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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Consider words of length n. The set of all periods of a word of length n is a subset of {0,1,2,…,n-1}. However, any subset of {0,1,2,…,n-1} is not necessarily a valid set of periods. In a seminal paper in 1981, Guibas and Odlyzko proposed to encode the set of periods of a word into an n long binary string, called an autocorrelation, where a one at position i denotes the period i. They considered the question of recognizing a valid period set, and also studied the number of valid period sets for strings of length n, denoted κ_n. They conjectured that ln(κ_n) asymptotically converges to a constant times ln²(n). Although improved lower bounds for ln(κ_n)/ln²(n) were proposed in 2001, the question of a tight upper bound has remained open since Guibas and Odlyzko’s paper. Here, we exhibit an upper bound for this fraction, which implies its convergence and closes this longstanding conjecture. Moreover, we extend our result to find similar bounds for the number of correlations: a generalization of autocorrelations which encodes the overlaps between two strings.

Eric Rivals, Michelle Sweering, and Pengfei Wang. Convergence of the Number of Period Sets in Strings. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 100:1-100:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{rivals_et_al:LIPIcs.ICALP.2023.100, author = {Rivals, Eric and Sweering, Michelle and Wang, Pengfei}, title = {{Convergence of the Number of Period Sets in Strings}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {100:1--100:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.100}, URN = {urn:nbn:de:0030-drops-181527}, doi = {10.4230/LIPIcs.ICALP.2023.100}, annote = {Keywords: Autocorrelation, period, border, combinatorics, correlation, periodicity, upper bound, asymptotic convergence} }

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**Published in:** LIPIcs, Volume 191, 32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021)

The hierarchical overlap graph (HOG) is a graph that encodes overlaps from a given set P of n strings, as the overlap graph does. A best known algorithm constructs HOG in O(||P|| log n) time and O(||P||) space, where ||P|| is the sum of lengths of the strings in P. In this paper we present a new algorithm to construct HOG in O(||P||) time and space. Hence, the construction time and space of HOG are better than those of the overlap graph, which are O(||P|| + n²).

Sangsoo Park, Sung Gwan Park, Bastien Cazaux, Kunsoo Park, and Eric Rivals. A Linear Time Algorithm for Constructing Hierarchical Overlap Graphs. In 32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 191, pp. 22:1-22:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{park_et_al:LIPIcs.CPM.2021.22, author = {Park, Sangsoo and Park, Sung Gwan and Cazaux, Bastien and Park, Kunsoo and Rivals, Eric}, title = {{A Linear Time Algorithm for Constructing Hierarchical Overlap Graphs}}, booktitle = {32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021)}, pages = {22:1--22:9}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-186-3}, ISSN = {1868-8969}, year = {2021}, volume = {191}, editor = {Gawrychowski, Pawe{\l} and Starikovskaya, Tatiana}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2021.22}, URN = {urn:nbn:de:0030-drops-139736}, doi = {10.4230/LIPIcs.CPM.2021.22}, annote = {Keywords: overlap graph, hierarchical overlap graph, shortest superstring problem, border array} }

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**Published in:** LIPIcs, Volume 128, 30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019)

The boom of genomic sequencing makes compression of sets of sequences inescapable. This underlies the need for multi-string indexing data structures that helps compressing the data. The most prominent example of such data structures is the Burrows-Wheeler Transform (BWT), a reversible permutation of a text that improves its compressibility. A similar data structure, the eXtended Burrows-Wheeler Transform (XBW), is able to index a tree labelled with alphabet symbols. A link between a multi-string BWT and the Aho-Corasick automaton has already been found and led to a way to build a XBW from a multi-string BWT. We exhibit a stronger link between a multi-string BWT and a XBW by using the order of the concatenation in the multi-string. This bijective link has several applications: first, it allows one to build one data structure from the other; second, it enables one to compute an ordering of the input strings that optimises a Run-Length measure (i.e., the compressibility) of the BWT or of the XBW.

Bastien Cazaux and Eric Rivals. Linking BWT and XBW via Aho-Corasick Automaton: Applications to Run-Length Encoding. In 30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 128, pp. 24:1-24:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{cazaux_et_al:LIPIcs.CPM.2019.24, author = {Cazaux, Bastien and Rivals, Eric}, title = {{Linking BWT and XBW via Aho-Corasick Automaton: Applications to Run-Length Encoding}}, booktitle = {30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019)}, pages = {24:1--24:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-103-0}, ISSN = {1868-8969}, year = {2019}, volume = {128}, editor = {Pisanti, Nadia and P. Pissis, Solon}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2019.24}, URN = {urn:nbn:de:0030-drops-104955}, doi = {10.4230/LIPIcs.CPM.2019.24}, annote = {Keywords: Data Structure, Algorithm, Aho-Corasick Tree, compression, RLE} }

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**Published in:** LIPIcs, Volume 103, 17th International Symposium on Experimental Algorithms (SEA 2018)

Given a set P of words, the Shortest Linear Superstring (SLS) problem is an optimisation problem that asks for a superstring of P of minimal length. SLS has applications in data compression, where a superstring is a compact representation of P, and in bioinformatics where it models the first step of genome assembly. Unfortunately SLS is hard to solve (NP-hard) and to closely approximate (MAX-SNP-hard). If numerous polynomial time approximation algorithms have been devised, few articles report on their practical performance. We lack knowledge about how closely an approximate superstring can be from an optimal one in practice. Here, we exhibit a linear time algorithm that reports an upper and a lower bound on the length of an optimal superstring. The upper bound is the length of an approximate superstring. This algorithm can be used to evaluate beforehand whether one can get an approximate superstring whose length is close to the optimum for a given instance. Experimental results suggest that its approximation performance is orders of magnitude better than previously reported practical values. Moreover, the proposed algorithm remainso efficient even on large instances and can serve to explore in practice the approximability of SLS.

Bastien Cazaux, Samuel Juhel, and Eric Rivals. Practical lower and upper bounds for the Shortest Linear Superstring. In 17th International Symposium on Experimental Algorithms (SEA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 103, pp. 18:1-18:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{cazaux_et_al:LIPIcs.SEA.2018.18, author = {Cazaux, Bastien and Juhel, Samuel and Rivals, Eric}, title = {{Practical lower and upper bounds for the Shortest Linear Superstring}}, booktitle = {17th International Symposium on Experimental Algorithms (SEA 2018)}, pages = {18:1--18:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-070-5}, ISSN = {1868-8969}, year = {2018}, volume = {103}, editor = {D'Angelo, Gianlorenzo}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2018.18}, URN = {urn:nbn:de:0030-drops-89530}, doi = {10.4230/LIPIcs.SEA.2018.18}, annote = {Keywords: greedy, approximation, overlap, Concat-Cycles, cyclic cover, linear time, text compression} }

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**Published in:** LIPIcs, Volume 105, 29th Annual Symposium on Combinatorial Pattern Matching (CPM 2018)

A superstring of a set of words P = {s_1, ..., s_p } is a string that contains each word of P as substring. Given P, the well known Shortest Linear Superstring problem (SLS), asks for a shortest superstring of P. In a variant of SLS, called Multi-SLS, each word s_i comes with an integer m(i), its multiplicity, that sets a constraint on its number of occurrences, and the goal is to find a shortest superstring that contains at least m(i) occurrences of s_i. Multi-SLS generalizes SLS and is obviously as hard to solve, but it has been studied only in special cases (with words of length 2 or with a fixed number of words). The approximability of Multi-SLS in the general case remains open. Here, we study the approximability of Multi-SLS and that of the companion problem Multi-SCCS, which asks for a shortest cyclic cover instead of shortest superstring. First, we investigate the approximation of a greedy algorithm for maximizing the compression offered by a superstring or by a cyclic cover: the approximation ratio is 1/2 for Multi-SLS and 1 for Multi-SCCS. Then, we exhibit a linear time approximation algorithm, Concat-Greedy, and show it achieves a ratio of 4 regarding the superstring length. This demonstrates that for both measures Multi-SLS belongs to the class of APX problems.

Bastien Cazaux and Eric Rivals. Superstrings with multiplicities. In 29th Annual Symposium on Combinatorial Pattern Matching (CPM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 105, pp. 21:1-21:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{cazaux_et_al:LIPIcs.CPM.2018.21, author = {Cazaux, Bastien and Rivals, Eric}, title = {{Superstrings with multiplicities}}, booktitle = {29th Annual Symposium on Combinatorial Pattern Matching (CPM 2018)}, pages = {21:1--21:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-074-3}, ISSN = {1868-8969}, year = {2018}, volume = {105}, editor = {Navarro, Gonzalo and Sankoff, David and Zhu, Binhai}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2018.21}, URN = {urn:nbn:de:0030-drops-86881}, doi = {10.4230/LIPIcs.CPM.2018.21}, annote = {Keywords: greedy algorithm, approximation, overlap, cyclic cover, APX, subset system} }

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