Streaming Algorithms with a Sorting Primitive

A&C Seminar

Speaker: Eric Chen

Abstract: We solve several fundamental geometric problems under a new streaming model recently proposed by Ruhl et al. In this model, in one pass the input stream can be scanned to generate an output stream or be sorted based on a user-defined comparator; all intermediate streams must be of size $O(n)$. We obtain the following geometric results for any fixed constant $\epsilon > 0$:

We can construct 2D convex hulls in $O(1)$ passes with $O(n^{\epsilon})$ extra space. We can construct 3D convex hulls in $O(1)$ expected number of passes with $O(n^{\epsilon})$ extra space. We can construct a triangulation of a simple polygon in $O(1)$ expected number of passes with $O(n^{\epsilon})$ extra space, where $n$ is the number of vertices on the polygon. We can report all $k$ intersections of a set of 2D line segments in $O(1)$ passes with $O(n^\epsilon)$ extra space, if an intermediate stream of size $O(n+k)$ is allowed.

We also consider a weaker model, where we do not have the sorting primitive but are allowed to choose a scan direction for every scan pass. Here we can construct a 2D convex hull from an $x$-ordered point set in $O(1)$ passes with $O(n^\epsilon)$ extra space.

November 30th, 2007: The Frobenius Problem in a Free Monoid

A&C Seminar

Speaker: Zhi Xu

Abstract: The classical Frobenius problem is to compute the largest integer $g$ not representable as non-negative integer linear combination of $x_1,x_2,\ldots,x_k$, where $x_1,x_2,\ldots,x_k$ are positive integers with $\gcd(x_1, x_2, \ldots, x_k)=1$. We generalize this problem to the non-commutative setting of a free monoid, where $S=\{u_1,u_2,\ldots,u_k\}$ is a set of words over a finite alphabet $\Sigma$ such that $S^*$, the set of all words factorizable into elements of $S$, is co-finite. We use techniques from automata theory and formal language theory to discuss the length of the longest word not in $S^*$.

Unlike the commutative case, where the bound on $g(x_1,x_2,\ldots,x_k)$ is quadratic, we are able to prove that the length of the longest word not in $S^*$ is bounded above by $$\frac{2(2^n|\Sigma|^n-1)}{2|\Sigma|-1},$$ where $n=\max_{1\leq i\leq k}|u_i|$. Furthermore, we are able to show a tight (exponential) lower bound for the worst case of the form $$g(m,m|\Sigma|^{n-m}+n-m)$$ in the case where $S\subseteq\Sigma^m\cup\Sigma^n$, where $0<2m$.

We obtain upper and lower bounds for other generalizations of the Frobenius problem, such as the state complexity of $S^*$, and we also obtain results on the total number of words not in $S^*$, generalizing an 1884 result of Sylvester.

November 21st, 2007: On the relative dominance of paging algorithms

A&C Seminar

Speaker: Reza Dorrigiv

Abstract: In this talk we give a finer separation of several known paging algorithms. This is accomplished using a new technique called relative interval analysis. This technique compares the fault rate of two paging algorithms across the entire range of inputs of a given size rather than in the worst case alone. Using this technique we characterize the relative performance of LRU and LRU-2, as well as LRU and FWF, among others. We also show that lookahead is beneficial for a paging algorithm, a fact that is well known in practice but it was, until recently, not verified by theory.

This work is joint with Alejandro Lopez-Ortiz and Ian Munro.

November 14th, 2007: Approximation Algorithms for Minimizing Segments in Radiation Therapy

A&C Seminar

Speaker: Maxwell Young

Abstract: Intensity modulated radiation therapy is one of the most effective modalities for modern cancer treatment. The key to successful treatment hinges on the delivery of a two-dimensional discrete radiation intensity matrix using a device called a multileaf collimator (MLC). Mathematically, the delivery of an intensity matrix using an MLC can be viewed as the problem of representing a non-negative integral matrix (i.e. the intensity matrix) by a linear combination of certain special non-negative integral matrices called segments, where each such segment corresponds to one of the allowed states of the MLC. The problem of representing the intensity matrix with the minimum number of segments is known to be NP-complete. Minimizing the number of segments reduces treatment time which can minimize machine wear and tear, increase patient throughput and thus reduce the cost of the procedure.

In this paper, we present two approximation algorithms for this matrix representation problem. To the best of our knowledge, these are the first algorithms to achieve non-trivial performance guarantees for multi-row intensity matrices. This is joint work with Shuang Luan and Jared Saia from the University of New Mexico.

October 31st, 2007: Properties of Shortest Descending Paths

A&C Seminar

Speaker: Mustaq Ahmed

Abstract: A path from s to t on a polyhedral terrain is descending if the height of a point p never increases while we move p along the path from s to t. No efficient algorithm is known to find an exact shortest descending path (SDP) from s to t in a general polyhedral terrain, although approximation algorithms for general terrains and exact algorithms for special classes of terrains have been devised. The crucial property that permits efficient shortest path algorithms on polyhedral surfaces is that a locally shortest (or geodesic) path unfolds to a straight line. Thus it is easy to: (1) extend a locally shortest path into the next face; and (2) find a shortest path from s to t through a given face sequence. Locally shortest descending paths (LSDPs) do not unfold to straight lines, and problems (1) and (2) become non-trivial. In this talk we will solve problem (1) efficiently by characterizing the bends in LSDPs. In particular, we will prove that the bend angles follow a generalized form of Snell’s law of refraction of light. We will also show that problem (2) is the only barrier to applying the approach of Chen and Han to find SDPs in polynomial time.

October 20th, 2007: Streaming Algorithms for Selection and Approximate Sorting

Joint A&C Seminar and CS Seminar

Speaker: Richard Karp, University of California, Berkeley

Abstract: Companies such as Yahoo, Google and Microsoft maintain extremely large data repositories within which searches are frequently conducted. There is interest in developing such repositories in the public sector and applying them to massive data set problems of importance to the scientific community and society in general.

It is of interest to develop streaming algorithms for basic information processing tasks within such data repositories. We present such algorithms for selecting the keys of given ranks in a totally ordered set of n keys, and for a related problem of approximate sorting. We derive bounds on the storage and time requirements of these algorithms under the assumption that blocks of keys arrive in a random order, and show that these bounds are close to information theoretic lower bounds for the problems. We assume random arrivals because these repositories support random access to disc blocks. The alpha-quantile of a totally ordered set of n keys is the (alpha n)th smallest element. We present near-optimal algorithms (simultaneously for time and storage), under the random arrivals assumption, for the following problems:

1. Selection: compute the alpha-quantile for a given alpha;.

2. Multiple selection: compute alpha-quantiles for many given values of alpha;.

3. Parallel selection, in which the input is divided into streams, each with its own buffer, and the different streams communicate by message passing.

4. Approximate selection: given alpha and epsilon;, find a key whose rank differs from (alpha n) by at most (epsilon n).

5. Approximate sorting: Given a small positive constant epsilon;, compute an ordering of the keys in which the rank assigned to each key agrees with its rank in the true ordering, within a relative error of epsilon.

Finally, as a byproduct of our analysis of approximate sorting, we give an elegant method for computing the expected number of comparisons for some classical randomized algorithms for selection and selection.

Note: Dr Karp will be receiving an honourary D Math from the University of Waterloo in the afternoon convocation of the same day

BIO:

One of the world’s most eminent scholars in theoretical computer science, Richard Karp received a PhD in Mathematics from Harvard University in 1959. He was a member of the Mathematical Sciences Department at IBM Research in Yorktown, New York until 1968, at which time he moved to the University of California at Berkeley. He has been there ever since (except for the period 1995-1999 when he was at the University of Washington). Dr. Karp currently holds the post of University Professor in the Department of Electrical Engineering and Computer Science, with additional appointments in Mathematics, Bioengineering, and Operations Research. Among numerous awards and medals he has won, perhaps most notable is the Turing Award (1985), the most prestigious award in computer science, awarded annually by the Association for Computing Machinery for contributions considered of lasting and major technical importance to the computing community.

October 17th, 2007: Vertex pursuit games on models of complex networks

A&C Seminar

Speaker: Anthony Bonato

Abstract: The game of Cops and Robber is one of the most well-known and widely studied examples of a discrete-time game played on graphs. Apart from its theoretical interest, Cops and Robber supplies a model for security in real-world networks. The web graph is one such network, whose vertices represent web pages and whose edges correspond to links between pages. Several stochastic models for the web graph were recently proposed that generalize the classical G(n,p) random graphs of Erdos and Rényi. An unexplored direction of research is to analyze the cop number in models for complex real-world networks such as the web graph. As a first step in this direction, we provide asymptotic bounds for the cop number of the random power law graph model introduced by Fan Chung Graham. We present concentration results for the cop number in G(n,p) random graphs, and discuss the cop number and density of the infinite random graph. This is joint work with Gena Hahn, Pawel Pralat, and Changping Wang.

October 10th, 2007: Succint data structures, Adaptive (analysis of) algorithms : overview, combination and perspective

A&C Seminar

Speaker: Jeremy Barbay

Abstract: Succinct data structures replace static instances of pointer based data structures, improving performance in both time and space in the word RAM model (a restriction of the RAM model where the size of a word is restricted). The adaptive analysis of algorithms considers the complexity in a finer way than merely grouping the instances by size, yielding more precise lower and upper bound on the complexity of a problem. We give a quick overview of those two techniques, some brief examples of how they can be combined on various search problems to obtain near optimal solutions, and some general perspective on the development and application of those techniques to other problems and in undert different models.

This survery will be presented at Knuth'07 on Oct 29 in Bordeaux.