9 Search Results for "Tiskin, Alexander"

Document
DOI: 10.4230/LIPIcs.CPM.2026.26
The Communication Complexity of Pattern Matching with Edits Revisited

Authors: Tomasz Kociumaka, Jakob Nogler, and Philip Wellnitz

Published in: LIPIcs, Volume 369, 37th Annual Symposium on Combinatorial Pattern Matching (CPM 2026)

Abstract
The decades-old Pattern Matching with Edits problem, given a length-n string T (the text), a length-m string P (the pattern), and a positive integer k (the threshold), asks to list the k-error occurrences of P in T, that is, all fragments of T whose edit distance to P is at most k. The one-way communication complexity of this problem is the minimum number of bits that Alice, given an instance (P,T,k) of the problem, must send to Bob so that Bob can reconstruct the answer solely from that message. In recent work [STOC'24], we showed that, in the natural parameter regime 0 < k < m < n/2, Ω(n/m ⋅ k log(m/k)) bits are necessary and 𝒪(n/m ⋅ k log² m) bits are sufficient for this problem. More generally, for strings over an alphabet Σ, we gave an 𝒪(n/m ⋅ k log m log(m|Σ|))-bit encoding that allows one to recover a shortest sequence of edits for every k-error occurrence of P in T. In this paper, we revisit the original proof and improve the encoding size to 𝒪(n/m ⋅ k log (m|Σ|/k)), which matches the lower bound for constant-sized alphabets. We further establish a new tight lower bound of Ω(n/m ⋅ k log(m|Σ|/k)) for the edit sequence reporting variant we solve. Our encoding size also matches the communication complexity established for the simpler Pattern Matching with Mismatches problem in the context of streaming algorithms [Clifford, Kociumaka, Porat; SODA'19].
Cite as

Tomasz Kociumaka, Jakob Nogler, and Philip Wellnitz. The Communication Complexity of Pattern Matching with Edits Revisited. In 37th Annual Symposium on Combinatorial Pattern Matching (CPM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 369, pp. 26:1-26:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{kociumaka_et_al:LIPIcs.CPM.2026.26,
  author =	{Kociumaka, Tomasz and Nogler, Jakob and Wellnitz, Philip},
  title =	{{The Communication Complexity of Pattern Matching with Edits Revisited}},
  booktitle =	{37th Annual Symposium on Combinatorial Pattern Matching (CPM 2026)},
  pages =	{26:1--26:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-420-8},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{369},
  editor =	{Bille, Philip and Prezza, Nicola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2026.26},
  URN =		{urn:nbn:de:0030-drops-259525},
  doi =		{10.4230/LIPIcs.CPM.2026.26},
  annote =	{Keywords: Edit distance, Pattern matching, Communication complexity}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.87
Range Longest Increasing Subsequence and Its Relatives

Authors: Karthik C. S. and Saladi Rahul

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
Longest increasing subsequence (LIS) is a classical textbook problem which is still actively studied in various computational models. In this work, we present a few results for the range longest increasing subsequence problem (Range-LIS) and its variants. The input to Range-LIS is a sequence 𝒮 of n real numbers and a collection 𝒬 of m query ranges and for each query in 𝒬, the goal is to report the LIS of the sequence 𝒮 restricted to that query. Our two main results are for the following generalizations of the Range-LIS problem: 2D Range Queries: In this variant of the Range-LIS problem, each query is a pair of ranges, one of indices and the other of values, and we provide a randomized algorithm with running time Õ(mn^{1/2}+ n^{3/2})+O(k), where k is the cumulative length of the m output subsequences. This improves on the elementary Õ(mn) runtime algorithm when m = Ω(√n). Previously, the only known result breaking the quadratic barrier was of Tiskin [SODA'10] which could only handle 1D range queries (i.e., each query was a range of indices) and also just outputted the length of the LIS (instead of reporting the subsequence achieving that length). Subsequent to our paper, Gawrychowski, Gorbachev, and Kociumaka in a preprint have extended Tiskin’s approach to handle reporting 1D range queries in O(n(log n)³+m+k) time. Colored Sequences: In this variant of the Range-LIS problem, each element in 𝒮 is colored and for each query in 𝒬, the goal is to report a monochromatic LIS contained in the sequence 𝒮 restricted to that query. For 2D queries, we provide a randomized algorithm for this colored version with running time Õ(mn^{2/3}+ n^{5/3})+O(k). Moreover, for 1D queries, we provide an improved algorithm with running time Õ(mn^{1/2}+ n^{3/2})+O(k). Thus, we again improve on the elementary Õ(mn) runtime algorithm. Additionally, we prove that assuming the well-known Combinatorial Boolean Matrix Multiplication Hypothesis, that the runtime for 1D queries is essentially tight for combinatorial algorithms. Our algorithms combine several tools such as dynamic programming (to precompute increasing subsequences with some desirable properties), geometric data structures (to efficiently compute the dynamic programming entries), random sampling (to capture elements which are part of the LIS), classification of query ranges into large LIS and small LIS, and classification of colors into light and heavy. We believe that our techniques will be of interest to tackle other variants of LIS problem and other range-searching problems.
Cite as

Karthik C. S. and Saladi Rahul. Range Longest Increasing Subsequence and Its Relatives. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 87:1-87:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{karthikc.s._et_al:LIPIcs.ITCS.2026.87,
  author =	{Karthik C. S. and Rahul, Saladi},
  title =	{{Range Longest Increasing Subsequence and Its Relatives}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{87:1--87:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.87},
  URN =		{urn:nbn:de:0030-drops-253740},
  doi =		{10.4230/LIPIcs.ITCS.2026.87},
  annote =	{Keywords: Longest Increasing Subsequence, Range Query, Fine-Grained Complexity}
}
Document
DOI: 10.4230/LIPIcs.ESA.2025.74
Core-Sparse Monge Matrix Multiplication: Improved Algorithm and Applications

Authors: Paweł Gawrychowski, Egor Gorbachev, and Tomasz Kociumaka

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract
Min-plus matrix multiplication is a fundamental tool for designing algorithms operating on distances in graphs and different problems solvable by dynamic programming. We know that, assuming the APSP hypothesis, no subcubic-time algorithm exists for the case of general matrices. However, in many applications the matrices admit certain structural properties that can be used to design faster algorithms. For example, when considering a planar graph, one often works with a Monge matrix A, meaning that the density matrix A^◻ has non-negative entries, that is, A^◻_{i,j} := A_{i+1,j} + A_{i,j+1} - A_{i,j} -A_{i+1,j+1} ≥ 0. The min-plus product of two n×n Monge matrices can be computed in 𝒪(n²) time using the famous SMAWK algorithm. In applications such as longest common subsequence, edit distance, and longest increasing subsequence, the matrices are even more structured, as observed by Tiskin [J. Discrete Algorithms, 2008]: they are (or can be converted to) simple unit-Monge matrices, meaning that the density matrix is a permutation matrix and, furthermore, the first column and the last row of the matrix consist of only zeroes. Such matrices admit an implicit representation of size 𝒪(n) and, as shown by Tiskin [SODA 2010 & Algorithmica, 2015], their min-plus product can be computed in 𝒪(nlog n) time. Russo [SPIRE 2010 & Theor. Comput. Sci., 2012] identified a general structural property of matrices that admit such efficient representation and min-plus multiplication algorithms: the core size δ, defined as the number of non-zero entries in the density matrices of the input and output matrices. He provided an adaptive implementation of the SMAWK algorithm that runs in 𝒪((n+δ)log³ n) or 𝒪((n+δ)log² n) time (depending on the representation of the input matrices). In this work, we further investigate the core size as the parameter that enables efficient min-plus matrix multiplication. On the combinatorial side, we provide a (linear) bound on the core size of the product matrix in terms of the core sizes of the input matrices. On the algorithmic side, we generalize Tiskin’s algorithm (but, arguably, with a more elementary analysis) to solve the core-sparse Monge matrix multiplication problem in 𝒪(n+δlog δ) ⊆ 𝒪(n + δ log n) time, matching the complexity for simple unit-Monge matrices. As witnessed by the recent work of Gorbachev and Kociumaka [STOC'25] for edit distance with integer weights, our generalization opens up the possibility of speed-ups for weighted sequence alignment problems. Furthermore, our multiplication algorithm is also capable of producing an efficient data structure for recovering the witness for any given entry of the output matrix. This allows us, for example, to preprocess an integer array of size n in Õ(n) time so that the longest increasing subsequence of any sub-array can be reconstructed in Õ(𝓁) time, where 𝓁 is the length of the reported subsequence. In comparison, Karthik C. S. and Rahul [arXiv, 2024] recently achieved 𝒪(𝓁+n^{1/2}polylog n)-time reporting after 𝒪(n^{3/2}polylog n)-time preprocessing.
Cite as

Paweł Gawrychowski, Egor Gorbachev, and Tomasz Kociumaka. Core-Sparse Monge Matrix Multiplication: Improved Algorithm and Applications. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 74:1-74:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{gawrychowski_et_al:LIPIcs.ESA.2025.74,
  author =	{Gawrychowski, Pawe{\l} and Gorbachev, Egor and Kociumaka, Tomasz},
  title =	{{Core-Sparse Monge Matrix Multiplication: Improved Algorithm and Applications}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{74:1--74:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.74},
  URN =		{urn:nbn:de:0030-drops-245427},
  doi =		{10.4230/LIPIcs.ESA.2025.74},
  annote =	{Keywords: Min-plus matrix multiplication, Monge matrix, longest increasing subsequence}
}
Document
DOI: 10.4230/LIPIcs.ESA.2025.45
Bounded Weighted Edit Distance: Dynamic Algorithms and Matching Lower Bounds

Authors: Itai Boneh, Egor Gorbachev, and Tomasz Kociumaka

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract
The edit distance ed(X,Y) of two strings X,Y ∈ Σ^* is the minimum number of character edits (insertions, deletions, and substitutions) needed to transform X into Y. Its weighted counterpart ed^w(X,Y) minimizes the total cost of edits, where the costs of individual edits, depending on the edit type and the characters involved, are specified using a function w, normalized so that each edit costs at least one. The textbook dynamic-programming procedure, given strings X,Y ∈ Σ^{≤ n} and oracle access to w, computes ed^w(X,Y) in 𝒪(n²) time. Nevertheless, one can achieve better running times if the computed distance, denoted k, is small: 𝒪(n+k²) for unit weights [Landau and Vishkin; JCSS'88] and Õ(n+√{nk³}) for arbitrary weights [Cassis, Kociumaka, Wellnitz; FOCS'23]. In this paper, we study the dynamic version of the weighted edit distance problem, where the goal is to maintain ed^w(X,Y) for strings X,Y ∈ Σ^{≤ n} that change over time, with each update specified as an edit in X or Y. Very recently, Gorbachev and Kociumaka [STOC'25] showed that the unweighted distance ed(X,Y) can be maintained in Õ(k) time per update after Õ(n+k²)-time preprocessing; here, k denotes the current value of ed(X,Y). Their algorithm generalizes to small integer weights, but the underlying approach is incompatible with large weights. Our main result is a dynamic algorithm that maintains ed^w(X,Y) in Õ(k^{3-γ}) time per update after Õ(nk^γ)-time preprocessing. Here, γ ∈ [0,1] is a real trade-off parameter and k ≥ 1 is an integer threshold fixed at preprocessing time, with ∞ returned whenever ed^w(X,Y) > k. We complement our algorithm with conditional lower bounds showing fine-grained optimality of our trade-off for γ ∈ [0.5,1) and justifying our choice to fix k. We also generalize our solution to a much more robust setting while preserving the fine-grained optimal trade-off. Our full algorithm maintains X ∈ Σ^{≤ n} subject not only to character edits but also substring deletions and copy-pastes, each supported in Õ(k²) time. Instead of dynamically maintaining Y, it answers queries that, given any string Y specified through a sequence of 𝒪(k) arbitrary edits transforming X into Y, in Õ(k^{3-γ}) time compute ed^w(X,Y) or report that ed^w(X,Y) > k.
Cite as

Itai Boneh, Egor Gorbachev, and Tomasz Kociumaka. Bounded Weighted Edit Distance: Dynamic Algorithms and Matching Lower Bounds. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 45:1-45:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{boneh_et_al:LIPIcs.ESA.2025.45,
  author =	{Boneh, Itai and Gorbachev, Egor and Kociumaka, Tomasz},
  title =	{{Bounded Weighted Edit Distance: Dynamic Algorithms and Matching Lower Bounds}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{45:1--45:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.45},
  URN =		{urn:nbn:de:0030-drops-245139},
  doi =		{10.4230/LIPIcs.ESA.2025.45},
  annote =	{Keywords: Edit distance, dynamic algorithms, conditional lower bounds}
}
Document
DOI: 10.4230/LIPIcs.CPM.2025.6
String Problems in the Congested Clique Model

Authors: Shay Golan and Matan Kraus

Published in: LIPIcs, Volume 331, 36th Annual Symposium on Combinatorial Pattern Matching (CPM 2025)

Abstract
In this paper we present algorithms for several string problems in the Congested Clique model. In the Congested Clique model, n nodes (computers) are used to solve some problem. The input to the problem is distributed among the nodes, and the communication between the nodes is conducted in rounds. In each round, every node is allowed to send an O(log n)-bit message to every other node in the network. We consider three fundamental string problems in the Congested Clique model. First, we present an O(1) rounds algorithm for string sorting that supports strings of arbitrary length. Second, we present an O(1) rounds combinatorial pattern matching algorithm. Finally, we present an O(log log n) rounds algorithm for the computation of the suffix array and the corresponding Longest Common Prefix array of a given string.
Cite as

Shay Golan and Matan Kraus. String Problems in the Congested Clique Model. In 36th Annual Symposium on Combinatorial Pattern Matching (CPM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 331, pp. 6:1-6:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{golan_et_al:LIPIcs.CPM.2025.6,
  author =	{Golan, Shay and Kraus, Matan},
  title =	{{String Problems in the Congested Clique Model}},
  booktitle =	{36th Annual Symposium on Combinatorial Pattern Matching (CPM 2025)},
  pages =	{6:1--6:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-369-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{331},
  editor =	{Bonizzoni, Paola and M\"{a}kinen, Veli},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2025.6},
  URN =		{urn:nbn:de:0030-drops-231003},
  doi =		{10.4230/LIPIcs.CPM.2025.6},
  annote =	{Keywords: String Sorting, Pattern Matching, Suffix Array, Congested Clique, Sorting}
}
Document
DOI: 10.4230/LIPIcs.CPM.2025.13
Doubly-Periodic String Comparison

Authors: Nikita Gaevoy, Boris Zolotov, and Alexander Tiskin

Published in: LIPIcs, Volume 331, 36th Annual Symposium on Combinatorial Pattern Matching (CPM 2025)

Abstract
The longest common subsequence (LCS) problem is a fundamental algorithmic problem. Given a pair of strings, the problem asks for the length of the longest string that is a subsequence in both input strings. Among the many relatives of this problem, there is its natural version where one or both of input strings have periodic structure. The case where only one of the input strings is periodic has been considered before; in this work, we develop an efficient algorithm for the more difficult case where both input strings are periodic. The algorithm is based on the existing algebraic framework for the LCS problem, developed by the third author; in particular, we extend this framework to dealing with affine (i.e. doubly-infinite periodic) permutations instead of finite ones. Given input strings that are a k-repeat of a period of length m and an 𝓁-repeat of a period of length n, the resulting algorithm runs in time O(mn+n log n log k), which is a substantial improvement over existing approaches. The algorithm has been implemented by the first author; by running his code, one can process pairs of periodic input strings with lengths far beyond the reach of all known alternative algorithms.
Cite as

Nikita Gaevoy, Boris Zolotov, and Alexander Tiskin. Doubly-Periodic String Comparison. In 36th Annual Symposium on Combinatorial Pattern Matching (CPM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 331, pp. 13:1-13:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{gaevoy_et_al:LIPIcs.CPM.2025.13,
  author =	{Gaevoy, Nikita and Zolotov, Boris and Tiskin, Alexander},
  title =	{{Doubly-Periodic String Comparison}},
  booktitle =	{36th Annual Symposium on Combinatorial Pattern Matching (CPM 2025)},
  pages =	{13:1--13:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-369-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{331},
  editor =	{Bonizzoni, Paola and M\"{a}kinen, Veli},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2025.13},
  URN =		{urn:nbn:de:0030-drops-231079},
  doi =		{10.4230/LIPIcs.CPM.2025.13},
  annote =	{Keywords: String Comparison, periodic Strings, Longest common Subsequence, affine Hecke Monoid, affine sticky Braids}
}
Document
DOI: 10.4230/LIPIcs.STACS.2025.45
Forbidden Patterns in Mixed Linear Layouts

Authors: Deborah Haun, Laura Merker, and Sergey Pupyrev

Published in: LIPIcs, Volume 327, 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)

Abstract
An ordered graph is a graph with a total order over its vertices. A linear layout of an ordered graph is a partition of the edges into sets of either non-crossing edges, called stacks, or non-nesting edges, called queues. The stack (queue) number of an ordered graph is the minimum number of required stacks (queues). Mixed linear layouts combine these layouts by allowing each set of edges to form either a stack or a queue. The minimum number of stacks plus queues is called the mixed page number. It is well known that ordered graphs with small stack number are characterized, up to a function, by the absence of large twists (that is, pairwise crossing edges). Similarly, ordered graphs with small queue number are characterized by the absence of large rainbows (that is, pairwise nesting edges). However, no such characterization via forbidden patterns is known for mixed linear layouts. We address this gap by introducing patterns similar to twists and rainbows, which we call thick patterns; such patterns allow a characterization, again up to a function, of mixed linear layouts of bounded-degree graphs. That is, we show that a family of ordered graphs with bounded maximum degree has bounded mixed page number if and only if the size of the largest thick pattern is bounded. In addition, we investigate an exact characterization of ordered graphs whose mixed page number equals a fixed integer k via a finite set of forbidden patterns. We show that for k = 2, there is no such characterization, which supports the nature of our first result.
Cite as

Deborah Haun, Laura Merker, and Sergey Pupyrev. Forbidden Patterns in Mixed Linear Layouts. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 45:1-45:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{haun_et_al:LIPIcs.STACS.2025.45,
  author =	{Haun, Deborah and Merker, Laura and Pupyrev, Sergey},
  title =	{{Forbidden Patterns in Mixed Linear Layouts}},
  booktitle =	{42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)},
  pages =	{45:1--45:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-365-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{327},
  editor =	{Beyersdorff, Olaf and Pilipczuk, Micha{\l} and Pimentel, Elaine and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2025.45},
  URN =		{urn:nbn:de:0030-drops-228717},
  doi =		{10.4230/LIPIcs.STACS.2025.45},
  annote =	{Keywords: Ordered Graphs, linear Layout, mixed linear Layout, Stack Layout, Queue Layout}
}
Document
DOI: 10.4230/LIPIcs.ESA.2022.86
Fast RSK Correspondence by Doubling Search

Authors: Alexander Tiskin

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Abstract
The Robinson-Schensted-Knuth (RSK) correspondence is a fundamental concept in combinatorics and representation theory. It is defined as a certain bijection between permutations and pairs of Young tableaux of a given order. We consider the RSK correspondence as an algorithmic problem, along with the closely related k-chain problem. We give a simple, direct description of the symmetric RSK algorithm, which is implied by the k-chain algorithms of Viennot and of Felsner and Wernisch. We also show how the doubling search of Bentley and Yao can be used as a subroutine by the symmetric RSK algorithm, replacing the default binary search. Surprisingly, such a straightforward replacement improves the asymptotic worst-case running time for the RSK correspondence that has been best known since 1998. A similar improvement also holds for the average running time of RSK on uniformly random permutations.
Cite as

Alexander Tiskin. Fast RSK Correspondence by Doubling Search. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 86:1-86:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{tiskin:LIPIcs.ESA.2022.86,
  author =	{Tiskin, Alexander},
  title =	{{Fast RSK Correspondence by Doubling Search}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{86:1--86:10},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.86},
  URN =		{urn:nbn:de:0030-drops-170249},
  doi =		{10.4230/LIPIcs.ESA.2022.86},
  annote =	{Keywords: combinatorics of permutations, Robinson-Schensted-Knuth correspondence, k-chains, RSK algorithm}
}
Document
DOI: 10.4230/LIPIcs.WABI.2019.16
Bounded-Length Smith-Waterman Alignment

Authors: Alexander Tiskin

Published in: LIPIcs, Volume 143, 19th International Workshop on Algorithms in Bioinformatics (WABI 2019)

Abstract
Given a fixed alignment scoring scheme, the bounded length (respectively, bounded total length) Smith-Waterman alignment problem on a pair of strings of lengths m, n, asks for the maximum alignment score across all substring pairs, such that the first substring’s length (respectively, the sum of the two substrings' lengths) is above the given threshold w. The latter problem was introduced by Arslan and Eğecioğlu under the name "local alignment with length threshold". They proposed a dynamic programming algorithm solving the problem in time O(mn^2), and also an approximation algorithm running in time O(rmn), where r is a parameter controlling the accuracy of approximation. We show that both these problems can be solved exactly in time O(mn), assuming a rational scoring scheme; furthermore, this solution can be used to obtain an exact algorithm for the normalised bounded total length Smith - Waterman alignment problem, running in time O(mn log n). Our algorithms rely on the techniques of fast window-substring alignment and implicit unit-Monge matrix searching, developed previously by the author and others.
Cite as

Alexander Tiskin. Bounded-Length Smith-Waterman Alignment. In 19th International Workshop on Algorithms in Bioinformatics (WABI 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 143, pp. 16:1-16:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{tiskin:LIPIcs.WABI.2019.16,
  author =	{Tiskin, Alexander},
  title =	{{Bounded-Length Smith-Waterman Alignment}},
  booktitle =	{19th International Workshop on Algorithms in Bioinformatics (WABI 2019)},
  pages =	{16:1--16:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-123-8},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{143},
  editor =	{Huber, Katharina T. and Gusfield, Dan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WABI.2019.16},
  URN =		{urn:nbn:de:0030-drops-110461},
  doi =		{10.4230/LIPIcs.WABI.2019.16},
  annote =	{Keywords: sequence alignment, local alignment, Smith, Waterman alignment, matrix searching}
}
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