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Volume

LIPIcs, Volume 161

31st Annual Symposium on Combinatorial Pattern Matching (CPM 2020)

CPM 2020, June 17-19, 2020, Copenhagen, Denmark

Editors: Inge Li Gørtz and Oren Weimann

Document

DOI: 10.4230/LIPIcs.STACS.2026.79

Mind the Gap. Doubling Constant Parametrization of Weighted Problems: TSP, Max-Cut, and More

Authors: Mihail Stoian

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

Abstract

Despite much research, hard weighted problems still resist super-polynomial improvements over their textbook solution. On the other hand, the unweighted versions of these problems have recently witnessed the sought-after speedups. Currently, the only way to repurpose the algorithm of the unweighted version for the weighted version is to employ a polynomial embedding of the input weights. This, however, introduces a pseudo-polynomial factor into the running time, which becomes impractical for arbitrarily weighted instances. In this paper, we introduce a new way to repurpose the algorithm of the unweighted problem. Specifically, we show that the time complexity of several well-known NP-hard problems operating over the (min, +) and (max, +) semirings, such as TSP, Weighted Max-Cut, and Edge-Weighted k-Clique, is proportional to that of their unweighted versions when the set of input weights has small doubling. We achieve this by a meta-algorithm that converts the input weights into polynomially bounded integers using the recent constructive Freiman’s theorem by Randolph and Węgrzycki [ESA 2024] before applying the polynomial embedding.

Cite as

Mihail Stoian. Mind the Gap. Doubling Constant Parametrization of Weighted Problems: TSP, Max-Cut, and More. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 79:1-79:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{stoian:LIPIcs.STACS.2026.79,
  author =	{Stoian, Mihail},
  title =	{{Mind the Gap. Doubling Constant Parametrization of Weighted Problems: TSP, Max-Cut, and More}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{79:1--79:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle 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.2026.79},
  URN =		{urn:nbn:de:0030-drops-255680},
  doi =		{10.4230/LIPIcs.STACS.2026.79},
  annote =	{Keywords: doubling constant parametrization, weighted problems, traveling salesman, weighted max-cut, edge-weighted k-clique}
}

Document

DOI: 10.4230/LIPIcs.STACS.2026.26

Approximate Cartesian Tree Matching with Substitutions

Authors: Panagiotis Charalampopoulos, Jonas Ellert, and Manal Mohamed

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

Abstract

The Cartesian tree of a sequence captures the relative order of the sequence’s elements. In recent years, Cartesian tree matching has attracted considerable attention, particularly due to its applications in time series analysis. Consider a text T of length n and a pattern P of length m. In the exact Cartesian tree matching problem, the task is to find all length-m fragments of T whose Cartesian tree coincides with the Cartesian tree CT(P) of the pattern. Although the exact version of the problem can be solved in linear time [Park et al., TCS 2020], it remains rather restrictive; for example, it is not robust to outliers in the pattern. To overcome this limitation, we consider the approximate setting, where the goal is to identify all fragments of T that are close to some string whose Cartesian tree matches CT(P). In this work, we quantify closeness via the widely used Hamming distance metric. For a given integer parameter k > 0, we present an algorithm that computes all fragments of T that are at Hamming distance at most k from a string whose Cartesian tree matches CT(P). Our algorithm runs in time 𝒪(n √m ⋅ k^{2.5}) for k ≤ m^{1/5} and in time 𝒪(nk⁵) for k ≥ m^{1/5}, thereby improving upon the state-of-the-art 𝒪(nmk)-time algorithm of Kim and Han [TCS 2025] in the regime k = o(m^{1/4}). On the way to our solution, we develop a toolbox of independent interest. First, we introduce a new notion of periodicity in Cartesian trees. Then, we lift multiple well-known combinatorial and algorithmic results for string matching and periodicity in strings to Cartesian tree matching and periodicity in Cartesian trees.

Cite as

Panagiotis Charalampopoulos, Jonas Ellert, and Manal Mohamed. Approximate Cartesian Tree Matching with Substitutions. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 26:1-26:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{charalampopoulos_et_al:LIPIcs.STACS.2026.26,
  author =	{Charalampopoulos, Panagiotis and Ellert, Jonas and Mohamed, Manal},
  title =	{{Approximate Cartesian Tree Matching with Substitutions}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{26:1--26:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle 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.2026.26},
  URN =		{urn:nbn:de:0030-drops-255151},
  doi =		{10.4230/LIPIcs.STACS.2026.26},
  annote =	{Keywords: Cartesian tree, Hamming distance, approximate pattern matching}
}

Document

DOI: 10.4230/LIPIcs.STACS.2026.62

Relative Compressed Reverse Suffix Array

Authors: Muhammed Oguzhan Kulekci, Mano Prakash Parthasarathi, Rahul Shah, and Sharma V. Thankachan

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

Abstract

Suffix trees and suffix arrays are two fundamental data structures in the field of string algorithms. For a string (a.k.a. text or sequence) of length n over an alphabet of size σ, these structures typically require O(nlog n) bits of space. The FM-index provides a compressed representation of the suffix array in ≈ nlog σ bits, allowing for efficient queries on both the suffix array and its inverse array in near logarithmic time. In certain applications, such as approximate pattern matching (i.e., with wildcards, mismatches, edits), there is a need to access the suffix array of a text, as well as the suffix array of text’s reverse. Motivated by this, we explore the possibility of encoding the suffix array of the reversed text in a compact form, assuming the availability of the FM-index for the original text. Our first solution is an O(n)-bit (relative) encoding of the suffix array of the reversed text, with the time for decoding an entry being only O(log^*n) times that of decoding an entry in the text’s suffix array using FM-index. We then demonstrate how to reduce the space to O(n/κ) bits for a parameter κ, while multiplicative factor in time becomes approximately O(κlog^*n+κ³). We can also support inverse suffix array and longest common extension queries on the reversed text. These results are achieved through some careful and non-trivial application of various succinct data structure techniques.

Cite as

Muhammed Oguzhan Kulekci, Mano Prakash Parthasarathi, Rahul Shah, and Sharma V. Thankachan. Relative Compressed Reverse Suffix Array. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 62:1-62:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{kulekci_et_al:LIPIcs.STACS.2026.62,
  author =	{Kulekci, Muhammed Oguzhan and Parthasarathi, Mano Prakash and Shah, Rahul and Thankachan, Sharma V.},
  title =	{{Relative Compressed Reverse Suffix Array}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{62:1--62:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle 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.2026.62},
  URN =		{urn:nbn:de:0030-drops-255512},
  doi =		{10.4230/LIPIcs.STACS.2026.62},
  annote =	{Keywords: String Matching, Text Indexing, Data Structures, Suffix Trees}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2026.5

Maximum-Flow and Minimum-Cut Sensitivity Oracles for Directed Graphs

Authors: Mridul Ahi, Keerti Choudhary, Shlok Pande, Pushpraj, and Lakshay Saggi

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

Abstract

This paper addresses the problem of designing fault-tolerant data structures for the (s,t)-max-flow and (s,t)-min-cut problems in unweighted directed graphs. Given a directed graph G = (V, E) with a designated source s, sink t, and an (s,t)-max-flow of value λ, we present constructions for max-flow and min-cut sensitivity oracles, and introduce the concept of a fault-tolerant flow family, which may be of independent interest. Our main contributions are as follows. 1) Fault-Tolerant Flow Family: We construct a family ℬ of 2λ+1 (s,t)-flows such that for every edge e, ℬ contains an (s,t)-max-flow of G-e. This covering property is tight up to constants for single failures and provably cannot extend to comparably small families for k ≥ 2, where we show an Ω(n) lower bound on the family size, independent of λ. 2) Max-Flow Sensitivity Oracle: Using the fault-tolerant flow family, we construct a single as well as dual-edge sensitivity oracle for (s,t)-max-flow that requires only O(λ n) space. Given any set F of up to two failing edges, the oracle reports the updated max-flow value in G-F in O(n) time. Additionally, for the single-failure case, the oracle can determine in constant time whether the flow through an edge x changes when another edge e fails. 3) Min-Cut Sensitivity Oracle for Dual Failures: Recently, Baswana et al. (ICALP’22) designed an O(n²)-sized oracle for answering (s,t)-min-cut size queries under dual edge failures in constant time, along with a matching lower bound. We extend this by focusing on graphs with small min-cut values λ, and present a more compact oracle of size O(λ n) that answers such min-cut size queries in constant time and reports the corresponding (s,t)-min-cut partition in O(n) time. We also show that the space complexity of our oracle is asymptotically optimal in this setting. 4) Min-Cut Sensitivity Oracle for Multiple Failures: We extend our results to the general case of k edge failures. For any graph with (s,t)-min-cut of size λ, we construct a k-fault-tolerant min-cut oracle with space complexity O_{λ,k}(n log n) that answers min-cut size queries in O_{λ,k}(log n) time. This also leads to improved fault-tolerant (s,t)-reachability oracles, achieving O(n log n) space and O(log n) query time for up to k = O(1) edge failures.

Cite as

Mridul Ahi, Keerti Choudhary, Shlok Pande, Pushpraj, and Lakshay Saggi. Maximum-Flow and Minimum-Cut Sensitivity Oracles for Directed Graphs. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 5:1-5:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{ahi_et_al:LIPIcs.ITCS.2026.5,
  author =	{Ahi, Mridul and Choudhary, Keerti and Pande, Shlok and Pushpraj and Saggi, Lakshay},
  title =	{{Maximum-Flow and Minimum-Cut Sensitivity Oracles for Directed Graphs}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{5:1--5:24},
  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.2026.5},
  URN =		{urn:nbn:de:0030-drops-252920},
  doi =		{10.4230/LIPIcs.ITCS.2026.5},
  annote =	{Keywords: Fault tolerance, Data structures, Minimum cuts, Maximum flows}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2026.4

Pseudodeterministic Algorithms for Minimum Cut Problems

Authors: Aryan Agarwala and Nithin Varma

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

Abstract

In this paper we present efficient pseudodeterministic algorithms for both the global minimum cut and minimum s-t cut problems. The running time of our algorithm for the global minimum cut problem is asymptotically better than the fastest sequential deterministic global minimum cut algorithm (Henzinger, Li, Rao, Wang; SODA 2024). Furthermore, we implement our algorithm in streaming, PRAM, and cut-query models, where no efficient deterministic global minimum cut algorithms are known.

Cite as

Aryan Agarwala and Nithin Varma. Pseudodeterministic Algorithms for Minimum Cut Problems. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 4:1-4:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{agarwala_et_al:LIPIcs.ITCS.2026.4,
  author =	{Agarwala, Aryan and Varma, Nithin},
  title =	{{Pseudodeterministic Algorithms for Minimum Cut Problems}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{4:1--4:15},
  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.2026.4},
  URN =		{urn:nbn:de:0030-drops-252917},
  doi =		{10.4230/LIPIcs.ITCS.2026.4},
  annote =	{Keywords: Minimum Cut, Pseudodeterministic Algorithms}
}

Document

DOI: 10.4230/LIPIcs.ISAAC.2025.12

Star-Based Separators for Intersection Graphs of c-Colored Pseudo-Segments

Authors: Mark de Berg, Bart M. P. Jansen, and Jeroen S. K. Lamme

Published in: LIPIcs, Volume 359, 36th International Symposium on Algorithms and Computation (ISAAC 2025)

Abstract

The Planar Separator Theorem, which states that any planar graph 𝒢 has a separator consisting of O(√n) nodes whose removal partitions 𝒢 into components of size at most 2n/3, is a widely used tool to obtain fast algorithms on planar graphs. Intersection graphs of disks, which generalize planar graphs, do not admit such separators. It has recently been shown that disk graphs do admit so-called clique-based separators that consist of O(√n) cliques. This result has been generalized to intersection graphs of various other types of disk-like objects. Unfortunately, segment intersection graphs do not admit small clique-based separators, because they can contain arbitrarily large bicliques. This is true even in the simple case of axis-aligned segments. In this paper we therefore introduce biclique-based separators (and, in particular, star-based separators), which are separators consisting of a small number of bicliques (or stars). We prove that any c-oriented set of n segments in the plane, where c is a constant, admits a star-based separator consisting of O(√n) stars. In fact, our result is more general, as it applies to any set of n pseudo-segments that is partitioned into c subsets such that the pseudo-segments in the same subset are pairwise disjoint. We extend our result to intersection graphs of c-oriented polygons. These results immediately lead to an almost-exact distance oracle for such intersection graphs, which has O(n√n) storage and O(√n) query time, and that can report the hop-distance between any two query nodes in the intersection graph with an additive error of at most 2. This is the first distance oracle for such types of intersection graphs that has subquadratic storage and sublinear query time and that only has an additive error.

Cite as

Mark de Berg, Bart M. P. Jansen, and Jeroen S. K. Lamme. Star-Based Separators for Intersection Graphs of c-Colored Pseudo-Segments. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 12:1-12:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{deberg_et_al:LIPIcs.ISAAC.2025.12,
  author =	{de Berg, Mark and Jansen, Bart M. P. and Lamme, Jeroen S. K.},
  title =	{{Star-Based Separators for Intersection Graphs of c-Colored Pseudo-Segments}},
  booktitle =	{36th International Symposium on Algorithms and Computation (ISAAC 2025)},
  pages =	{12:1--12:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-408-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{359},
  editor =	{Chen, Ho-Lin and Hon, Wing-Kai and Tsai, Meng-Tsung},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2025.12},
  URN =		{urn:nbn:de:0030-drops-249207},
  doi =		{10.4230/LIPIcs.ISAAC.2025.12},
  annote =	{Keywords: Computational geometry, intersection graphs, biclique-based separators, distance oracles}
}

Document

DOI: 10.4230/LIPIcs.ESA.2025.94

Faster Algorithm for Bounded Tree Edit Distance in the Low-Distance Regime

Authors: Tomasz Kociumaka and Ali Shahali

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

Abstract

The tree edit distance is a natural dissimilarity measure between rooted ordered trees whose nodes are labeled over an alphabet Σ. It is defined as the minimum number of node edits - insertions, deletions, and relabelings - required to transform one tree into the other. The weighted variant assigns costs ≥ 1 to edits (based on node labels), minimizing total cost rather than edit count. The unweighted tree edit distance between two trees of total size n can be computed in 𝒪(n^{2.6857}) time; in contrast, determining the weighted tree edit distance is fine-grained equivalent to the All-Pairs Shortest Paths (APSP) problem and requires n³/2^Ω(√{log n}) time [Nogler, Polak, Saha, Vassilevska Williams, Xu, Ye; STOC'25]. These impractical super-quadratic times for large, similar trees motivate the bounded version, parameterizing runtime by the distance k to enable faster algorithms for k ≪ n. Prior algorithms for bounded unweighted edit distance achieve 𝒪(nk²log n) [Akmal & Jin; ICALP’21] and 𝒪(n + k⁷log k) [Das, Gilbert, Hajiaghayi, Kociumaka, Saha; STOC'23]. For weighted, only 𝒪(n + k^{15}) is known [Das, Gilbert, Hajiaghayi, Kociumaka, Saha; STOC'23]. We present an 𝒪(n + k⁶ log k)-time algorithm for bounded tree edit distance in both weighted/unweighted settings. First, we devise a simpler weighted 𝒪(nk² log n)-time algorithm. Next, we exploit periodic structures in input trees via an optimized universal kernel: modifying prior 𝒪(n)-time 𝒪(k⁵)-size kernels to generate such structured instances, enabling efficient analysis.

Cite as

Tomasz Kociumaka and Ali Shahali. Faster Algorithm for Bounded Tree Edit Distance in the Low-Distance Regime. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 94:1-94:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kociumaka_et_al:LIPIcs.ESA.2025.94,
  author =	{Kociumaka, Tomasz and Shahali, Ali},
  title =	{{Faster Algorithm for Bounded Tree Edit Distance in the Low-Distance Regime}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{94:1--94:16},
  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.2025.94},
  URN =		{urn:nbn:de:0030-drops-245634},
  doi =		{10.4230/LIPIcs.ESA.2025.94},
  annote =	{Keywords: tree edit distance, edit distance, kernelization, dynamic programming}
}

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.2025.72

Color Distance Oracles and Snippets: Separation Between Exact and Approximate Solutions

Authors: Noam Horowicz and Tsvi Kopelowitz

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

Abstract

In the snippets problem, the goal is to preprocess a text T so that given two pattern queries, P₁ and P₂, one can quickly locate the occurrences of the two patterns in T that are closest to each other, or report the distance between these occurrences. Kopelowitz and Krauthgamer [CPM2016] showed upper bound tradeoffs and conditional lower bounds tradeoffs for the snippets problem, by utilizing connections between the snippets problem and the problem of constructing a color distance oracle (CDO), which is a data structure that preprocess a set of points with associated colors so that given two colors c and c' one can quickly find the (distance between the) closest pair of points where one has color c and the other has color c'. However, the existing upper bound and lower bound curves are not tight. Inspired by recent advances by Kopelowitz and Vassilevska-Williams [ICALP2020] regarding tradeoff curves for Set-disjointness data structures, in this paper we introduce new conditionally optimal algorithms for a (1+ε) approximation version of the snippets problem and a (1+ε) approximation version of the CDO problem, by applying fast matrix multiplication. For example, for CDO on n points in an array, if the preprocessing time is Õ(n^a) and the query time is Õ(n^b) then, assuming that ω = 2 (where ω is the exponent of n in the runtime of the fastest matrix multiplication algorithm on two squared matrices of size n× n), we show that approximate CDO can be solved with the following tradeoff a + 2b = 2 (if 0 ≤ b ≤ 1/3) 2a + b = 3 (if 1/3 ≤ b ≤ 1). Moreover, we prove that for exact CDO on points in an array, the algorithm of Kopelowitz and Krauthgamer [CPM2016], which obtains a tradeoff of a+b = 2, is essentially optimal assuming that the strong all-pairs shortest paths hypothesis holds for randomized algorithms. Thus, we demonstrate that the exact version of CDO is strictly harder than the approximate version. Moreover, this separation carries over to the snippets problem.

Cite as

Noam Horowicz and Tsvi Kopelowitz. Color Distance Oracles and Snippets: Separation Between Exact and Approximate Solutions. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 72:1-72:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{horowicz_et_al:LIPIcs.ESA.2025.72,
  author =	{Horowicz, Noam and Kopelowitz, Tsvi},
  title =	{{Color Distance Oracles and Snippets: Separation Between Exact and Approximate Solutions}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{72:1--72:17},
  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.2025.72},
  URN =		{urn:nbn:de:0030-drops-245403},
  doi =		{10.4230/LIPIcs.ESA.2025.72},
  annote =	{Keywords: data structures, fast matrix multiplication, fine-grained complexity, pattern matching, distance oracles}
}

Document

DOI: 10.4230/LIPIcs.ESA.2025.44

Near-Optimal Vertex Fault-Tolerant Labels for Steiner Connectivity

Authors: Koustav Bhanja and Asaf Petruschka

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

Abstract

We present a compact labeling scheme for determining whether a designated set of terminals in a graph remains connected after any f (or less) vertex failures occur. An f-FT Steiner connectivity labeling scheme for an n-vertex graph G = (V,E) with terminal set U ⊆ V provides labels to the vertices of G, such that given only the labels of any subset F ⊆ V with |F| ≤ f, one can determine if U remains connected in G-F. The main complexity measure is the maximum label length. The special case U = V of global connectivity has been recently studied by Jiang, Parter, and Petruschka [Yonggang Jiang et al., 2025], who provided labels of n^{1-1/f} ⋅ poly(f,log n) bits. This is near-optimal (up to poly(f,log n) factors) by a lower bound of Long, Pettie and Saranurak [Yaowei Long et al., 2025]. Our scheme achieves labels of |U|^{1-1/f} ⋅ poly(f, log n) for general U ⊆ V, which is near-optimal for any given size |U| of the terminal set. To handle terminal sets, our approach differs from [Yonggang Jiang et al., 2025]. We use a well-structured Steiner tree for U produced by a decomposition theorem of Duan and Pettie [Ran Duan and Seth Pettie, 2020], and bypass the need for Nagamochi-Ibaraki sparsification [Hiroshi Nagamochi and Toshihide Ibaraki, 1992].

Cite as

Koustav Bhanja and Asaf Petruschka. Near-Optimal Vertex Fault-Tolerant Labels for Steiner Connectivity. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 44:1-44:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bhanja_et_al:LIPIcs.ESA.2025.44,
  author =	{Bhanja, Koustav and Petruschka, Asaf},
  title =	{{Near-Optimal Vertex Fault-Tolerant Labels for Steiner Connectivity}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{44:1--44:13},
  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.2025.44},
  URN =		{urn:nbn:de:0030-drops-245123},
  doi =		{10.4230/LIPIcs.ESA.2025.44},
  annote =	{Keywords: Fault Tolerance, Labeling Schemes, Steiner Connectivity}
}

Document

DOI: 10.4230/LIPIcs.ESA.2025.39

Going Beyond Surfaces in Diameter Approximation

Authors: Michał Włodarczyk

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

Abstract

Calculating the diameter of an undirected graph requires quadratic running time under the Strong Exponential Time Hypothesis and this barrier works even against any approximation better than 3/2. For planar graphs with positive edge weights, there are known (1+ε)-approximation algorithms with running time poly(1/ε, log n)⋅ n. However, these algorithms rely on shortest path separators and this technique falls short to yield efficient algorithms beyond graphs of bounded genus. In this work we depart from embedding-based arguments and obtain diameter approximations relying on VC set systems and the local treewidth property. We present two orthogonal extensions of the planar case by giving (1+ε)-approximation algorithms with the following running times: - 𝒪_h((1/ε)^𝒪(h) ⋅ nlog² n)-time algorithm for graphs excluding an apex graph of size h as a minor, - 𝒪_d((1/ε)^𝒪(d) ⋅ nlog² n)-time algorithm for the class of d-apex graphs. As a stepping stone, we obtain efficient (1+ε)-approximate distance oracles for graphs excluding an apex graph of size h as a minor. Our oracle has preprocessing time 𝒪_h((1/ε)⁸⋅ nlog nlog W) and query time 𝒪_h((1/ε)²⋅log n log W), where W is the metric stretch. Such oracles have been so far only known for bounded genus graphs. All our algorithms are deterministic.

Cite as

Michał Włodarczyk. Going Beyond Surfaces in Diameter Approximation. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 39:1-39:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{wlodarczyk:LIPIcs.ESA.2025.39,
  author =	{W{\l}odarczyk, Micha{\l}},
  title =	{{Going Beyond Surfaces in Diameter Approximation}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{39:1--39:19},
  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.2025.39},
  URN =		{urn:nbn:de:0030-drops-245076},
  doi =		{10.4230/LIPIcs.ESA.2025.39},
  annote =	{Keywords: diameter, approximation, distance oracles, graph minors, treewidth}
}

Document

RANDOM

DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.42

On the Spectral Expansion of Monotone Subsets of the Hypercube

Authors: Yumou Fei and Renato Ferreira Pinto Jr.

Published in: LIPIcs, Volume 353, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)

Abstract

We study the spectral gap of subgraphs of the hypercube induced by monotone subsets of vertices. For a monotone subset A ⊆ {0,1}ⁿ of density μ(A), the previous best lower bound on the spectral gap, due to Cohen [Cohen, 2016], was γ ≳ μ(A)/n², improving upon the earlier bound γ ≳ μ(A)²/n² established by Ding and Mossel [Ding and Mossel, 2014]. In this paper, we prove the optimal lower bound γ ≳ μ(A)/n. As a corollary, we improve the mixing time upper bound of the random walk on constant-density monotone sets from O(n³), as shown by Ding and Mossel, to O(n²). Along the way, we develop two new inequalities that may be of independent interest: (1) a directed L²-Poincaré inequality on the hypercube, and (2) an "approximate" FKG inequality for monotone sets.

Cite as

Yumou Fei and Renato Ferreira Pinto Jr.. On the Spectral Expansion of Monotone Subsets of the Hypercube. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 42:1-42:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{fei_et_al:LIPIcs.APPROX/RANDOM.2025.42,
  author =	{Fei, Yumou and Ferreira Pinto Jr., Renato},
  title =	{{On the Spectral Expansion of Monotone Subsets of the Hypercube}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{42:1--42:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-397-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{353},
  editor =	{Ene, Alina and Chattopadhyay, Eshan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.42},
  URN =		{urn:nbn:de:0030-drops-244081},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.42},
  annote =	{Keywords: Random walks, mixing time, FKG inequality, Poincar\'{e} inequality, directed isoperimetry}
}

Document

DOI: 10.4230/LIPIcs.WADS.2025.12

Testing Whether a Subgraph Is Convex or Isometric

Authors: Sergio Cabello

Published in: LIPIcs, Volume 349, 19th International Symposium on Algorithms and Data Structures (WADS 2025)

Abstract

We consider the following two algorithmic problems: given a graph G and a subgraph H ⊆ G, decide whether H is an isometric or a geodesically convex subgraph of G. It is relatively easy to see that the problems can be solved by computing the distances between all pairs of vertices. We provide a conditional lower bound showing that, for sparse graphs with n vertices and Θ(n) edges, we cannot expect to solve the problem in O(n^{2-ε}) time for any constant ε > 0. We also show that the problem can be solved in subquadratic time for planar graphs and in near-linear time for graphs of bounded treewidth. Finally, we provide a near-linear time algorithm for the setting where G is a plane graph and H is defined by a few cycles in G.

Cite as

Sergio Cabello. Testing Whether a Subgraph Is Convex or Isometric. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 12:1-12:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{cabello:LIPIcs.WADS.2025.12,
  author =	{Cabello, Sergio},
  title =	{{Testing Whether a Subgraph Is Convex or Isometric}},
  booktitle =	{19th International Symposium on Algorithms and Data Structures (WADS 2025)},
  pages =	{12:1--12:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-398-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{349},
  editor =	{Morin, Pat and Oh, Eunjin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WADS.2025.12},
  URN =		{urn:nbn:de:0030-drops-242439},
  doi =		{10.4230/LIPIcs.WADS.2025.12},
  annote =	{Keywords: convex subgraph, isometric subgraph, plane graph}
}

Document

DOI: 10.4230/LIPIcs.MFCS.2025.9

Linear Time Subsequence and Supersequence Regex Matching

Authors: Antoine Amarilli, Florin Manea, Tina Ringleb, and Markus L. Schmid

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract

It is well-known that checking whether a given string w matches a given regular expression r can be done in quadratic time O(|w|⋅ |r|) and that this cannot be improved to a truly subquadratic running time of O((|w|⋅ |r|)^{1-ε}) assuming the strong exponential time hypothesis (SETH). We study a different matching paradigm where we ask instead whether w has a subsequence that matches r, and show that regex matching in this sense can be solved in linear time O(|w| + |r|). Further, the same holds if we ask for a supersequence. We show that the quantitative variants where we want to compute a longest or shortest subsequence or supersequence of w that matches r can be solved in O(|w|⋅ |r|), i. e., asymptotically no worse than classical regex matching; and we show that O(|w| + |r|) is conditionally not possible for these problems. We also investigate these questions with respect to other natural string relations like the infix, prefix, left-extension or extension relation instead of the subsequence and supersequence relation. We further study the complexity of the universal problem where we ask if all subsequences (or supersequences, infixes, prefixes, left-extensions or extensions) of an input string satisfy a given regular expression.

Cite as

Antoine Amarilli, Florin Manea, Tina Ringleb, and Markus L. Schmid. Linear Time Subsequence and Supersequence Regex Matching. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 9:1-9:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{amarilli_et_al:LIPIcs.MFCS.2025.9,
  author =	{Amarilli, Antoine and Manea, Florin and Ringleb, Tina and Schmid, Markus L.},
  title =	{{Linear Time Subsequence and Supersequence Regex Matching}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{9:1--9:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.9},
  URN =		{urn:nbn:de:0030-drops-241162},
  doi =		{10.4230/LIPIcs.MFCS.2025.9},
  annote =	{Keywords: subsequence, supersequence, regular language, regular expression, automata}
}

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