Clustering is a fundamental task in machine learning and data analysis, but it frequently fails to provide fair representation for various marginalized communities defined by multiple protected attributes -- a shortcoming often caused by biases in the training data. As a result, there is a growing need to enhance the fairness of clustering outcomes, ideally by making minimal modifications, possibly as a post-processing step after conventional clustering. A recent work initiated the study of closest fair clustering, though in a restricted scenario where data points belong to only two groups. In practice, however, data points are typically characterized by many groups, reflecting diverse protected attributes such as age, ethnicity, gender, etc. In this work, we generalize the study of the closest fair clustering problem to settings with an arbitrary number (more than two) of groups. We begin by showing that the problem is NP-hard even when all groups are of equal size -- a stark contrast with the two-group case, for which an exact algorithm exists. Next, we propose near-linear time approximation algorithms that efficiently handle arbitrary-sized multiple groups. Leveraging our closest fair clustering algorithms, we further achieve improved approximation guarantees for the fair correlation clustering problem, advancing the state-of-the-art results. Additionally, we are the first to provide approximation algorithms for the fair consensus clustering problem involving multiple (more than two) groups.

Given a set of strings over a specified alphabet, identifying a median or consensus string that minimizes the total distance to all input strings is a fundamental data aggregation problem. When the Hamming distance is considered as the underlying metric, this problem has extensive applications, ranging from bioinformatics to pattern recognition. However, modern applications often require the generation of multiple (near-)optimal yet diverse median strings to enhance flexibility and robustness in decision-making. In this study, we address this need by focusing on two prominent diversity measures: sum dispersion and min dispersion. We first introduce an exact algorithm for the diameter variant of the problem, which identifies pairs of near-optimal medians that are maximally diverse. Subsequently, we propose a (1-ε)-approximation algorithm (for any ε > 0) for sum dispersion, as well as a bi-criteria approximation algorithm for the more challenging min dispersion case, allowing the generation of multiple (more than two) diverse near-optimal Hamming medians. Our approach primarily leverages structural insights into the Hamming median space and also draws on techniques from error-correcting code construction to establish these results.

Consensus clustering, a fundamental task in machine learning and data analysis, aims to aggregate multiple input clusterings of a dataset, potentially based on different non-sensitive attributes, into a single clustering that best represents the collective structure of the data. In this work, we study this fundamental problem through the lens of fair clustering, as introduced by Chierichetti et al. [NeurIPS’17], which incorporates the disparate impact doctrine to ensure proportional representation of each protected group in the dataset within every cluster. Our objective is to find a consensus clustering that is not only representative but also fair with respect to specific protected attributes. To the best of our knowledge, we are the first to address this problem and provide a constant-factor approximation. As part of our investigation, we examine how to minimally modify an existing clustering to enforce fairness – an essential postprocessing step in many clustering applications that require fair representation. We develop an optimal algorithm for datasets with equal group representation and near-linear time constant factor approximation algorithms for more general scenarios with different proportions of two group sizes. We complement our approximation result by showing that the problem is NP-hard for two unequal-sized groups. Given the fundamental nature of this problem, we believe our results on Closest Fair Clustering could have broader implications for other clustering problems, particularly those for which no prior approximation guarantees exist for their fair variants.

Aggregating multiple input rankings into a consensus ranking is essential in various fields such as social choice theory, hiring, college admissions, web search, and databases. A major challenge is that the optimal consensus ranking might be biased against individual candidates or groups, especially those from marginalized communities. This concern has led to recent studies focusing on fairness in rank aggregation. The goal is to ensure that candidates from different groups are fairly represented in the top-k positions of the aggregated ranking. We study this fair rank aggregation problem by considering the Kendall tau as the underlying metric. While we know of a polynomial-time approximation scheme (PTAS) for the classical rank aggregation problem, the corresponding fair variant only possesses a quite straightforward 3-approximation algorithm due to Wei et al., SIGMOD'22, and Chakraborty et al., NeurIPS'22, which finds closest fair ranking for each input ranking and then simply outputs the best one. In this paper, we first provide a novel algorithm that achieves (2+ε)-approximation (for any ε > 0), significantly improving over the 3-approximation bound. Next, we provide a 2.881-approximation fair rank aggregation algorithm that works irrespective of the fairness notion, given one can find a closest fair ranking, beating the 3-approximation bound. We complement our theoretical guarantee by performing extensive experiments on various real-world datasets to establish the effectiveness of our algorithm further by comparing it with the performance of state-of-the-art algorithms.

In this paper, we consider the question of computing sparse subgraphs for any input directed graph \(G=(V,E)\) on \(n\) vertices and \(m\) edges, that preserves reachability and/or strong connectivity structures. We show \(O(n+\min\{|{\mathcal{P}}|\sqrt{n},n\sqrt{|{\mathcal{P} }|}\})\) bound on a subgraph that is an \(1\) -fault-tolerant reachability preserver for a given vertex-pair set \({\mathcal{P}}\subseteq V\times V\) , i.e., it preserves reachability between any pair of vertices in \({\mathcal{P}}\) under single edge (or vertex) failure. Our result is a significant improvement over the previous best \(O(n|{\mathcal{P}}|)\) bound obtained as a corollary of single-source reachability preserver construction. We prove our upper bound by exploiting the special structure of single fault-tolerant reachability preserver for any pair, and then considering the interaction among such structures for different pairs. We also present the first sub-quadratic bound of at most \(\tilde{O}(k2^{k}n^{2-1/k})\) size, for strong-connectivity preservers of directed graphs under \(k\) failures. To the best of our knowledge no non-trivial bound for this problem was known before, for a general \(k\) . We get our result by adopting the color-coding technique of Alon, Yuster, and Zwick [JACM’95].

Equivalence testing, a fundamental problem in the field of distribution testing, seeks to infer if two unknown distributions on $[n]$ are the same or far apart in the total variation distance. Conditional sampling has emerged as a powerful query model and has been investigated by theoreticians and practitioners alike, leading to the design of optimal algorithms albeit in a sequential setting (also referred to as adaptive tester). Given the profound impact of parallel computing over the past decades, there has been a strong desire to design algorithms that enable high parallelization. Despite significant algorithmic advancements over the last decade, parallelizable techniques (also termed non-adaptive testers) have $\tilde{O}(\log^{12}n)$ query complexity, a prohibitively large complexity to be of practical usage. Therefore, the primary challenge is whether it is possible to design algorithms that enable high parallelization while achieving efficient query complexity. Our work provides an affirmative answer to the aforementioned challenge: we present a highly parallelizable tester with a query complexity of $\tilde{O}(\log n)$, achieved through a single round of adaptivity, marking a significant stride towards harmonizing parallelizability and efficiency in equivalence testing.

We study the equivalence testing problem where the goal is to determine if the given two unknown distributions on [n] are equal or ɛ-far in the total variation distance in the conditional sampling model (CFGM, SICOMP16; CRS, SICOMP15) wherein a tester can get a sample from the distribution conditioned on any subset. Equivalence testing is a central problem in distribution testing, and there has been a plethora of work on this topic in various sampling models.

The fundamental task of rank aggregation is to combine multiple rankings on a group of candidates into a single ranking to mitigate biases inherent in individual input rankings. This task has a myriad of applications, such as in social choice theory, collaborative filtering, web search, statistics, databases, sports, and admission systems. One popular version of this task, maximum rank aggregation (or the center ranking problem), aims to find a ranking (not necessarily from the input set) that minimizes the maximum distance to the input rankings. However, even for four input rankings, this problem is NP-hard (Dwork et al., WWW'01, and Biedl et al., Discrete Math.'09), and only a (folklore) polynomial-time 2-approximation algorithm is known for finding an optimal aggregate ranking under the commonly used Kendall-tau distance metric. Achieving a better approximation factor in polynomial time, ideally, a polynomial time approximation scheme (PTAS), is one of the major challenges. This paper presents significant progress in solving this problem by considering the Mallows model, a classical probabilistic model. Our proposed algorithm outputs an (1+ε)-approximate aggregate ranking for any ε > 0, with high probability, as long as the input rankings come from a Mallows model, even in a streaming fashion. Furthermore, the same approximation guarantee is achieved even in the presence of outliers, presumably a more challenging task.

Given a Boolean formula ϕ over n variables, the problem of model counting is to compute the number of solutions of ϕ. Model counting is a fundamental problem in computer science with wide-ranging applications in domains such as quantified information leakage, probabilistic reasoning, network reliability, neural network verification, and more. Owing to the #P-hardness of the problems, Stockmeyer initiated the study of the complexity of approximate counting. Stockmeyer showed that log n calls to an NP oracle are necessary and sufficient to achieve (ε,δ) guarantees. The hashing-based framework proposed by Stockmeyer has been very influential in designing practical counters over the past decade, wherein the SAT solver substitutes the NP oracle calls in practice. It is well known that an NP oracle does not fully capture the behavior of SAT solvers, as SAT solvers are also designed to provide satisfying assignments when a formula is satisfiable, without additional overhead. Accordingly, the notion of SAT oracle has been proposed to capture the behavior of SAT solver wherein given a Boolean formula, an SAT oracle returns a satisfying assignment if the formula is satisfiable or returns unsatisfiable otherwise. Since the practical state-of-the-art approximate counting techniques use SAT solvers, a natural question is whether an SAT oracle is more powerful than an NP oracle in the context of approximate model counting. The primary contribution of this work is to study the relative power of the NP oracle and SAT oracle in the context of approximate model counting. The previous techniques proposed in the context of an NP oracle are weak to provide strong bounds in the context of SAT oracle since, in contrast to an NP oracle that provides only one bit of information, a SAT oracle can provide n bits of information. We therefore develop a new methodology to achieve the main result: a SAT oracle is no more powerful than an NP oracle in the context of approximate model counting.

We study the classical metric $k$-median clustering problem over a set of input rankings (i.e., permutations), which has myriad applications, from social-choice theory to web search and databases. A folklore algorithm provides a $2$-approximate solution in polynomial time for all $k=O(1)$, and works irrespective of the underlying distance measure, so long it is a metric; however, going below the $2$-factor is a notorious challenge. We consider the Ulam distance, a variant of the well-known edit-distance metric, where strings are restricted to be permutations. For this metric, Chakraborty, Das, and Krauthgamer [SODA, 2021] provided a $(2-δ)$-approximation algorithm for $k=1$, where $δ\approx 2^{-40}$. Our primary contribution is a new algorithmic framework for clustering a set of permutations. Our first result is a $1.999$-approximation algorithm for the metric $k$-median problem under the Ulam metric, that runs in time $(k \log (nd))^{O(k)}n d^3$ for an input consisting of $n$ permutations over $[d]$. In fact, our framework is powerful enough to extend this result to the streaming model (where the $n$ input permutations arrive one by one) using only polylogarithmic (in $n$) space. Additionally, we show that similar results can be obtained even in the presence of outliers, which is presumably a more difficult problem.

We consider the problem of estimating the support size of a distribution $D$. Our investigations are pursued through the lens of distribution testing and seek to understand the power of conditional sampling (denoted as COND), wherein one is allowed to query the given distribution conditioned on an arbitrary subset $S$. The primary contribution of this work is to introduce a new approach to lower bounds for the COND model that relies on using powerful tools from information theory and communication complexity. Our approach allows us to obtain surprisingly strong lower bounds for the COND model and its extensions. 1) We bridge the longstanding gap between the upper ($O(\log \log n + \frac{1}{ε^2})$) and the lower bound $Ω(\sqrt{\log \log n})$ for COND model by providing a nearly matching lower bound. Surprisingly, we show that even if we get to know the actual probabilities along with COND samples, still $Ω(\log \log n + \frac{1}{ε^2 \log (1/ε)})$ queries are necessary. 2) We obtain the first non-trivial lower bound for COND equipped with an additional oracle that reveals the conditional probabilities of the samples (to the best of our knowledge, this subsumes all of the models previously studied): in particular, we demonstrate that $Ω(\log \log \log n + \frac{1}{ε^2 \log (1/ε)})$ queries are necessary.

In this paper, we consider reachability oracles and reachability preservers for directed graphs/networks prone to edge/node failures. Let G = (V, E) be a directed graph on n-nodes, and P ⊆ V× V be a set of vertex pairs in G. We present the first non-trivial constructions of single and dual fault-tolerant pairwise reachability oracle with constant query time. Furthermore, we provide extremal bounds for sparse fault-tolerant reachability preservers, resilient to two or more failures. Prior to this work, such oracles and reachability preservers were widely studied for the special scenario of single-source and all-pairs settings. However, for the scenario of arbitrary pairs, no prior (non-trivial) results were known for dual (or more) failures, except those implied from the single-source setting. One of the main questions is whether it is possible to beat the O(n |P|) size bound (derived from the single-source setting) for reachability oracle and preserver for dual failures (or O(2^k n|P|) bound for k failures). We answer this question affirmatively. Below we summarize our contributions. - For an n-vertex directed graph G = (V, E) and P ⊆ V× V, we present a construction of O(n √{|P|}) sized dual fault-tolerant pairwise reachability oracle with constant query time. We further provide a matching (up to the word size) lower bound of Ω(n √{|P|}) on the size (in bits) of the oracle for the dual fault setting, thereby proving that our oracle is (near-)optimal. - Next, we provide a construction of O(n + min{|P|√ n,~n√{|P|}}) sized oracle with O(1) query time, resilient to single node/edge failure. In particular, for |P| bounded by O(√n) this yields an oracle of just O(n) size. We complement the upper bound with a lower bound of Ω(n^{2/3}|P|^{1/2}) (in bits), refuting the possibility of a linear-sized oracle for P of size ω(n^{2/3}). - We also present a construction of O(n^{4/3} |P|^{1/3}) sized pairwise reachability preservers resilient to dual edge/vertex failures. Previously, such preservers were known to exist only under single failure and had O(n+min{|P|√n,~n√ {|P|}}) size [Chakraborty and Choudhary, ICALP'20]. We also show a lower bound of Ω(n √{|P|}) edges on the size of dual fault-tolerant reachability preservers, thereby providing a sharp gap between single and dual fault-tolerant reachability preservers for |P| = o(n). - Finally, we provide a generic pairwise reachability preserver construction that provides a o(2^k n |P|) sized subgraph resilient to k failures, for any k ≥ 1. Before this work, we only knew of an O(2^k n |P|) bound implied from the single-source setting [Baswana, Choudhary, and Roditty, STOC'16].

Ranking algorithms find extensive usage in diverse areas such as web search, employment, college admission, voting, etc. The related rank aggregation problem deals with combining multiple rankings into a single aggregate ranking. However, algorithms for both these problems might be biased against some individuals or groups due to implicit prejudice or marginalization in the historical data. We study ranking and rank aggregation problems from a fairness or diversity perspective, where the candidates (to be ranked) may belong to different groups and each group should have a fair representation in the final ranking. We allow the designer to set the parameters that define fair representation. These parameters specify the allowed range of the number of candidates from a particular group in the top-$k$ positions of the ranking. Given any ranking, we provide a fast and exact algorithm for finding the closest fair ranking for the Kendall tau metric under {\em strong fairness}, i.e., when the final ranking is fair for all values of $k$. We also provide an exact algorithm for finding the closest fair ranking for the Ulam metric under strong fairness when there are only $O(1)$ number of groups. Our algorithms are simple, fast, and might be extendable to other relevant metrics. We also give a novel meta-algorithm for the general rank aggregation problem under the fairness framework. Surprisingly, this meta-algorithm works for any generalized mean objective (including center and median problems) and any fairness criteria. As a byproduct, we obtain 3-approximation algorithms for both center and median problems, under both Kendall tau and Ulam metrics. Furthermore, using sophisticated techniques we obtain a $(3-\varepsilon)$-approximation algorithm, for a constant $\varepsilon>0$, for the Ulam metric under strong fairness.

We consider an \emph{approximate} version of the trace reconstruction problem, where the goal is to recover an unknown string $s\in\{0,1\}^n$ from $m$ traces (each trace is generated independently by passing $s$ through a probabilistic insertion-deletion channel with rate $p$). We present a deterministic near-linear time algorithm for the average-case model, where $s$ is random, that uses only \emph{three} traces. It runs in near-linear time $\tilde O(n)$ and with high probability reports a string within edit distance $O(εp n)$ from $s$ for $ε=\tilde O(p)$, which significantly improves over the straightforward bound of $O(pn)$. Technically, our algorithm computes a $(1+ε)$-approximate median of the three input traces. To prove its correctness, our probabilistic analysis shows that an approximate median is indeed close to the unknown $s$. To achieve a near-linear time bound, we have to bypass the well-known dynamic programming algorithm that computes an optimal median in time $O(n^3)$.

We study the problem of approximating a center under the Ulam metric. The Ulam metric, defined over a set of permutations over [n], is the minimum number of move operations (deletion plus insertion) to transform one permutation into another. The Ulam metric is a simpler variant of the general edit distance metric. It provides a measure of dissimilarity over a set of rankings/permutations. In the center problem, given a set of permutations, we are asked to find a permutation (not necessarily from the input set) that minimizes the maximum distance to the input permutations. This problem is also referred to as maximum rank aggregation under Ulam. So far, we only know of a folklore 2-approximation algorithm for this NP-hard problem. Even for constantly many permutations, we do not know anything better than an exhaustive search over all n! permutations. In this paper, we achieve a (3/2 - 1/(3m))-approximation of the Ulam center in time n^O(m² ln m), for m input permutations over [n]. We therefore get a polynomial time bound while achieving better than a 3/2-approximation for constantly many permutations. This problem is of special interest even for constantly many permutations because under certain dissimilarity measures over rankings, even for four permutations, the problem is NP-hard. In proving our result, we establish a surprising connection between the approximate Ulam center problem and the closest string with wildcards problem (the center problem over the Hamming metric, allowing wildcards). We further study the closest string with wildcards problem and show that there cannot exist any (2-ε)-approximation algorithm (for any ε > 0) for it unless 𝖯 = NP. This inapproximability result is in sharp contrast with the same problem without wildcards, where we know of a PTAS.

We study approximation algorithms for variants of the median string problem, which asks for a string that minimizes the sum of edit distances from a given set of m strings of length n. Only the straightforward 2-approximation is known for this NP-hard problem. This problem is motivated e.g. by computational biology, and belongs to the class of median problems (over different metric spaces), which are fundamental tasks in data analysis. Our main result is for the Ulam metric, where all strings are permutations over [n] and each edit operation moves a symbol (deletion plus insertion). We devise for this problem an algorithms that breaks the 2-approximation barrier, i.e., computes a (2 – δ)-approximate median permutation for some constant δ > 0 in time Õ(nm2 + n3). We further use these techniques to achieve a (2 – δ) approximation for the median string problem in the special case where the median is restricted to length n and the optimal objective is large Ω(mn). We also design an approximation algorithm for the following probabilistic model of the Ulam median: the input consists of m perturbations of an (unknown) permutation x, each generated by moving every symbol to a random position with probability (a parameter) ∊ > 0. Our algorithm computes with high probability a (1 + o(1/∊))-approximate median permutation in time O(mn2 + n3).

The problem of finding longest common subsequence (LCS) is one of the fundamental problems in computer science, which finds application in fields such as computational biology, text processing, information retrieval, data compression etc. It is well known that (decision version of) the problem of finding the length of a LCS of an arbitrary number of input sequences (which we refer to as Multi-LCS problem) is NP-complete. Jiang and Li [SICOMP'95] showed that if Max-Clique is hard to approximate within a factor of $s$ then Multi-LCS is also hard to approximate within a factor of $Θ(s)$. By the NP-hardness of the problem of approximating Max-Clique by Zuckerman [ToC'07], for any constant $δ>0$, the length of a LCS of arbitrary number of input sequences of length $n$ each, cannot be approximated within an $n^{1-δ}$-factor in polynomial time unless {\tt{P}}$=${\NP}. However, the reduction of Jiang and Li assumes the alphabet size to be $Ω(n)$. So far no hardness result is known for the problem of approximating Multi-LCS over sub-linear sized alphabet. On the other hand, it is easy to get $1/|Σ|$-factor approximation for strings of alphabet $Σ$. In this paper, we make a significant progress towards proving hardness of approximation over small alphabet by showing a polynomial-time reduction from the well-studied \emph{densest $k$-subgraph} problem with {\em perfect completeness} to approximating Multi-LCS over alphabet of size $poly(n/k)$. As a consequence, from the known hardness result of densest $k$-subgraph problem (e.g. [Manurangsi, STOC'17]) we get that no polynomial-time algorithm can give an $n^{-o(1)}$-factor approximation of Multi-LCS over an alphabet of size $n^{o(1)}$, unless the Exponential Time Hypothesis is false.

In this paper, we consider the question of computing sparse subgraphs for any input directed graph $G=(V,E)$ on $n$ vertices and $m$ edges, that preserves reachability and/or strong connectivity structures. We show $O(n+\min\{|{\cal P}|\sqrt{n},n\sqrt{|{\cal P}|}\})$ bound on a subgraph that is an $1$-fault-tolerant reachability preserver for a given vertex-pair set ${\cal P}\subseteq V\times V$, i.e., it preserves reachability between any pair of vertices in ${\cal P}$ under single edge (or vertex) failure. Our result is a significant improvement over the previous best $O(n |{\cal P}|)$ bound obtained as a corollary of single-source reachability preserver construction. We prove our upper bound by exploiting the special structure of single fault-tolerant reachability preserver for any pair, and then considering the interaction among such structures for different pairs. In the lower bound side, we show that a 2-fault-tolerant reachability preserver for a vertex-pair set ${\cal P}\subseteq V\times V$ of size $Ω(n^ε)$, for even any arbitrarily small $ε$, requires at least $Ω(n^{1+ε/8})$ edges. This refutes the existence of linear-sized dual fault-tolerant preservers for reachability for any polynomial sized vertex-pair set. We also present the first sub-quadratic bound of at most $\tilde{O}(k 2^k n^{2-1/k})$ size, for strong-connectivity preservers of directed graphs under $k$ failures. To the best of our knowledge no non-trivial bound for this problem was known before, for a general $k$. We get our result by adopting the color-coding technique of Alon, Yuster, and Zwick [JACM'95].

Edit distance is a measure of similarity of two strings based on the minimum number of character insertions, deletions, and substitutions required to transform one string into the other. The edit distance can be computed exactly using a dynamic programming algorithm that runs in quadratic time. Andoni, Krauthgamer, and Onak (2010) gave a nearly linear time algorithm that approximates edit distance within approximation factor poly(log n ). In this article, we provide an algorithm with running time Õ( n 2−2/7 ) that approximates the edit distance within a constant factor.

We consider the approximate pattern matching problem under edit distance. In this problem we are given a pattern P of length m and a text T of length n over some alphabet Σ, and a positive integer k. The goal is to find all the positions j in T such that there is a substring of T ending at j which has edit distance at most k from the pattern P. Recall, the edit distance between two strings is the minimum number of character insertions, deletions, and substitutions required to transform one string into the other. For a position t in {1,...,n}, let kt be the smallest edit distance between P and any substring of T ending at t. In this paper we give a constant factor approximation to the sequence k1,k2,...,kn. We consider both offline and online settings. In the offline setting, where both P and T are available, we present an algorithm that for all t in {1,...,n}, computes the value of kt approximately within a constant factor. The worst case running time of our algorithm is Õ(nm3/4). In the online setting, we are given P and then T arrives one symbol at a time. We design an algorithm that upon arrival of the t-th symbol of T computes kt approximately within O(1)multiplicative factor and m8/9-additive error. Our algorithm takes Õ(m1−(7/54)) amortized time per symbol arrival and takes Õ(m1−(1/54)) additional space apart from storing the pattern P. Both of our algorithms are randomized and produce correct answer with high probability. To the best of our knowledge this is the first algorithm that takes worst-case sublinear (in the length of the pattern) time and sublinear extra space for the online approximate pattern matching problem. To get our result we build on the technique of Chakraborty, Das, Goldenberg, Koucký and Saks [FOCS'18] for computing a constant factor approximation of edit distance in sub-quadratic time.

A quasi-Gray code of dimension n and length l over an alphabet Sigma is a sequence of distinct words w_1,w_2,...,w_l from Sigma^n such that any two consecutive words differ in at most c coordinates, for some fixed constant c>0. In this paper we are interested in the read and write complexity of quasi-Gray codes in the bit-probe model, where we measure the number of symbols read and written in order to transform any word w_i into its successor w_{i+1}. We present construction of quasi-Gray codes of dimension n and length 3^n over the ternary alphabet {0,1,2} with worst-case read complexity O(log n) and write complexity 2. This generalizes to arbitrary odd-size alphabets. For the binary alphabet, we present quasi-Gray codes of dimension n and length at least 2^n - 20n with worst-case read complexity 6+log n and write complexity 2. This complements a recent result by Raskin [Raskin '17] who shows that any quasi-Gray code over binary alphabet of length 2^n has read complexity Omega(n). Our results significantly improve on previously known constructions and for the odd-size alphabets we break the Omega(n) worst-case barrier for space-optimal (non-redundant) quasi-Gray codes with constant number of writes. We obtain our results via a novel application of algebraic tools together with the principles of catalytic computation [Buhrman et al. '14, Ben-Or and Cleve '92, Barrington '89, Coppersmith and Grossman '75].

Edit distance is a measure of similarity of two strings based on the minimum number of character insertions, deletions, and substitutions required to transform one string into the other. The edit distance can be computed exactly using a dynamic programming algorithm that runs in quadratic time. Andoni, Krauthgamer and Onak (2010) gave a nearly linear time algorithm that approximates edit distance within approximation factor poly(log n). In this paper, we provide an algorithm with running time Õ(n^2-2/7) that approximates the edit distance within a constant factor.

The conjectured hardness of Boolean matrix-vector multiplication has been used with great success to prove conditional lower bounds for numerous important data structure problems, see Henzinger et al. [STOC’15]. In recent work, Larsen and Williams [SODA’17] attacked the problem from the upper bound side and gave a surprising cell probe data structure (that is, we only charge for memory accesses, while computation is free). Their cell probe data structure answers queries in Õ(n7/4) time and is succinct in the sense that it stores the input matrix in read-only memory, plus an additional Õ(n7/4) bits on the side. In this paper, we essentially settle the cell probe complexity of succinct Boolean matrix-vector multiplication. We present a new cell probe data structure with query time Õ(n3/2) storing just Õ(n3/2) bits on the side. We then complement our data structure with a lower bound showing that any data structure storing r bits on the side, with n < r < n2 must have query time t satisfying t r = Ω(n3). For r ≤ n, any data structure must have t = Ω(n2). Since lower bounds in the cell probe model also apply to classic word-RAM data structures, the lower bounds naturally carry over. We also prove similar lower bounds for matrix-vector multiplication over F2.

The Hamming and the edit metrics are two common notions of measuring distances between pairs of strings x,y lying in the Boolean hypercube. The edit distance between x and y is defined as the minimum number of character insertion, deletion, and bit flips needed for converting x into y. Whereas, the Hamming distance between x and y is the number of bit flips needed for converting x to y. In this paper we study a randomized injective embedding of the edit distance into the Hamming distance with a small distortion. We show a randomized embedding with quadratic distortion. Namely, for any x,y satisfying that their edit distance equals k, the Hamming distance between the embedding of x and y is O(k2) with high probability. This improves over the distortion ratio of O( n * n) obtained by Jowhari (2012) for small values of k. Moreover, the embedding output size is linear in the input size and the embedding can be computed using a single pass over the input. We provide several applications for this embedding. Among our results we provide a one-pass (streaming) algorithm for edit distance running in space O(s) and computing edit distance exactly up-to distance s1/6. This algorithm is based on kernelization for edit distance that is of independent interest.

In this paper we propose a quantification of distributions on a set of strings, in terms of how close to pseudorandom a distribution is. The quantification is an adaptation of the theory of dimension of sets of infinite sequences introduced by Lutz. Adapting Hitchcock's work, we also show that the logarithmic loss incurred by a predictor on a distribution is quantitatively equivalent to the notion of dimension we define. Roughly, this captures the equivalence between pseudorandomness defined via indistinguishability and via unpredictability. Later we show some natural properties of our notion of dimension. We also do a comparative study among our proposed notion of dimension and two well known notions of computational analogue of entropy, namely HILL-type pseudo min-entropy and next-bit pseudo Shannon entropy. Further, we apply our quantification to the following problem. If we know that the dimension of a distribution on the set of n-length strings is s in (0,1], can we extract out O(sn) pseudorandom bits out of the distribution? We show that to construct such extractor, one need at least Omega(log n) bits of pure randomness. However, it is still open to do the same using O(log n) random bits. We show that deterministic extraction is possible in a special case - analogous to the bit-fixing sources introduced by Chor et al., which we term nonpseudorandom bit-fixing source. We adapt the techniques of Gabizon, Raz and Shaltiel to construct a deterministic pseudorandom extractor for this source. By the end, we make a little progress towards P vs. BPP problem by showing that existence of optimal stretching function that stretches O(log n) input bits to produce n output bits such that output distribution has dimension s in (0,1], implies P=BPP.

We obtain the following new simultaneous time-space upper bounds for the directed reachability problem. (1) A polynomial-time, O(n^{2/3} * g^{1/3})-space algorithm for directed graphs embedded on orientable surfaces of genus g. (2) A polynomial-time, O(n^{2/3})-space algorithm for all H-minor-free graphs given the tree decomposition, and (3) for K_{3,3}-free and K_5-free graphs, a polynomial-time, O(n^{1/2 + epsilon})-space algorithm, for every epsilon > 0. For the general directed reachability problem, the best known simultaneous time-space upper bound is the BBRS bound, due to Barnes, Buss, Ruzzo, and Schieber, which achieves a space bound of O(n/2^{k * sqrt(log(n))}) with polynomial running time, for any constant k. It is a significant open question to improve this bound for reachability over general directed graphs. Our algorithms beat the BBRS bound for graphs embedded on surfaces of genus n/2^{omega(sqrt(log(n))}, and for all H-minor-free graphs. This significantly broadens the class of directed graphs for which the BBRS bound can be improved.