DROPS

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DOI: 10.4230/LIPIcs.SEA.2026.4

Integer Programming Models for the Median of a 0-1 String Set Under Levenshtein Distance

Authors: Claudio Arbib, Andrea D'Ascenzo, Oya E. Karaşan, and Andrea Pizzuti

Published in: LIPIcs, Volume 371, 24th International Symposium on Experimental Algorithms (SEA 2026)

Abstract

The Median String Problem calls for finding a string that minimizes the average distance from a given set of strings. Under the Levenshtein (or edit) metric, the problem is NP-hard even for binary strings. We devised two novel integer linear programming models for this case and tested them against the only formulation we are aware of in the literature. Our numerical experiments attest to the efficacy of the proposed approach.

Cite as

Claudio Arbib, Andrea D'Ascenzo, Oya E. Karaşan, and Andrea Pizzuti. Integer Programming Models for the Median of a 0-1 String Set Under Levenshtein Distance. In 24th International Symposium on Experimental Algorithms (SEA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 371, pp. 4:1-4:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{arbib_et_al:LIPIcs.SEA.2026.4,
  author =	{Arbib, Claudio and D'Ascenzo, Andrea and Kara\c{s}an, Oya E. and Pizzuti, Andrea},
  title =	{{Integer Programming Models for the Median of a 0-1 String Set Under Levenshtein Distance}},
  booktitle =	{24th International Symposium on Experimental Algorithms (SEA 2026)},
  pages =	{4:1--4:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-422-2},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{371},
  editor =	{Aum\"{u}ller, Martin and Finocchi, Irene},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2026.4},
  URN =		{urn:nbn:de:0030-drops-260081},
  doi =		{10.4230/LIPIcs.SEA.2026.4},
  annote =	{Keywords: Levenshtein Distance, Median String Problem, Integer Programming}
}

Document

DOI: 10.4230/LIPIcs.CPM.2026.3

Computing k-mers in Graphs

Authors: Jarno N. Alanko and Máximo Pérez-López

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

Abstract

We initiate the study of computational problems on k-mers (strings of length k) in labeled graphs. As a starting point, we consider the problem of counting the number of distinct k-mers found on the walks of a graph. We establish that this is #P-hard, even on connected deterministic DAGs. However, in the class of deterministic Wheeler graphs (Gagie, Manzini, and Sirén, TCS 2017), we show that distinct k-mers of such a graph W = (V, E) can be counted using O(|W|k) or O(n⁴ log k) arithmetic operations, where n = |V|, m = |E| and |W| = n+m. The latter result uses a new generalization of the technique of prefix doubling to Wheeler graphs. To generalize our results beyond Wheeler graphs, we discuss ways to transform a graph into a Wheeler graph in a manner that preserves the k-mers. As an application of our k-mer counting algorithms, we construct a representation of the de Bruijn graph of the k-mers that occupies O(n_k + |W|k log(max_{1 ≤ 𝓁 ≤ k} n_𝓁) + σlog m) bits of space, where n_𝓁 is the number of distinct 𝓁-mers in the Wheeler graph, and σ is the size of the alphabet. We show how to construct it in the same time complexity. Given that the Wheeler graph can be exponentially smaller than the de Bruijn graph, for large k this provides a theoretical improvement over previous de Bruijn graph construction methods from graphs, which must spend Ω(k) time per k-mer in the graph.

Cite as

Jarno N. Alanko and Máximo Pérez-López. Computing k-mers in Graphs. In 37th Annual Symposium on Combinatorial Pattern Matching (CPM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 369, pp. 3:1-3:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{alanko_et_al:LIPIcs.CPM.2026.3,
  author =	{Alanko, Jarno N. and P\'{e}rez-L\'{o}pez, M\'{a}ximo},
  title =	{{Computing k-mers in Graphs}},
  booktitle =	{37th Annual Symposium on Combinatorial Pattern Matching (CPM 2026)},
  pages =	{3:1--3:17},
  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.2026.3},
  URN =		{urn:nbn:de:0030-drops-259294},
  doi =		{10.4230/LIPIcs.CPM.2026.3},
  annote =	{Keywords: Wheeler graph, Wheeler language, de Bruijn graph, graph, k-mer, q-gram, DFA, #P-hard}
}

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DOI: 10.4230/LIPIcs.CPM.2026.14

Hardness Results on Characteristics for Elastic-Degenerate Strings

Authors: Dominik Köppl and Jannik Olbrich

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

Abstract

Generalizations of plain strings have been proposed as a compact way to represent a collection of nearly identical sequences or to express uncertainty at specific text positions by enumerating all possibilities. While a plain string stores a character at each of its positions, generalizations consider a set of characters (indeterminate strings), a set of strings of equal length (generalized degenerate strings, or shortly GD strings), or a set of strings of arbitrary lengths (elastic-degenerate strings, or shortly ED strings). These generalizations are of importance to compactly represent such type of data, and find applications in bioinformatics for representing and maintaining a set of genetic sequences of the same taxonomy or a multiple sequence alignment. To be of use, attention has been drawn to answering various query types such as pattern matching or measuring similarity of ED strings by generalizing techniques known to plain strings. However, for some types of queries, it has been shown that a generalization of a polynomial-time solvable query on classic strings becomes NP-hard on ED strings, e.g. [Russo et al., 2022]. In that light, we wonder about other types of queries that are of particular interest to bioinformatics: unique substrings, absent words, anti-powers, longest previous factors, and Lempel-Ziv-like compression schemes. While we obtain a polynomial time algorithm for a variation of longest previous factors, we show that all other problems are NP-hard to compute, some of them even under the restriction that the input can be modeled as an indeterminate or GD string.

Cite as

Dominik Köppl and Jannik Olbrich. Hardness Results on Characteristics for Elastic-Degenerate Strings. In 37th Annual Symposium on Combinatorial Pattern Matching (CPM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 369, pp. 14:1-14:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{koppl_et_al:LIPIcs.CPM.2026.14,
  author =	{K\"{o}ppl, Dominik and Olbrich, Jannik},
  title =	{{Hardness Results on Characteristics for Elastic-Degenerate Strings}},
  booktitle =	{37th Annual Symposium on Combinatorial Pattern Matching (CPM 2026)},
  pages =	{14:1--14:25},
  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.2026.14},
  URN =		{urn:nbn:de:0030-drops-259409},
  doi =		{10.4230/LIPIcs.CPM.2026.14},
  annote =	{Keywords: Elastic-degenerate strings, NP-hardness, longest common factor, minimal unique substring, minimal absent word, anti-power, longest previous factor}
}

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.CPM.2026.27

Exploring the Gap Between LCS and LCStr

Authors: Shay Golan, Matan Kraus, Ely Porat, and B. Riva Shalom

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

Abstract

The Longest Common Subsequence (LCS) problem and the Longest Common Substring (LCStr) problem are classical string problems with broad theoretical and practical significance. The former has a quadratic conditional lower bound [FOCS, 2015], while the latter admits a linear-time solution. In this paper, we study a natural variation of these problems, the Longest Common Subsequence-Substring (LCSS) problem. The LCSS problem seeks the longest string that is simultaneously a subsequence of one input string and a substring of the other. This variant bridges LCS and LCStr, raising intriguing algorithmic questions: Does the complexity of computing LCSS interpolate between the linear time of LCStr and the quadratic time of LCS? What about approximability? We also examine a natural extension of LCSS to multiple strings, parameterizing the balance between subsequence and substring requirements. Our results reveal several insights. First, under the SETH conjecture, the inherent complexity of LCSS is quadratic, similar to LCS. In contrast, we provide a linear-time approximation for LCSS. Finally, for the multi-string variant, unlike both problems, we design a quadratic-time algorithm, uncovering deeper structural properties of the problem. By studying the complexity of the LCSS problem, we aim to gain some understanding of what influences whether a variant of the LCS problem behaves more like the standard LCS or like LCStr. Our findings suggest that hybrid constraints can create computational "sweet spots," where problems become more tractable than their pure counterparts. This opens a broader research direction in constraint-mediated algorithm design. Beyond LCSS itself, our work highlights unexpected connections between subsequence and substring constraints, advancing the theoretical understanding of string problems and laying the foundation for new algorithmic techniques and complexity-theoretic insights in the rich space between classical string comparison paradigms.

Cite as

Shay Golan, Matan Kraus, Ely Porat, and B. Riva Shalom. Exploring the Gap Between LCS and LCStr. In 37th Annual Symposium on Combinatorial Pattern Matching (CPM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 369, pp. 27:1-27:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{golan_et_al:LIPIcs.CPM.2026.27,
  author =	{Golan, Shay and Kraus, Matan and Porat, Ely and Shalom, B. Riva},
  title =	{{Exploring the Gap Between LCS and LCStr}},
  booktitle =	{37th Annual Symposium on Combinatorial Pattern Matching (CPM 2026)},
  pages =	{27:1--27:21},
  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.2026.27},
  URN =		{urn:nbn:de:0030-drops-259535},
  doi =		{10.4230/LIPIcs.CPM.2026.27},
  annote =	{Keywords: Longest Common Subsequence, Longest Common Substring, Conditional Lower Bound}
}

Document

DOI: 10.4230/LIPIcs.SWAT.2026.27

New Results on Three-Sided Skyline Range Counting and Reporting

Authors: Suruchi Kushwaha and Yakov Nekrich

Published in: LIPIcs, Volume 370, 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)

Abstract

In the orthogonal skyline range counting (resp. reporting) problem we store the set of points P in a data structure so that for any query range Q the number of points (resp. the list of all points) on the skyline of Q∩ P can be found efficiently. In this paper we study two-dimensional range counting and reporting problems in the case when the query range is bounded on three sides. We describe a linear-space data structure that answers top-open three-sided skyline counting queries in O(log log N) time, where N is the number of points stored in the data structure. We also show that bottom-open three-sided skyline counting queries are as difficult as general four-sided queries and any data structure that uses O(Nlog^c N) space for a constant c requires Ω(log N/log log N) time to answer such queries. Next, we turn to skyline color range queries. In this variant of the problem each point in P is assigned a color and we must count (resp. report) the distinct colors of point on the skyline of Q∩ P. We describe an O(N)-space data structure that answers top-open three-sided color counting queries in O(log N/log log N) time. Finally, we study top-open three-sided skyline color reporting in the EM model and describe a data structure that uses linear space and answers queries in O(k/B+1) I/Os where k is the number of colors on the skyline. This is the first external-memory data structure with optimal query cost and space usage for this problem.

Cite as

Suruchi Kushwaha and Yakov Nekrich. New Results on Three-Sided Skyline Range Counting and Reporting. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 27:1-27:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{kushwaha_et_al:LIPIcs.SWAT.2026.27,
  author =	{Kushwaha, Suruchi and Nekrich, Yakov},
  title =	{{New Results on Three-Sided Skyline Range Counting and Reporting}},
  booktitle =	{20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
  pages =	{27:1--27:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-421-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{370},
  editor =	{Fraigniaud, Pierre},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2026.27},
  URN =		{urn:nbn:de:0030-drops-260631},
  doi =		{10.4230/LIPIcs.SWAT.2026.27},
  annote =	{Keywords: Data Structures, Range Searching, Skyline Queries}
}

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.WABI.2025.7

Sequence Similarity Estimation by Random Subsequence Sketching

Authors: Ke Chen, Vinamratha Pattar, and Mingfu Shao

Published in: LIPIcs, Volume 344, 25th International Conference on Algorithms for Bioinformatics (WABI 2025)

Abstract

Sequence similarity estimation is essential for many bioinformatics tasks, including functional annotation, phylogenetic analysis, and overlap graph construction. Alignment-free methods aim to solve large-scale sequence similarity estimation by mapping sequences to more easily comparable features that can approximate edit distances efficiently. Substrings or k-mers, as the dominant choice of features, face an unavoidable compromise between sensitivity and specificity when selecting the proper k-value. Recently, subsequence-based features have shown improved performance, but they are computationally demanding, and determining the ideal subsequence length remains an intricate art. In this work, we introduce SubseqSketch, a novel alignment-free scheme that maps a sequence to an integer vector, where the entries correspond to dynamic, rather than fixed, lengths of random subsequences. The cosine similarity between these vectors exhibits a strong correlation with the edit similarity between the original sequences. Through experiments on benchmark datasets, we demonstrate that SubseqSketch is both efficient and effective across various alignment-free tasks, including nearest neighbor search and phylogenetic clustering. A C++ implementation of SubseqSketch is openly available at https://github.com/Shao-Group/SubseqSketch.

Cite as

Ke Chen, Vinamratha Pattar, and Mingfu Shao. Sequence Similarity Estimation by Random Subsequence Sketching. In 25th International Conference on Algorithms for Bioinformatics (WABI 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 344, pp. 7:1-7:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chen_et_al:LIPIcs.WABI.2025.7,
  author =	{Chen, Ke and Pattar, Vinamratha and Shao, Mingfu},
  title =	{{Sequence Similarity Estimation by Random Subsequence Sketching}},
  booktitle =	{25th International Conference on Algorithms for Bioinformatics (WABI 2025)},
  pages =	{7:1--7:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-386-7},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{344},
  editor =	{Brejov\'{a}, Bro\v{n}a and Patro, Rob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WABI.2025.7},
  URN =		{urn:nbn:de:0030-drops-239332},
  doi =		{10.4230/LIPIcs.WABI.2025.7},
  annote =	{Keywords: Alignment-free sequence comparison, Phylogenetic clustering, Nearest neighbor search, Edit distance embedding}
}

Document

DOI: 10.4230/LIPIcs.WABI.2025.4

Haplotype-Aware Long-Read Error Correction

Authors: Parvesh Barak, Daniel Gibney, and Chirag Jain

Published in: LIPIcs, Volume 344, 25th International Conference on Algorithms for Bioinformatics (WABI 2025)

Abstract

Error correction of long reads is an important initial step in genome assembly workflows. For organisms with ploidy greater than one, it is important to preserve haplotype-specific variation during read correction. This challenge has driven the development of several haplotype-aware correction methods. However, existing methods are based on either ad-hoc heuristics or deep learning approaches. In this paper, we introduce a rigorous formulation for this problem. Our approach builds on the minimum error correction framework used in reference-based haplotype phasing. We prove that the proposed formulation for error correction of reads in de novo context, i.e., without using a reference genome, is NP-hard. To make our exact algorithm scale to large datasets, we introduce practical heuristics. Experiments using PacBio HiFi sequencing datasets from human and plant genomes show that our approach achieves accuracy comparable to state-of-the-art methods. The software is freely available at https://github.com/at-cg/HALE.

Cite as

Parvesh Barak, Daniel Gibney, and Chirag Jain. Haplotype-Aware Long-Read Error Correction. In 25th International Conference on Algorithms for Bioinformatics (WABI 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 344, pp. 4:1-4:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{barak_et_al:LIPIcs.WABI.2025.4,
  author =	{Barak, Parvesh and Gibney, Daniel and Jain, Chirag},
  title =	{{Haplotype-Aware Long-Read Error Correction}},
  booktitle =	{25th International Conference on Algorithms for Bioinformatics (WABI 2025)},
  pages =	{4:1--4:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-386-7},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{344},
  editor =	{Brejov\'{a}, Bro\v{n}a and Patro, Rob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WABI.2025.4},
  URN =		{urn:nbn:de:0030-drops-239300},
  doi =		{10.4230/LIPIcs.WABI.2025.4},
  annote =	{Keywords: Genome assembly, phasing, clustering, overlap graph, consensus}
}

Document

DOI: 10.4230/LIPIcs.WABI.2025.17

An Efficient Data Structure and Algorithm for Long-Match Query in Run-Length Compressed BWT

Authors: Ahsan Sanaullah, Degui Zhi, and Shaojie Zhang

Published in: LIPIcs, Volume 344, 25th International Conference on Algorithms for Bioinformatics (WABI 2025)

Abstract

String matching problems in bioinformatics are typically for finding exact substring matches between a query and a reference text. Previous formulations often focus on maximum exact matches (MEMs). However, multiple occurrences of substrings of the query in the text that are long enough but not maximal may not be captured by MEMs. Such long matches can be informative, especially when the text is a collection of similar sequences such as genomes. In this paper, we describe a new type of match between a pattern and a text that aren't necessarily maximal in the query, but still contain useful matching information: locally maximal exact matches (LEMs). There are usually a large amount of LEMs, so we only consider those above some length threshold ℒ. These are referred to as long LEMs. The purpose of long LEMs is to capture substring matches between a query and a text that are not necessarily maximal in the pattern but still long enough to be important. Therefore efficient long LEMs finding algorithms are desired for these datasets. However, these datasets are too large to query on traditional string indexes. Fortunately, these datasets are very repetitive. Recently, compressed string indexes that take advantage of the redundancy in the data but retain efficient querying capability have been proposed as a solution. We therefore give an efficient algorithm for computing all the long LEMs of a query and a text in a BWT runs compressed string index. We describe an O(m+occ) expected time algorithm that relies on an O(r) words space string index for outputting all long LEMs of a pattern with respect to a text given the matching statistics of the pattern with respect to the text. Here m is the length of the query, occ is the number of long LEMs outputted, and r is the number of runs in the BWT of the text. The O(r) space string index we describe relies on an adaptation of the move data structure by Nishimoto and Tabei. We are able to support LCP[i] queries in constant time given SA[i]. In other words, we answer PLCP[i] queries in constant time. These PLCP queries enable the efficient long LEM query. Long LEMs may provide useful similarity information between a pattern and a text that MEMs may ignore. This information is particularly useful in pangenome and biobank scale haplotype panel contexts.

Cite as

Ahsan Sanaullah, Degui Zhi, and Shaojie Zhang. An Efficient Data Structure and Algorithm for Long-Match Query in Run-Length Compressed BWT. In 25th International Conference on Algorithms for Bioinformatics (WABI 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 344, pp. 17:1-17:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{sanaullah_et_al:LIPIcs.WABI.2025.17,
  author =	{Sanaullah, Ahsan and Zhi, Degui and Zhang, Shaojie},
  title =	{{An Efficient Data Structure and Algorithm for Long-Match Query in Run-Length Compressed BWT}},
  booktitle =	{25th International Conference on Algorithms for Bioinformatics (WABI 2025)},
  pages =	{17:1--17:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-386-7},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{344},
  editor =	{Brejov\'{a}, Bro\v{n}a and Patro, Rob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WABI.2025.17},
  URN =		{urn:nbn:de:0030-drops-239433},
  doi =		{10.4230/LIPIcs.WABI.2025.17},
  annote =	{Keywords: BWT, LEM, Long LEM, MEM, Run Length Compressed BWT, Move Data Structure, Pangenome}
}

Document

Research

DOI: 10.4230/OASIcs.Grossi.7

Conditional Lower Bounds for String Matching in Labelled Graphs

Authors: Massimo Equi

Published in: OASIcs, Volume 132, From Strings to Graphs, and Back Again: A Festschrift for Roberto Grossi's 60th Birthday (2025)

Abstract

The problem of String Matching in Labelled Graphs (SMLG) is one possible generalization of the classic problem of finding a string inside another of greater length. In its most general form, SMLG asks to find a match for a string into a graph, which can be directed or undirected. As for string matching, many different variations are possible. For example, the match could be exact or approximate, and the match could lie on a path or a walk. Some of these variations easily fall into the NP-hard realm, while other variants are solvable in polynomial time. For the latter ones, fine-grained complexity has been a game changer in proving quadratic conditional lower bounds, allowing to finally close the gap with those upper bounds that remained unmatched for almost two decades. If the match is allowed to be approximate, SMLG enjoys the same conditional quadratic lower bounds shown for example for edit distance (Backurs and Indyk, STOC '15). The case that really requires ad hoc conditional lower bounds is the one of finding an exact match that lies on a walk. In this work, we focus on explaining various conditional lower bounds for this version of SMLG, with the goal of giving an overall perspective that could help understand which aspects of the problem make it quadratic. We will introduce the reader to the field of fine-grained complexity and show how it can successfully provide the exact type of lower bounds needed for polynomial problems such as SMLG.

Cite as

Massimo Equi. Conditional Lower Bounds for String Matching in Labelled Graphs. In From Strings to Graphs, and Back Again: A Festschrift for Roberto Grossi's 60th Birthday. Open Access Series in Informatics (OASIcs), Volume 132, pp. 7:1-7:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{equi:OASIcs.Grossi.7,
  author =	{Equi, Massimo},
  title =	{{Conditional Lower Bounds for String Matching in Labelled Graphs}},
  booktitle =	{From Strings to Graphs, and Back Again: A Festschrift for Roberto Grossi's 60th Birthday},
  pages =	{7:1--7:13},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-391-1},
  ISSN =	{2190-6807},
  year =	{2025},
  volume =	{132},
  editor =	{Conte, Alessio and Marino, Andrea and Rosone, Giovanna and Vitter, Jeffrey Scott},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.Grossi.7},
  URN =		{urn:nbn:de:0030-drops-238063},
  doi =		{10.4230/OASIcs.Grossi.7},
  annote =	{Keywords: conditional lower bounds, strong exponential time hypothesis, fine-grained complexity, string matching, graphs}
}

Document

DOI: 10.4230/OASIcs.Manzini.3

BWT for String Collections

Authors: Davide Cenzato, Zsuzsanna Lipták, Nadia Pisanti, Giovanna Rosone, and Marinella Sciortino

Published in: OASIcs, Volume 131, The Expanding World of Compressed Data: A Festschrift for Giovanni Manzini's 60th Birthday (2025)

Abstract

We survey the different methods used for extending the BWT to collections of strings, following largely [Cenzato and Lipták, CPM 2022, Bioinformatics 2024]. We analyze the specific aspects and combinatorial properties of the resulting BWT variants and give a categorization of publicly available tools for computing the BWT of string collections. We show how the specific method used impacts on the resulting transform, including the number of runs, and on the dynamicity of the transform with respect to adding or removing strings from the collection. We then focus on the number of runs of these BWT variants and present the optimal BWT introduced in [Cenzato et al., DCC 2023], which implements an algorithm originally proposed by [Bentley et al., ESA 2020] to minimize the number of BWT-runs. We also discuss several recent heuristics and study their impact on the compression of biological sequences. We conclude with an overview of the applications and the impact of the BWT of string collections in bioinformatics.

Cite as

Davide Cenzato, Zsuzsanna Lipták, Nadia Pisanti, Giovanna Rosone, and Marinella Sciortino. BWT for String Collections. In The Expanding World of Compressed Data: A Festschrift for Giovanni Manzini's 60th Birthday. Open Access Series in Informatics (OASIcs), Volume 131, pp. 3:1-3:29, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{cenzato_et_al:OASIcs.Manzini.3,
  author =	{Cenzato, Davide and Lipt\'{a}k, Zsuzsanna and Pisanti, Nadia and Rosone, Giovanna and Sciortino, Marinella},
  title =	{{BWT for String Collections}},
  booktitle =	{The Expanding World of Compressed Data: A Festschrift for Giovanni Manzini's 60th Birthday},
  pages =	{3:1--3:29},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-390-4},
  ISSN =	{2190-6807},
  year =	{2025},
  volume =	{131},
  editor =	{Ferragina, Paolo and Gagie, Travis and Navarro, Gonzalo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.Manzini.3},
  URN =		{urn:nbn:de:0030-drops-239113},
  doi =		{10.4230/OASIcs.Manzini.3},
  annote =	{Keywords: Burrows-Wheeler transform, Extended Burrows-Wheeler transform, compressed text indexes, text compression, string collections, bioinformatics}
}

Document

DOI: 10.4230/OASIcs.Manzini.1

BWT and Combinatorics on Words

Authors: Gabriele Fici, Sabrina Mantaci, Antonio Restivo, Giuseppe Romana, Giovanna Rosone, and Marinella Sciortino

Published in: OASIcs, Volume 131, The Expanding World of Compressed Data: A Festschrift for Giovanni Manzini's 60th Birthday (2025)

Abstract

The Burrows-Wheeler Transform (BWT) is a reversible transformation on words (strings) introduced in 1994 in the context of data compression, which is a permutation of the characters in the word. Its clustering effect, i.e., the remarkable property of grouping identical characters (BWT runs) when they share common contexts, has made it a powerful tool for boosting compression performances and enabling efficient pattern searching in highly repetitive string collections. In this chapter, we analyze the Burrows-Wheeler transform under the combinatorial point of view, and we survey known properties and connections with different aspects of combinatorics on words. In particular, we focus on the properties of words in relation to the number of their BWT runs. The value r, which counts the number of BWT runs, impacts both compression performance and indexing efficiency, and is considered a measure to evaluate the above-mentioned clustering effect and, consequently, the repetitiveness of a word. We give an overview of the results relating r to other combinatorial repetitiveness measures related to the factor complexity. The chapter also explores extremal cases of the clustering effect. Finally, some results on the sensitivity of the measure r are considered, where the effects of combinatorial operations are studied, such as reversal, edits, and the application of morphisms.

Cite as

Gabriele Fici, Sabrina Mantaci, Antonio Restivo, Giuseppe Romana, Giovanna Rosone, and Marinella Sciortino. BWT and Combinatorics on Words. In The Expanding World of Compressed Data: A Festschrift for Giovanni Manzini's 60th Birthday. Open Access Series in Informatics (OASIcs), Volume 131, pp. 1:1-1:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{fici_et_al:OASIcs.Manzini.1,
  author =	{Fici, Gabriele and Mantaci, Sabrina and Restivo, Antonio and Romana, Giuseppe and Rosone, Giovanna and Sciortino, Marinella},
  title =	{{BWT and Combinatorics on Words}},
  booktitle =	{The Expanding World of Compressed Data: A Festschrift for Giovanni Manzini's 60th Birthday},
  pages =	{1:1--1:23},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-390-4},
  ISSN =	{2190-6807},
  year =	{2025},
  volume =	{131},
  editor =	{Ferragina, Paolo and Gagie, Travis and Navarro, Gonzalo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.Manzini.1},
  URN =		{urn:nbn:de:0030-drops-239090},
  doi =		{10.4230/OASIcs.Manzini.1},
  annote =	{Keywords: Burrows-Wheeler Transform, Combinatorics on Words, Clustering Effect, BWT Runs}
}

Document

DOI: 10.4230/OASIcs.Manzini.12

Wheeler Graphs and Wheeler Languages

Authors: Nicola Cotumaccio, Giovanna D'Agostino, Daniel Gibney, Alberto Policriti, Nicola Prezza, and Sharma V. Thankachan

Published in: OASIcs, Volume 131, The Expanding World of Compressed Data: A Festschrift for Giovanni Manzini's 60th Birthday (2025)

Abstract

Suffix sorting stands at the core of the most efficient solutions for indexed pattern matching: the suffix tree, the suffix array, compressed indexes based on the Burrows-Wheeler transform, and so on. In [Gagie, Manzini, Sirén, TCS 2017] this concept was extended to labeled graphs, obtaining the rich class of Wheeler graphs. This work opened a very fruitful line of research, ultimately generating results able to bridge the fields of compressed data structures, graph theory, and regular language theory. In a Wheeler graph, nodes are sorted according to the alphabetic order of their incoming labels, propagating this order through pairs of equally-labeled edges. This apparently-simple definition makes it possible to solve on Wheeler graphs problems (including, but not limited to: compression, subpath queries, NFA equivalence, determinization, minimization) that on general labeled graphs are extremely hard to solve, and induces a rich structure in the class of regular languages (Wheeler languages) recognized by automata whose state transition is a Wheeler graph. The goal of this survey is to provide a summary of (and intuitions behind) the results on Wheeler graphs that appeared in the literature since their introduction, in addition to a discussion of interesting problems that are still open in the field.

Cite as

Nicola Cotumaccio, Giovanna D'Agostino, Daniel Gibney, Alberto Policriti, Nicola Prezza, and Sharma V. Thankachan. Wheeler Graphs and Wheeler Languages. In The Expanding World of Compressed Data: A Festschrift for Giovanni Manzini's 60th Birthday. Open Access Series in Informatics (OASIcs), Volume 131, pp. 12:1-12:28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{cotumaccio_et_al:OASIcs.Manzini.12,
  author =	{Cotumaccio, Nicola and D'Agostino, Giovanna and Gibney, Daniel and Policriti, Alberto and Prezza, Nicola and Thankachan, Sharma V.},
  title =	{{Wheeler Graphs and Wheeler Languages}},
  booktitle =	{The Expanding World of Compressed Data: A Festschrift for Giovanni Manzini's 60th Birthday},
  pages =	{12:1--12:28},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-390-4},
  ISSN =	{2190-6807},
  year =	{2025},
  volume =	{131},
  editor =	{Ferragina, Paolo and Gagie, Travis and Navarro, Gonzalo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.Manzini.12},
  URN =		{urn:nbn:de:0030-drops-239205},
  doi =		{10.4230/OASIcs.Manzini.12},
  annote =	{Keywords: Wheeler languages, Wheeler graphs, pattern matching, indexing, compressed data structures}
}

Document

DOI: 10.4230/OASIcs.Manzini.13

Graph Indexing Beyond Wheeler Graphs

Authors: Jarno N. Alanko, Elena Biagi, Massimo Equi, Veli Mäkinen, Simon J. Puglisi, Nicola Rizzo, Kunihiko Sadakane, and Jouni Sirén

Published in: OASIcs, Volume 131, The Expanding World of Compressed Data: A Festschrift for Giovanni Manzini's 60th Birthday (2025)

Abstract

After the discovery of the FM index, which linked the Burrows-Wheeler transform (BWT) to pattern matching on strings, several contemporaneous strands of research began on indexing more complex structures with the BWT, such as tries, finite languages, de Bruijn graphs, and aligned sequences. These directions can now be viewed as culminating in the theory of Wheeler Graphs, but sometimes they go beyond. This chapter reviews the significant body of "proto Wheeler Graph" indexes, many of which exploit characteristics of their specific case to outperform Wheeler graphs, especially in practice.

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Jarno N. Alanko, Elena Biagi, Massimo Equi, Veli Mäkinen, Simon J. Puglisi, Nicola Rizzo, Kunihiko Sadakane, and Jouni Sirén. Graph Indexing Beyond Wheeler Graphs. In The Expanding World of Compressed Data: A Festschrift for Giovanni Manzini's 60th Birthday. Open Access Series in Informatics (OASIcs), Volume 131, pp. 13:1-13:29, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{alanko_et_al:OASIcs.Manzini.13,
  author =	{Alanko, Jarno N. and Biagi, Elena and Equi, Massimo and M\"{a}kinen, Veli and Puglisi, Simon J. and Rizzo, Nicola and Sadakane, Kunihiko and Sir\'{e}n, Jouni},
  title =	{{Graph Indexing Beyond Wheeler Graphs}},
  booktitle =	{The Expanding World of Compressed Data: A Festschrift for Giovanni Manzini's 60th Birthday},
  pages =	{13:1--13:29},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-390-4},
  ISSN =	{2190-6807},
  year =	{2025},
  volume =	{131},
  editor =	{Ferragina, Paolo and Gagie, Travis and Navarro, Gonzalo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.Manzini.13},
  URN =		{urn:nbn:de:0030-drops-239215},
  doi =		{10.4230/OASIcs.Manzini.13},
  annote =	{Keywords: indexing, compression, compressed data structures, string algorithms, pattern matching}
}

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