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.

The rank aggregation problem, seeks to combine multiple rank orderings of the same set of candidates into a single consensus ordering. Such problems arise in diverse domains, including web search, employment, college admissions, and voting. In this work we focus on the 1-median objective: given a set of m rankings over [n], the goal is to compute a ranking that minimizes the sum of its distances to all input rankings. We study rank aggregation under several classical distance metrics: Ulam distance, Spearman’s footrule, Hamming distance, and Kendall-tau, as well as their weighted variants. Our contributions begin with a novel unified framework that identifies a key structural property: it suffices to focus on a small subset of rankings (of size three or five), where the corresponding local one-median provides a good approximation to the global median. This principle extends across these distance measures, yielding a general algorithmic framework for weighted rank aggregation. Building on this, we present a new approximation algorithm for rank aggregation under the Ulam distance that scales in the Massively Parallel Computation (MPC) model. Our algorithm computes a (2-α)-approximation, for a constant α > 0, to the 1-median in a constant number of rounds, using local memory sublinear in n (the size of a ranking) and total memory near linear in n. We further design new MPC approximation algorithms for Spearman’s footrule and for the element-weighted variants of Hamming and Kendall-tau distances. For each metric, we obtain a (2-ζ)-approximation, for a constant ζ > 0 (which may differ across metrics), to the 1-median in a constant number of rounds, using local memory sublinear in n and total memory linear or near-linear in n. Moreover, for the Ulam distance, where computing the 1-median is NP-hard [Fischer et al., ESA, 2025], we simplify and strengthen the analysis of Chakraborty et al. [ITCS 2023], obtaining an improved 1.968-approximation that further extends to the weighted setting.

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.

In this paper, we present the first dynamic algorithms for Dyck edit distance and tree edit distance that achieve subpolynomial update times. Dyck edit distance measures how far a parenthesis string is from a wellparenthesized expression (i.e., the Dyck language), while tree edit distance quantifies the minimum number of node insertions, deletions, and substitutions required to transform one rooted, ordered, and labeled tree into another. These problems have been studied extensively since the 1970s, with recent advances in both algorithmic efficiency and fine-grained complexity lower bounds. Despite this progress, no prior work has addressed efficient dynamic algorithms for these problems, even though many real-world applications involve evolving structured data such as LaTeX, JSON, XML, HTML, hierarchical datasets, and RNA secondary structures. We take the first step in this direction by designing new approximation algorithms for Dyck and tree edit distances in the dynamic setting. Our key technical contribution is a set of novel reduction and decomposition techniques that transform instances of Dyck and tree edit distance into efficiently maintainable instances of string edit distance. Leveraging existing dynamic algorithms for string edit distance, we obtain an $n^{o(1)}$ approximation for Dyck edit distance with $n^{o(1)}$ update time. This builds upon and significantly extends prior work on Dyck language decomposition ([Saha, FOCS’14] and [Koucký & Saks; SODA’23]). For tree edit distance, we introduce a new static reduction that improves the best-known approximation bound from $O\left(n^{3 / 4}\right)$ [Akutsu, Fukagawa, and Takasu; Algorithmica, 2010] to $\tilde{O}(\sqrt{n})$. Moreover, while the previous result was restricted to constant-degree trees, ours holds for arbitrary trees. We then extend our reduction dynamically, yielding a dynamic tree edit distance algorithm with an approximation factor of $n^{1 / 2+o(1)}$ and update time $n^{o(1)}$. A core component of our approach is a new dynamic maintenance algorithm for heavy-light decomposition, a widely used technique in tree algorithms. Given its broad applicability, we believe this result is of independent interest. Finally, we introduce a novel static and dynamic decomposition method that achieves an $\tilde{O}(k)$-approximation for tree edit distance when the tree edit distance is at most k; combined with the trivial bound $k \leq n$, this yields a deterministic $\tilde{O}(\sqrt{n})$-approximation. While similar decompositions exist for strings, no prior work has successfully extended them to trees. Our approach breaks this barrier, improving the best-known approximations for tree edit distance both in the static and dynamic setting. In the static setting, our algorithm runs in $\tilde{O}(n)$ time; in the dynamic setting, it only requires a polylogarithmic worst-case update time. The state-of-the-art near-lineartime static algorithm for tree edit distance previously achieved an $O(\sqrt{n})$-approximation [Boroujeni, Ghodsi, Hajiaghayi, and Seddighin; STOC’19].

Fitting distances to tree metrics and ultrametrics are two widely used methods in hierarchical clustering, primarily explored within the context of numerical taxonomy. Given a positive distance function $D:\binom{V}{2}\rightarrow\mathbb{R}_{>0}$, the goal is to find a tree (or ultrametric) $T$ including all elements of set $V$ such that the difference between the distances among vertices in $T$ and those specified by $D$ is minimized. In this paper, we initiate the study of ultrametric and tree metric fitting problems in the semi-streaming model, where the distances between pairs of elements from $V$ (with $|V|=n$), defined by the function $D$, can arrive in an arbitrary order. We study these problems under various distance norms: For the $\ell_0$ objective, we provide a single-pass polynomial-time $\tilde{O}(n)$-space $O(1)$ approximation algorithm for ultrametrics and prove that no single-pass exact algorithm exists, even with exponential time. Next, we show that the algorithm for $\ell_0$ implies an $O(Δ/δ)$ approximation for the $\ell_1$ objective, where $Δ$ is the maximum and $δ$ is the minimum absolute difference between distances in the input. This bound matches the best-known approximation for the RAM model using a combinatorial algorithm when $Δ/δ=O(n)$. For the $\ell_\infty$ objective, we provide a complete characterization of the ultrametric fitting problem. We present a single-pass polynomial-time $\tilde{O}(n)$-space 2-approximation algorithm and show that no better than 2-approximation is possible, even with exponential time. We also show that, with an additional pass, it is possible to achieve a polynomial-time exact algorithm for ultrametrics. Finally, we extend the results for all these objectives to tree metrics by using only one additional pass through the stream and without asymptotically increasing the approximation factor.

Consensus clustering is a method used to combine the results of multiple clustering algorithms applied to the same dataset. The goal is to find a single clustering solution that represents the common structure present in the various input clusterings, thereby improving the overall quality and robustness of the clustering outcome.

In the Massively Parallel Computation (MPC) model, data is distributed across multiple processors, and we call an algorithm space-efficient if each machine has n^1-ε + o(1) memory with a machine count of Ømega(n^ε).

We consider the numerical taxonomy problem of fitting a positive distance function \({\mathcal {D}:{S\choose 2}\rightarrow \mathbb {R}_{\gt 0}}\) by a tree metric. We want a tree T with positive edge weights and including S among the vertices so that their distances in T match those in \(\mathcal {D}\) . A nice application is in evolutionary biology where the tree T aims to approximate thebranching process leading to the observed distances in \(\mathcal {D}\) [Cavalli-Sforza and Edwards 1967]. We consider the total error, that is, the sum of distance errors over all pairs of points. We present a deterministic polynomial time algorithm minimizing the total error within a constant factor. We can do this both for general trees and for the special case of ultrametrics with a root having the same distance to all vertices in S . The problems are APX-hard, so a constant factor is the best we can hope for in polynomial time. The best previous approximation factor was O ((log n )(log log n )) by Ailon and Charikar [2005], who wrote “determining whether an O (1) approximation can be obtained is a fascinating question.”

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.

Given two strings of length n over alphabet Σ, and an upper bound k on their edit distance, the algorithm of Myers (Algorithmica’86) and Landau and Vishkin (JCSS’88) from almost forty years back computes the unweighted string edit distance in O(n+k2) time. To date, it remains the fastest algorithm for exact edit distance computation, and it is optimal under the Strong Exponential Hypothesis (Backurs and Indyk; STOC’15). Over the years, this result has inspired many developments, including fast approximation algorithms for string edit distance as well as similar Õ(n+poly(k))-time algorithms for generalizations to tree and Dyck edit distances. Surprisingly, all these results hold only for unweighted instances.

We consider the classic problem of computing the Longest Common Subsequence (LCS) of two strings of length n . The 40-year-old quadratic-time dynamic programming algorithm has recently been shown to be near-optimal by Abboud, Backurs, and Vassilevska Williams [FOCS’15] and Bringmann and Künnemann [FOCS’15] assuming the Strong Exponential Time Hypothesis. This has led the community to look for subquadratic approximation algorithms for the problem. Yet, unlike the edit distance problem for which a constant-factor approximation in almost-linear time is known, very little progress has been made on LCS, making it a notoriously difficult problem also in the realm of approximation. For the general setting, only a naive O ( n ɛ /2-approximation algorithm with running time OŠ ( n 2-ɛ has been known, for any constant 0 < ɛ ≤ 1. Recently, a breakthrough result by Hajiaghayi, Seddighin, Seddighin, and Sun [SODA’19] provided a linear-time algorithm that yields a O ( n 0.497956 -approximation in expectation; improving upon the naive \(O(\sqrt {n})\) -approximation for the first time. In this paper, we provide an algorithm that in time O ( n 2-ɛ ) computes an OŠ ( n 2ɛ/5 -approximation with high probability, for any 0 < ɛ ≤ 1. Our result (1) gives an OŠ ( n 0.4 -approximation in linear time, improving upon the bound of Hajiaghayi, Seddighin, Seddighin, and Sun, (2) provides an algorithm whose approximation scales with any subquadratic running time O ( n 2-ɛ ), improving upon the naive bound of O ( n ɛ/2 ) for any ɛ, and (3) instead of only in expectation, succeeds with high probability.

We study the problem of aligning multiple sequences with the goal of finding an alignment that either maximizes the number of aligned symbols (the longest common subsequence (LCS) problem), or minimizes the number of unaligned symbols (the alignment distance aka the complement of LCS). Multiple sequence alignment is a well-studied problem in bioinformatics and is used routinely to identify regions of similarity among DNA, RNA, or protein sequences to detect functional, structural, or evolutionary relationships among them. It is known that exact computation of LCS or alignment distance of m sequences each of length n requires Θ(n^m) time unless the Strong Exponential Time Hypothesis is false. However, unlike the case of two strings, fast algorithms to approximate LCS and alignment distance of multiple sequences are lacking in the literature. A major challenge in this area is to break the triangle inequality. Specifically, by splitting m sequences into two (roughly) equal sized groups, then computing the alignment distance in each group and finally combining them by using triangle inequality, it is possible to achieve a 2-approximation in Õ_m(n^⌈m/2⌉) time. But, an approximation factor below 2 which would need breaking the triangle inequality barrier is not known in O(n^{α m}) time for any α < 1. We make significant progress in this direction. First, we consider a semi-random model where, we show if just one out of m sequences is (p,B)-pseudorandom then, we can get a below-two approximation in Õ_m(nB^{m-1}+n^{⌊m/2⌋+3}) time. Such semi-random models are very well-studied for two strings scenario, however directly extending those works require one but all sequences to be pseudorandom, and would only give an O(1/p) approximation. We overcome these with significant new ideas. Specifically an ingredient to this proof is a new algorithm that achives below 2 approximations when alignment distance is large in Õ_m(n^{⌊m/2⌋+2}) time. This could be of independent interest. Next, for LCS of m sequences each of length n, we show if the optimum LCS is λ n for some λ ∈ [0,1], then in Õ_m(n^{⌊m/2⌋+1}) time, we can return a common subsequence of length at least λ²n/(2+ε) for any arbitrary constant ε > 0. In contrast, for two strings, the best known subquadratic algorithm may return a common subsequence of length Θ(λ⁴ n).

Computing the edit distance of two strings is one of the most basic problems in computer science and combinatorial optimization. Tree edit distance is a natural generalization of edit distance in which the task is to compute a measure of dissimilarity between two (unweighted) rooted trees with node labels. Perhaps the most notable recent application of tree edit distance is in NoSQL big databases, such as MongoDB, where each row of the database is a JSON document represented as a labeled rooted tree and finding dissimilarity between two rows is a basic operation. Until recently, the fastest algorithm for tree edit distance ran in cubic time (Demaine, Mozes, Rossman, Weimann; TALG’10); however, Mao (FOCS’21) broke the cubic barrier for the tree edit distance problem using fast matrix multiplication.Given a parameter k as an upper bound on the distance, an $\mathcal{O}(n+k^{2})$-time algorithm for edit distance has been known since the 1980s due to works of Myers (Algorithmica’86) and Landau and Vishkin (JCSS’88). The existence of an $\tilde{\mathcal{O}}(n+poly(k))$-time algorithm for tree edit distance has been posed as open question, e.g., by Akmal and Jin (ICALP’21), who give a stateof-the-art $O(nk^{2})$-time algorithm. In this paper, we answer this question positively.

The Dyck language, which consists of well-balanced sequences of parentheses, is one of the most fundamental context-free languages. The Dyck edit distance quantifies the number of edits (character insertions, deletions, and substitutions) required to make a given length-n parenthesis sequence well-balanced. RNA Folding involves a similar problem, where a closing parenthesis can match an opening parenthesis of the same type irrespective of their ordering. For example, in RNA Folding, both () and )( are valid matches, whereas the Dyck language only allows () as a match. Both of these problems have been studied extensively in the literature. Using fast matrix multiplication, it is possible to compute their exact solutions in time O(n^2.687) (Chi, Duan, Xie, Zhang, STOC'22), and a (1+ε)-multiplicative approximation is known with a running time of Ω(n^2.372). The impracticality of fast matrix multiplication often makes combinatorial algorithms much more desirable. Unfortunately, it is known that the problems of (exactly) computing the Dyck edit distance and the folding distance are at least as hard as Boolean matrix multiplication. Thereby, they are unlikely to admit truly subcubic-time combinatorial algorithms. In terms of fast approximation algorithms that are combinatorial in nature, the state of the art for Dyck edit distance is an O(log n)-factor approximation algorithm that runs in near-linear time (Saha, FOCS'14), whereas for RNA Folding only an ε n-additive approximation in Õ(n²/ε) time (Saha, FOCS'17) is known. In this paper, we make substantial improvements to the state of the art for Dyck edit distance (with any number of parenthesis types). We design a constant-factor approximation algorithm that runs in Õ(n^1.971) time (the first constant-factor approximation in subquadratic time). Moreover, we develop a (1+ε)-factor approximation algorithm running in Õ(n²/ε) time, which improves upon the earlier additive approximation. Finally, we design a (3+ε)-approximation that takes Õ(nd/ε) time, where d ≥ 1 is an upper bound on the sought distance. As for RNA folding, for any s ≥ 1, we design a factor-s approximation algorithm that runs in O(n+(n/s)³) time. To the best of our knowledge, this is the first nontrivial approximation algorithm for RNA Folding that can go below the n² barrier. All our algorithms are combinatorial in nature.

In the planar, dynamic All-Pairs Shortest Paths (APSP) problem, a planar, weighted digraph G undergoes a sequence of edge weight updates and the goal is to maintain a data structure on G, that can quickly answer distance queries between any two vertices x,y ∊ V(G). The currently best algorithms [FOCS'01, SODA'05] for this problem require Õ(n2/3) worst-case update and query time, while conditional lower bounds [FOCS'16] show that either update or query time is needed1. In this article, we present the first algorithm with near-optimal worst-case update and query time for the offline setting, where the update sequence is given initially. This result is obtained by giving the first offline dynamic algorithm for maintaining dense distance graphs (DDGs) faster than recomputing from scratch after each update. Further, we also present an online algorithm for the incremental APSP problem with worst-case update/query time. This allows us to reduce the online dynamic APSP problem to the online decremental APSP problem, which constitutes partial progress even for the online version of this notorious problem.

Stochastic sample-based estimators are among the most fundamental and universally applied tools in statistics. Such estimators are particularly important when processing huge amounts of data, where we need to be able to answer a wide range of statistical queries reliably, yet cannot afford to store the data in its full length. In many applications we need the sampling to be coordinated which is typically attained using hashing. In previous work, a common strategy to obtain reliable sample-based estimators that work within certain error bounds with high probability has been to design one that works with constant probability, and then boost the probability by taking the median over r independent repetitions. Aamand et al. (STOC'20) recently proposed a fast and practical hashing scheme with strong concentration bounds , Tabulation-1Permutation, the first of its kind. In this paper, we demonstrate that using such a hash family for the sampling, we achieve the same high probability bounds without any need for repetitions. Using the same space, this saves a factor r in time, and simplifies the overall algorithms. We validate our approach experimentally on both real and synthetic data. We compare Tabulation-1Permutation with other hash functions such as strongly universal hash functions and various other hash functions such as MurmurHash3 and BLAKE3, both with and without resorting to repetitions. We see that if we want reliability in terms of small error probabilities, then Tabulation-1Permutation is significantly faster.

We consider the numerical taxonomy problem of fitting a positive distance function$\mathcal{D}:\binom{S}{2}\rightarrow \mathbb{R}_{> 0}$by a tree metric. We want a tree$T$with positive edge weights and including$S$among the vertices so that their distances in$T$match those in$\mathcal{D}$. A nice application is in evolutionary biology where the tree$T$aims to approximate the branching process leading to the observed distances in$\mathcal{D}$[Cavalli-Sforza and Edwards 1967]. We consider the total error, that is the sum of distance errors over all pairs of points. We present a deterministic polynomial time algorithm minimizing the total error within a constant factor. We can do this both for general trees, and for the special case of ultrametrics with a root having the same distance to all vertices in$S$. The problems are APX-hard, so a constant factor is the best we can hope for in polynomial time. The best previous approximation factor was$O((\log n)(\log\log n)$) by Ailon and Charikar [2005] who wrote “Determining whether an$O(1)$approximation can be obtained is a fascinating question”.

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 consider the classic problem of computing the Longest Common Subsequence (LCS) of two strings of length n. While a simple quadratic algorithm has been known for the problem for more than 40 years, no faster algorithm has been found despite an extensive effort. The lack of progress on the problem has recently been explained by Abboud, Backurs, and Vassilevska Williams [FOCS'15] and Bringmann and Künnemann [FOCS'15] who proved that there is no subquadratic algorithm unless the Strong Exponential Time Hypothesis fails. This major roadblock for getting faster exact algorithms has led the community to look for subquadratic approximation algorithms for the problem. Yet, unlike the edit distance problem for which a constant-factor approximation in almost-linear time is known, very little progress has been made on LCS, making it a notoriously difficult problem also in the realm of approximation. For the general setting (where we make no assumption on the length of the optimum solution or the alphabet size), only a naive O(n^{ε/2})-approximation algorithm with running time Õ(n^{2-ε}) has been known, for any constant 0 < ε ≤ 1. Recently, a breakthrough result by Hajiaghayi, Seddighin, Seddighin, and Sun [SODA'19] provided a linear-time algorithm that yields a O(n^{0.497956})-approximation in expectation; improving upon the naive O(√n)-approximation for the first time. In this paper, we provide an algorithm that in time O(n^{2-ε}) computes an Õ(n^{2ε/5})-approximation with high probability, for any 0 < ε ≤ 1. Our result (1) gives an Õ(n^{0.4})-approximation in linear time, improving upon the bound of Hajiaghayi, Seddighin, Seddighin, and Sun, (2) provides an algorithm whose approximation scales with any subquadratic running time O(n^{2-ε}), improving upon the naive bound of O(n^{ε/2}) for any ε, and (3) instead of only in expectation, succeeds with high probability.

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).

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.

In this paper we propose models of combinatorial algorithms for the Boolean Matrix Multiplication (BMM), and prove lower bounds on computing BMM in these models. First, we give a relatively relaxed combinatorial model which is an extension of the model by Angluin (1976), and we prove that the time required by any algorithm for the BMM is at least $Ω(n^3 / 2^{O( \sqrt{ \log n })})$. Subsequently, we propose a more general model capable of simulating the "Four Russians Algorithm". We prove a lower bound of $Ω(n^{7/3} / 2^{O(\sqrt{ \log n })})$ for the BMM under this model. We use a special class of graphs, called $(r,t)$-graphs, originally discovered by Rusza and Szemeredi (1978), along with randomization, to construct matrices that are hard instances for our combinatorial models.

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.