Document

**Published in:** LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

A Lyndon word is a string that is lexicographically smaller than all of its non-trivial suffixes. For example, airbus is a Lyndon word, but amtrak is not a Lyndon word due to its suffix ak. The Lyndon array stores the length of the longest Lyndon prefix of each suffix of a string. For a length-n string over a general ordered alphabet, the array can be computed in O(n) time (Bille et al., ICALP 2020). However, on a word-RAM of word-width w ≥ log₂ n, linear time is not optimal if the string is over integer alphabet {0, … , σ} with σ ≪ n. In this case, the string can be stored in O(n log σ) bits (or O(n / log_σ n) words) of memory, and reading it takes only O(n / log_σ n) time. We show that O(n / log_σ n) time and words of space suffice to compute the succinct 2n-bit version of the Lyndon array. The time is optimal for w = O(log n). The algorithm uses precomputed lookup tables to perform significant parts of the computation in constant time. This is possible due to properties of periodic substrings, which we carefully analyze to achieve the desired result. We envision that the algorithm has applications in the computation of runs (maximal periodic substrings), where the Lyndon array plays a central role in both theoretically and practically fast algorithms.

Hideo Bannai and Jonas Ellert. Lyndon Arrays in Sublinear Time. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 14:1-14:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bannai_et_al:LIPIcs.ESA.2023.14, author = {Bannai, Hideo and Ellert, Jonas}, title = {{Lyndon Arrays in Sublinear Time}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {14:1--14:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. 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.2023.14}, URN = {urn:nbn:de:0030-drops-186670}, doi = {10.4230/LIPIcs.ESA.2023.14}, annote = {Keywords: Lyndon forest, Lyndon table, Lyndon array, sublinear time algorithms, word RAM algorithms, word packing, tabulation, lookup tables, periodicity} }

Document

**Published in:** LIPIcs, Volume 273, 23rd International Workshop on Algorithms in Bioinformatics (WABI 2023)

FM-indexes are a crucial data structure in DNA alignment, but searching with them usually takes at least one random access per character in the query pattern. Ferragina and Fischer [Ferragina and Fischer, 2007] observed in 2007 that word-based indexes often use fewer random accesses than character-based indexes, and thus support faster searches. Since DNA lacks natural word-boundaries, however, it is necessary to parse it somehow before applying word-based FM-indexing. Last year, Deng et al. [Deng et al., 2022] proposed parsing genomic data by induced suffix sorting, and showed the resulting word-based FM-indexes support faster counting queries than standard FM-indexes when patterns are a few thousand characters or longer. In this paper we show that using prefix-free parsing - which takes parameters that let us tune the average length of the phrases - instead of induced suffix sorting, gives a significant speedup for patterns of only a few hundred characters. We implement our method and demonstrate it is between 3 and 18 times faster than competing methods on queries to GRCh38. And was consistently faster on queries made to 25,000, 50,000 and 100,000 SARS-CoV-2 genomes. Hence, it is very clear that our method accelerates the performance of count over all state-of-the-art methods with a minor increase in the memory.

Aaron Hong, Marco Oliva, Dominik Köppl, Hideo Bannai, Christina Boucher, and Travis Gagie. Acceleration of FM-Index Queries Through Prefix-Free Parsing. In 23rd International Workshop on Algorithms in Bioinformatics (WABI 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 273, pp. 13:1-13:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{hong_et_al:LIPIcs.WABI.2023.13, author = {Hong, Aaron and Oliva, Marco and K\"{o}ppl, Dominik and Bannai, Hideo and Boucher, Christina and Gagie, Travis}, title = {{Acceleration of FM-Index Queries Through Prefix-Free Parsing}}, booktitle = {23rd International Workshop on Algorithms in Bioinformatics (WABI 2023)}, pages = {13:1--13:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-294-5}, ISSN = {1868-8969}, year = {2023}, volume = {273}, editor = {Belazzougui, Djamal and Ouangraoua, A\"{i}da}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WABI.2023.13}, URN = {urn:nbn:de:0030-drops-186390}, doi = {10.4230/LIPIcs.WABI.2023.13}, annote = {Keywords: FM-index, pangenomics, scalability, word-based indexing, random access} }

Document

**Published in:** LIPIcs, Volume 259, 34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023)

LZ-End is a variant of the well-known Lempel-Ziv parsing family such that each phrase of the parsing has a previous occurrence, with the additional constraint that the previous occurrence must end at the end of a previous phrase. LZ-End was initially proposed as a greedy parsing, where each phrase is determined greedily from left to right, as the longest factor that satisfies the above constraint [Kreft & Navarro, 2010]. In this work, we consider an optimal LZ-End parsing that has the minimum number of phrases in such parsings. We show that a decision version of computing the optimal LZ-End parsing is NP-complete by showing a reduction from the vertex cover problem. Moreover, we give a MAX-SAT formulation for the optimal LZ-End parsing adapting an approach for computing various NP-hard repetitiveness measures recently presented by [Bannai et al., 2022]. We also consider the approximation ratio of the size of greedy LZ-End parsing to the size of the optimal LZ-End parsing, and give a lower bound of the ratio which asymptotically approaches 2.

Hideo Bannai, Mitsuru Funakoshi, Kazuhiro Kurita, Yuto Nakashima, Kazuhisa Seto, and Takeaki Uno. Optimal LZ-End Parsing Is Hard. In 34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 259, pp. 3:1-3:11, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bannai_et_al:LIPIcs.CPM.2023.3, author = {Bannai, Hideo and Funakoshi, Mitsuru and Kurita, Kazuhiro and Nakashima, Yuto and Seto, Kazuhisa and Uno, Takeaki}, title = {{Optimal LZ-End Parsing Is Hard}}, booktitle = {34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023)}, pages = {3:1--3:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-276-1}, ISSN = {1868-8969}, year = {2023}, volume = {259}, editor = {Bulteau, Laurent and Lipt\'{a}k, Zsuzsanna}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2023.3}, URN = {urn:nbn:de:0030-drops-179571}, doi = {10.4230/LIPIcs.CPM.2023.3}, annote = {Keywords: Data Compression, LZ-End, Repetitiveness measures} }

Document

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

Repetitiveness measures reveal profound characteristics of datasets, and give rise to compressed data structures and algorithms working in compressed space. Alas, the computation of some of these measures is NP-hard, and straight-forward computation is infeasible for datasets of even small sizes. Three such measures are the smallest size of a string attractor, the smallest size of a bidirectional macro scheme, and the smallest size of a straight-line program. While a vast variety of implementations for heuristically computing approximations exist, exact computation of these measures has received little to no attention. In this paper, we present MAX-SAT formulations that provide the first non-trivial implementations for exact computation of smallest string attractors, smallest bidirectional macro schemes, and smallest straight-line programs. Computational experiments show that our implementations work for texts of length up to a few hundred for straight-line programs and bidirectional macro schemes, and texts even over a million for string attractors.

Hideo Bannai, Keisuke Goto, Masakazu Ishihata, Shunsuke Kanda, Dominik Köppl, and Takaaki Nishimoto. Computing NP-Hard Repetitiveness Measures via MAX-SAT. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 12:1-12:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bannai_et_al:LIPIcs.ESA.2022.12, author = {Bannai, Hideo and Goto, Keisuke and Ishihata, Masakazu and Kanda, Shunsuke and K\"{o}ppl, Dominik and Nishimoto, Takaaki}, title = {{Computing NP-Hard Repetitiveness Measures via MAX-SAT}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {12:1--12:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.12}, URN = {urn:nbn:de:0030-drops-169505}, doi = {10.4230/LIPIcs.ESA.2022.12}, annote = {Keywords: repetitiveness measures, string attractor, bidirectional macro scheme} }

Document

Complete Volume

**Published in:** LIPIcs, Volume 223, 33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022)

LIPIcs, Volume 223, CPM 2022, Complete Volume

Hideo Bannai and Jan Holub. LIPIcs, Volume 223, CPM 2022, Complete Volume. In 33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 223, pp. 1-470, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@Proceedings{bannai_et_al:LIPIcs.CPM.2022, title = {{LIPIcs, Volume 223, CPM 2022, Complete Volume}}, booktitle = {33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022)}, pages = {1--470}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-234-1}, ISSN = {1868-8969}, year = {2022}, volume = {223}, editor = {Bannai, Hideo and Holub, Jan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2022}, URN = {urn:nbn:de:0030-drops-161265}, doi = {10.4230/LIPIcs.CPM.2022}, annote = {Keywords: LIPIcs, Volume 223, CPM 2022, Complete Volume} }

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Front Matter

**Published in:** LIPIcs, Volume 223, 33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022)

Front Matter, Table of Contents, Preface, Conference Organization

Hideo Bannai and Jan Holub. Front Matter, Table of Contents, Preface, Conference Organization. In 33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 223, pp. 0:i-0:xviii, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bannai_et_al:LIPIcs.CPM.2022.0, author = {Bannai, Hideo and Holub, Jan}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022)}, pages = {0:i--0:xviii}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-234-1}, ISSN = {1868-8969}, year = {2022}, volume = {223}, editor = {Bannai, Hideo and Holub, Jan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2022.0}, URN = {urn:nbn:de:0030-drops-161271}, doi = {10.4230/LIPIcs.CPM.2022.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

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Invited Talk

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

Repeating structures in strings is one of the most fundamental characteristics of strings, and has been an important topic in the field of combinatorics on words and combinatorial pattern matching since their beginnings. In this talk, I will focus on squares and maximal repetitions and review the "runs" theorem [Hideo Bannai et al., 2017] as well as related results (e.g. [Aviezri S. Fraenkel and Jamie Simpson, 1998; Yuta Fujishige et al., 2017; Ryo Sugahara et al., 2019; Philip Bille et al., 2020; Hideo Bannai et al., 2020; Jonas Ellert and Johannes Fischer, 2021]) which address the two main questions: how many of them can be contained in a string of given length, and algorithms for computing them.

Hideo Bannai. Repetitions in Strings: A "Constant" Problem (Invited Talk). In 32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 191, p. 1:1, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bannai:LIPIcs.CPM.2021.1, author = {Bannai, Hideo}, title = {{Repetitions in Strings: A "Constant" Problem}}, booktitle = {32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021)}, pages = {1:1--1:1}, 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.2021.1}, URN = {urn:nbn:de:0030-drops-139523}, doi = {10.4230/LIPIcs.CPM.2021.1}, annote = {Keywords: Maximal repetitions, Squares, Lyndon words} }

Document

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

The Burrows-Wheeler transform (BWT) is a permutation whose applications are prevalent in data compression and text indexing. The bijective BWT (BBWT) is a bijective variant of it. Although it is known that the BWT can be constructed in linear time for integer alphabets by using a linear time suffix array construction algorithm, it was up to now only conjectured that the BBWT can also be constructed in linear time. We confirm this conjecture in the word RAM model by proposing a construction algorithm that is based on SAIS, improving the best known result of O(n lg n / lg lg n) time to linear. Since we can reduce the problem of constructing the extended BWT to constructing the BBWT in linear time, we obtain a linear-time algorithm computing the extended BWT at the same time.

Hideo Bannai, Juha Kärkkäinen, Dominik Köppl, and Marcin Piątkowski. Constructing the Bijective and the Extended Burrows-Wheeler Transform in Linear Time. In 32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 191, pp. 7:1-7:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bannai_et_al:LIPIcs.CPM.2021.7, author = {Bannai, Hideo and K\"{a}rkk\"{a}inen, Juha and K\"{o}ppl, Dominik and Pi\k{a}tkowski, Marcin}, title = {{Constructing the Bijective and the Extended Burrows-Wheeler Transform in Linear Time}}, booktitle = {32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021)}, pages = {7:1--7:16}, 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.2021.7}, URN = {urn:nbn:de:0030-drops-139588}, doi = {10.4230/LIPIcs.CPM.2021.7}, annote = {Keywords: Burrows-Wheeler Transform, Lyndon words, Circular Suffix Array, Suffix Array Construction Algorithm} }

Document

**Published in:** LIPIcs, Volume 161, 31st Annual Symposium on Combinatorial Pattern Matching (CPM 2020)

The equidistant subsequence pattern matching problem is considered. Given a pattern string P and a text string T, we say that P is an equidistant subsequence of T if P is a subsequence of the text such that consecutive symbols of P in the occurrence are equally spaced. We can consider the problem of equidistant subsequences as generalizations of (sub-)cadences. We give bit-parallel algorithms that yield o(n²) time algorithms for finding k-(sub-)cadences and equidistant subsequences. Furthermore, O(nlog² n) and O(nlog n) time algorithms, respectively for equidistant and Abelian equidistant matching for the case |P| = 3, are shown. The algorithms make use of a technique that was recently introduced which can efficiently compute convolutions with linear constraints.

Mitsuru Funakoshi, Yuto Nakashima, Shunsuke Inenaga, Hideo Bannai, Masayuki Takeda, and Ayumi Shinohara. Detecting k-(Sub-)Cadences and Equidistant Subsequence Occurrences. In 31st Annual Symposium on Combinatorial Pattern Matching (CPM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 161, pp. 12:1-12:11, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{funakoshi_et_al:LIPIcs.CPM.2020.12, author = {Funakoshi, Mitsuru and Nakashima, Yuto and Inenaga, Shunsuke and Bannai, Hideo and Takeda, Masayuki and Shinohara, Ayumi}, title = {{Detecting k-(Sub-)Cadences and Equidistant Subsequence Occurrences}}, booktitle = {31st Annual Symposium on Combinatorial Pattern Matching (CPM 2020)}, pages = {12:1--12:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-149-8}, ISSN = {1868-8969}, year = {2020}, volume = {161}, editor = {G{\o}rtz, Inge Li and Weimann, Oren}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2020.12}, URN = {urn:nbn:de:0030-drops-121375}, doi = {10.4230/LIPIcs.CPM.2020.12}, annote = {Keywords: string algorithms, pattern matching, bit parallelism, subsequences, cadences} }

Document

**Published in:** LIPIcs, Volume 161, 31st Annual Symposium on Combinatorial Pattern Matching (CPM 2020)

Two strings x and y over Σ ∪ Π of equal length are said to parameterized match (p-match) if there is a renaming bijection f:Σ ∪ Π → Σ ∪ Π that is identity on Σ and transforms x to y (or vice versa). The p-matching problem is to look for substrings in a text that p-match a given pattern. In this paper, we propose parameterized suffix automata (p-suffix automata) and parameterized directed acyclic word graphs (PDAWGs) which are the p-matching versions of suffix automata and DAWGs. While suffix automata and DAWGs are equivalent for standard strings, we show that p-suffix automata can have Θ(n²) nodes and edges but PDAWGs have only O(n) nodes and edges, where n is the length of an input string. We also give O(n |Π| log (|Π| + |Σ|))-time O(n)-space algorithm that builds the PDAWG in a left-to-right online manner. As a byproduct, it is shown that the parameterized suffix tree for the reversed string can also be built in the same time and space, in a right-to-left online manner.

Katsuhito Nakashima, Noriki Fujisato, Diptarama Hendrian, Yuto Nakashima, Ryo Yoshinaka, Shunsuke Inenaga, Hideo Bannai, Ayumi Shinohara, and Masayuki Takeda. DAWGs for Parameterized Matching: Online Construction and Related Indexing Structures. In 31st Annual Symposium on Combinatorial Pattern Matching (CPM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 161, pp. 26:1-26:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{nakashima_et_al:LIPIcs.CPM.2020.26, author = {Nakashima, Katsuhito and Fujisato, Noriki and Hendrian, Diptarama and Nakashima, Yuto and Yoshinaka, Ryo and Inenaga, Shunsuke and Bannai, Hideo and Shinohara, Ayumi and Takeda, Masayuki}, title = {{DAWGs for Parameterized Matching: Online Construction and Related Indexing Structures}}, booktitle = {31st Annual Symposium on Combinatorial Pattern Matching (CPM 2020)}, pages = {26:1--26:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-149-8}, ISSN = {1868-8969}, year = {2020}, volume = {161}, editor = {G{\o}rtz, Inge Li and Weimann, Oren}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2020.26}, URN = {urn:nbn:de:0030-drops-121512}, doi = {10.4230/LIPIcs.CPM.2020.26}, annote = {Keywords: parameterized matching, suffix trees, DAWGs, suffix automata} }

Document

**Published in:** LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)

For a set D of documents and a positive integer d, a string w is said to be d-left-right maximal, if (1) w occurs in at least d documents in D, and (2) any proper superstring of w occurs in less than d documents. The left-right-maximal generic words problem is, given a set D of documents, to preprocess D so that for any string p and for any positive integer d, all the superstrings of p that are d-left-right maximal can be answered quickly. In this paper, we present an O(n log m) space data structure (in words) which answers queries in O(|p| + o log log m) time, where n is the total length of documents in D, m is the number of documents in D and o is the number of outputs. Our solution improves the previous one by Nishimoto et al. (PSC 2015), which uses an O(n log n) space data structure answering queries in O(|p|+ r * log n + o * log^2 n) time, where r is the number of right-extensions q of p occurring in at least d documents such that any proper right extension of q occurs in less than d documents.

Yuta Fujishige, Yuto Nakashima, Shunsuke Inenaga, Hideo Bannai, and Masayuki Takeda. An Improved Data Structure for Left-Right Maximal Generic Words Problem. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 40:1-40:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{fujishige_et_al:LIPIcs.ISAAC.2019.40, author = {Fujishige, Yuta and Nakashima, Yuto and Inenaga, Shunsuke and Bannai, Hideo and Takeda, Masayuki}, title = {{An Improved Data Structure for Left-Right Maximal Generic Words Problem}}, booktitle = {30th International Symposium on Algorithms and Computation (ISAAC 2019)}, pages = {40:1--40:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-130-6}, ISSN = {1868-8969}, year = {2019}, volume = {149}, editor = {Lu, Pinyan and Zhang, Guochuan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.40}, URN = {urn:nbn:de:0030-drops-115366}, doi = {10.4230/LIPIcs.ISAAC.2019.40}, annote = {Keywords: generic words, suffix trees, string processing algorithms} }

Document

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

Recent large-scale community sequencing efforts allow at an unprecedented level of detail the identification of genomic regions that show signatures of natural selection. Traditional methods for identifying such regions from individuals' haplotype data, however, require excessive computing times and therefore are not applicable to current datasets. In 2019, Cunha et al. (Proceedings of BSB 2019) suggested the maximal perfect haplotype block as a very simple combinatorial pattern, forming the basis of a new method to perform rapid genome-wide selection scans. The algorithm they presented for identifying these blocks, however, had a worst-case running time quadratic in the genome length. It was posed as an open problem whether an optimal, linear-time algorithm exists. In this paper we give two algorithms that achieve this time bound, one conceptually very simple one using suffix trees and a second one using the positional Burrows-Wheeler Transform, that is very efficient also in practice.

Jarno Alanko, Hideo Bannai, Bastien Cazaux, Pierre Peterlongo, and Jens Stoye. Finding All Maximal Perfect Haplotype Blocks in Linear Time. In 19th International Workshop on Algorithms in Bioinformatics (WABI 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 143, pp. 8:1-8:9, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{alanko_et_al:LIPIcs.WABI.2019.8, author = {Alanko, Jarno and Bannai, Hideo and Cazaux, Bastien and Peterlongo, Pierre and Stoye, Jens}, title = {{Finding All Maximal Perfect Haplotype Blocks in Linear Time}}, booktitle = {19th International Workshop on Algorithms in Bioinformatics (WABI 2019)}, pages = {8:1--8:9}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-123-8}, ISSN = {1868-8969}, year = {2019}, volume = {143}, editor = {Huber, Katharina T. and Gusfield, Dan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WABI.2019.8}, URN = {urn:nbn:de:0030-drops-110388}, doi = {10.4230/LIPIcs.WABI.2019.8}, annote = {Keywords: Population genomics, selection coefficient, haplotype block, positional Burrows-Wheeler Transform} }

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

The Burrows-Wheeler transform (BWT) is a permutation whose applications are prevalent in data compression and text indexing. The bijective BWT is a bijective variant of it that has not yet been studied for text indexing applications. We fill this gap by proposing a self-index built on the bijective BWT . The self-index applies the backward search technique of the FM-index to find a pattern P with O(|P| lg|P|) backward search steps.

Hideo Bannai, Juha Kärkkäinen, Dominik Köppl, and Marcin Pia̧tkowski. Indexing the Bijective BWT. In 30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 128, pp. 17:1-17:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bannai_et_al:LIPIcs.CPM.2019.17, author = {Bannai, Hideo and K\"{a}rkk\"{a}inen, Juha and K\"{o}ppl, Dominik and Pia̧tkowski, Marcin}, title = {{Indexing the Bijective BWT}}, booktitle = {30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019)}, pages = {17:1--17:14}, 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.2019.17}, URN = {urn:nbn:de:0030-drops-104887}, doi = {10.4230/LIPIcs.CPM.2019.17}, annote = {Keywords: Burrows-Wheeler Transform, Lyndon words, Text Indexing} }

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

A maximal repetition, or run, in a string, is a maximal periodic substring whose smallest period is at most half the length of the substring. In this paper, we consider runs that correspond to a path on a trie, or in other words, on a rooted edge-labeled tree where the endpoints of the path must be a descendant/ancestor of the other. For a trie with n edges, we show that the number of runs is less than n. We also show an O(n sqrt{log n}log log n) time and O(n) space algorithm for counting and finding the shallower endpoint of all runs. We further show an O(n log n) time and O(n) space algorithm for finding both endpoints of all runs. We also discuss how to improve the running time even more.

Ryo Sugahara, Yuto Nakashima, Shunsuke Inenaga, Hideo Bannai, and Masayuki Takeda. Computing Runs on a Trie. In 30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 128, pp. 23:1-23:11, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{sugahara_et_al:LIPIcs.CPM.2019.23, author = {Sugahara, Ryo and Nakashima, Yuto and Inenaga, Shunsuke and Bannai, Hideo and Takeda, Masayuki}, title = {{Computing Runs on a Trie}}, booktitle = {30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019)}, pages = {23:1--23:11}, 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.2019.23}, URN = {urn:nbn:de:0030-drops-104943}, doi = {10.4230/LIPIcs.CPM.2019.23}, annote = {Keywords: runs, Lyndon words} }

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

Palindromes are important objects in strings which have been extensively studied from combinatorial, algorithmic, and bioinformatics points of views. Manacher [J. ACM 1975] proposed a seminal algorithm that computes the longest substring palindromes (LSPals) of a given string in O(n) time, where n is the length of the string. In this paper, we consider the problem of finding the LSPal after the string is edited. We present an algorithm that uses O(n) time and space for preprocessing, and answers the length of the LSPals in O(l + log log n) time, after a substring in T is replaced by a string of arbitrary length l. This outperforms the query algorithm proposed in our previous work [CPM 2018] that uses O(l + log n) time for each query.

Mitsuru Funakoshi, Yuto Nakashima, Shunsuke Inenaga, Hideo Bannai, and Masayuki Takeda. Faster Queries for Longest Substring Palindrome After Block Edit. In 30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 128, pp. 27:1-27:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{funakoshi_et_al:LIPIcs.CPM.2019.27, author = {Funakoshi, Mitsuru and Nakashima, Yuto and Inenaga, Shunsuke and Bannai, Hideo and Takeda, Masayuki}, title = {{Faster Queries for Longest Substring Palindrome After Block Edit}}, booktitle = {30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019)}, pages = {27:1--27:13}, 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.2019.27}, URN = {urn:nbn:de:0030-drops-104989}, doi = {10.4230/LIPIcs.CPM.2019.27}, annote = {Keywords: palindromes, string algorithm, periodicity} }

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

Lempel-Ziv (LZ) factorization and Lyndon factorization are well-known factorizations of strings. Recently, Kärkkäinen et al. studied the relation between the sizes of the two factorizations, and showed that the size of the Lyndon factorization is always smaller than twice the size of the non-overlapping LZ factorization [STACS 2017]. In this paper, we consider a similar problem for the overlapping version of the LZ factorization. Since the size of the overlapping LZ factorization is always smaller than the size of the non-overlapping LZ factorization and, in fact, can even be an O(log n) factor smaller, it is not immediately clear whether a similar bound as in previous work would hold. Nevertheless, in this paper, we prove that the size of the Lyndon factorization is always smaller than four times the size of the overlapping LZ factorization.

Yuki Urabe, Yuto Nakashima, Shunsuke Inenaga, Hideo Bannai, and Masayuki Takeda. On the Size of Overlapping Lempel-Ziv and Lyndon Factorizations. In 30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 128, pp. 29:1-29:11, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{urabe_et_al:LIPIcs.CPM.2019.29, author = {Urabe, Yuki and Nakashima, Yuto and Inenaga, Shunsuke and Bannai, Hideo and Takeda, Masayuki}, title = {{On the Size of Overlapping Lempel-Ziv and Lyndon Factorizations}}, booktitle = {30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019)}, pages = {29:1--29:11}, 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.2019.29}, URN = {urn:nbn:de:0030-drops-105008}, doi = {10.4230/LIPIcs.CPM.2019.29}, annote = {Keywords: Lyndon factorization, Lyndon words, Lempel-Ziv factorization} }

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

Given an indeterminate string pattern p and an indeterminate string text t, the problem of order-preserving pattern matching with character uncertainties (muOPPM) is to find all substrings of t that satisfy one of the possible orderings defined by p. When the text and pattern are determinate strings, we are in the presence of the well-studied exact order-preserving pattern matching (OPPM) problem with diverse applications on time series analysis. Despite its relevance, the exact OPPM problem suffers from two major drawbacks: 1) the inability to deal with indetermination in the text, thus preventing the analysis of noisy time series; and 2) the inability to deal with indetermination in the pattern, thus imposing the strict satisfaction of the orders among all pattern positions. In this paper, we provide the first polynomial algorithms to answer the muOPPM problem when: 1) indetermination is observed on the pattern or text; and 2) indetermination is observed on both the pattern and the text and given by uncertainties between pairs of characters. First, given two strings with the same length m and O(r) uncertain characters per string position, we show that the muOPPM problem can be solved in O(mr lg r) time when one string is indeterminate and r in N^+ and in O(m^2) time when both strings are indeterminate and r=2. Second, given an indeterminate text string of length n, we show that muOPPM can be efficiently solved in polynomial time and linear space.

Rui Henriques, Alexandre P. Francisco, Luís M. S. Russo, and Hideo Bannai. Order-Preserving Pattern Matching Indeterminate Strings. In 29th Annual Symposium on Combinatorial Pattern Matching (CPM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 105, pp. 2:1-2:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{henriques_et_al:LIPIcs.CPM.2018.2, author = {Henriques, Rui and Francisco, Alexandre P. and Russo, Lu{\'\i}s M. S. and Bannai, Hideo}, title = {{Order-Preserving Pattern Matching Indeterminate Strings}}, booktitle = {29th Annual Symposium on Combinatorial Pattern Matching (CPM 2018)}, pages = {2:1--2:15}, 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.2018.2}, URN = {urn:nbn:de:0030-drops-87087}, doi = {10.4230/LIPIcs.CPM.2018.2}, annote = {Keywords: Order-preserving pattern matching, Indeterminate string analysis, Generic pattern matching, Satisfiability} }

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

Lempel-Ziv 1977 (LZ77) parsing, matching statistics and the Burrows-Wheeler Transform (BWT) are all fundamental elements of stringology. In a series of recent papers, Policriti and Prezza (DCC 2016 and Algorithmica, CPM 2017) showed how we can use an augmented run-length compressed BWT (RLBWT) of the reverse T^R of a text T, to compute offline the LZ77 parse of T in O(n log r) time and O(r) space, where n is the length of T and r is the number of runs in the BWT of T^R. In this paper we first extend a well-known technique for updating an unaugmented RLBWT when a character is prepended to a text, to work with Policriti and Prezza's augmented RLBWT. This immediately implies that we can build online the LZ77 parse of T while still using O(n log r) time and O(r) space; it also seems likely to be of independent interest. Our experiments, using an extension of Ohno, Takabatake, I and Sakamoto's (IWOCA 2017) implementation of updating, show our approach is both time- and space-efficient for repetitive strings. We then show how to augment the RLBWT further - albeit making it static again and increasing its space by a factor proportional to the size of the alphabet - such that later, given another string S and O(log log n)-time random access to T, we can compute the matching statistics of S with respect to T in O(|S| log log n) time.

Hideo Bannai, Travis Gagie, and Tomohiro I. Online LZ77 Parsing and Matching Statistics with RLBWTs. In 29th Annual Symposium on Combinatorial Pattern Matching (CPM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 105, pp. 7:1-7:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bannai_et_al:LIPIcs.CPM.2018.7, author = {Bannai, Hideo and Gagie, Travis and I, Tomohiro}, title = {{Online LZ77 Parsing and Matching Statistics with RLBWTs}}, booktitle = {29th Annual Symposium on Combinatorial Pattern Matching (CPM 2018)}, pages = {7:1--7:12}, 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.2018.7}, URN = {urn:nbn:de:0030-drops-87025}, doi = {10.4230/LIPIcs.CPM.2018.7}, annote = {Keywords: Lempel-Ziv 1977, Matching Statistics, Run-Length Compressed Burrows-Wheeler Transform} }

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

An Elastic-Degenerate String [Iliopoulus et al., LATA 2017] is a sequence of sets of strings, which was recently proposed as a way to model a set of similar sequences. We give an online algorithm for the Elastic-Degenerate String Matching (EDSM) problem that runs in O(nm sqrt{m log m} + N) time and O(m) working space, where n is the number of elastic degenerate segments of the text, N is the total length of all strings in the text, and m is the length of the pattern. This improves the previous algorithm by Grossi et al. [CPM 2017] that runs in O(nm^2 + N) time.

Kotaro Aoyama, Yuto Nakashima, Tomohiro I, Shunsuke Inenaga, Hideo Bannai, and Masayuki Takeda. Faster Online Elastic Degenerate String Matching. In 29th Annual Symposium on Combinatorial Pattern Matching (CPM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 105, pp. 9:1-9:10, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{aoyama_et_al:LIPIcs.CPM.2018.9, author = {Aoyama, Kotaro and Nakashima, Yuto and I, Tomohiro and Inenaga, Shunsuke and Bannai, Hideo and Takeda, Masayuki}, title = {{Faster Online Elastic Degenerate String Matching}}, booktitle = {29th Annual Symposium on Combinatorial Pattern Matching (CPM 2018)}, pages = {9:1--9:10}, 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.2018.9}, URN = {urn:nbn:de:0030-drops-87016}, doi = {10.4230/LIPIcs.CPM.2018.9}, annote = {Keywords: elastic degenerate pattern matching, boolean convolution} }

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

It is known that the length of the longest substring palindromes (LSPals) of a given string T of length n can be computed in O(n) time by Manacher's algorithm [J. ACM '75]. In this paper, we consider the problem of finding the LSPal after the string is edited. We present an algorithm that uses O(n) time and space for preprocessing, and answers the length of the LSPals in O(log (min {sigma, log n })) time after single character substitution, insertion, or deletion, where sigma denotes the number of distinct characters appearing in T. We also propose an algorithm that uses O(n) time and space for preprocessing, and answers the length of the LSPals in O(l + log n) time, after an existing substring in T is replaced by a string of arbitrary length l.

Mitsuru Funakoshi, Yuto Nakashima, Shunsuke Inenaga, Hideo Bannai, and Masayuki Takeda. Longest substring palindrome after edit. In 29th Annual Symposium on Combinatorial Pattern Matching (CPM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 105, pp. 12:1-12:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{funakoshi_et_al:LIPIcs.CPM.2018.12, author = {Funakoshi, Mitsuru and Nakashima, Yuto and Inenaga, Shunsuke and Bannai, Hideo and Takeda, Masayuki}, title = {{Longest substring palindrome after edit}}, booktitle = {29th Annual Symposium on Combinatorial Pattern Matching (CPM 2018)}, pages = {12:1--12:14}, 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.2018.12}, URN = {urn:nbn:de:0030-drops-86977}, doi = {10.4230/LIPIcs.CPM.2018.12}, annote = {Keywords: maximal palindromes, edit operations, periodicity, suffix trees} }

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

A square is a non-empty string of form YY. The longest common square subsequence (LCSqS) problem is to compute a longest square occurring as a subsequence in two given strings A and B. We show that the problem can easily be solved in O(n^6) time or O(|M|n^4) time with O(n^4) space, where n is the length of the strings and M is the set of matching points between A and B. Then, we show that the problem can also be solved in O(sigma |M|^3 + n) time and O(|M|^2 + n) space, or in O(|M|^3 log^2 n log log n + n) time with O(|M|^3 + n) space, where sigma is the number of distinct characters occurring in A and B. We also study lower bounds for the LCSqS problem for two or more strings.

Takafumi Inoue, Shunsuke Inenaga, Heikki Hyyrö, Hideo Bannai, and Masayuki Takeda. Computing longest common square subsequences. In 29th Annual Symposium on Combinatorial Pattern Matching (CPM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 105, pp. 15:1-15:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{inoue_et_al:LIPIcs.CPM.2018.15, author = {Inoue, Takafumi and Inenaga, Shunsuke and Hyyr\"{o}, Heikki and Bannai, Hideo and Takeda, Masayuki}, title = {{Computing longest common square subsequences}}, booktitle = {29th Annual Symposium on Combinatorial Pattern Matching (CPM 2018)}, pages = {15:1--15:13}, 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.2018.15}, URN = {urn:nbn:de:0030-drops-86946}, doi = {10.4230/LIPIcs.CPM.2018.15}, annote = {Keywords: squares, subsequences, matching rectangles, dynamic programming} }

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

The longest Lyndon substring of a string T is the longest substring of T which is a Lyndon word. LLS(T) denotes the length of the longest Lyndon substring of a string T. In this paper, we consider computing LLS(T') where T' is an edited string formed from T. After O(n) time and space preprocessing, our algorithm returns LLS(T') in O(log n) time for any single character edit. We also consider a version of the problem with block edits, i.e., a substring of T is replaced by a given string of length l. After O(n) time and space preprocessing, our algorithm returns LLS(T') in O(l log sigma + log n) time for any block edit where sigma is the number of distinct characters in T. We can modify our algorithm so as to output all the longest Lyndon substrings of T' for both problems.

Yuki Urabe, Yuto Nakashima, Shunsuke Inenaga, Hideo Bannai, and Masayuki Takeda. Longest Lyndon Substring After Edit. In 29th Annual Symposium on Combinatorial Pattern Matching (CPM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 105, pp. 19:1-19:10, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{urabe_et_al:LIPIcs.CPM.2018.19, author = {Urabe, Yuki and Nakashima, Yuto and Inenaga, Shunsuke and Bannai, Hideo and Takeda, Masayuki}, title = {{Longest Lyndon Substring After Edit}}, booktitle = {29th Annual Symposium on Combinatorial Pattern Matching (CPM 2018)}, pages = {19:1--19:10}, 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.2018.19}, URN = {urn:nbn:de:0030-drops-86913}, doi = {10.4230/LIPIcs.CPM.2018.19}, annote = {Keywords: Lyndon word, Lyndon factorization, Lyndon tree, Edit operation} }

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

We revisit the problem of computing the Lyndon factorization of a string w of length N which is given as a straight line program (SLP) of size n. For this problem, we show a new algorithm which runs in O(P(n, N) + Q(n, N)n log log N) time and O(n log N + S(n, N)) space where P(n, N), S(n,N), Q(n,N) are respectively the pre-processing time, space, and query time of a data structure for longest common extensions (LCE) on SLPs. Our algorithm improves the algorithm proposed by I et al. (TCS '17), and can be more efficient than the O(N)-time solution by Duval (J. Algorithms '83) when w is highly compressible.

Isamu Furuya, Yuto Nakashima, Tomohiro I, Shunsuke Inenaga, Hideo Bannai, and Masayuki Takeda. Lyndon Factorization of Grammar Compressed Texts Revisited. In 29th Annual Symposium on Combinatorial Pattern Matching (CPM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 105, pp. 24:1-24:10, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{furuya_et_al:LIPIcs.CPM.2018.24, author = {Furuya, Isamu and Nakashima, Yuto and I, Tomohiro and Inenaga, Shunsuke and Bannai, Hideo and Takeda, Masayuki}, title = {{Lyndon Factorization of Grammar Compressed Texts Revisited}}, booktitle = {29th Annual Symposium on Combinatorial Pattern Matching (CPM 2018)}, pages = {24:1--24:10}, 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.2018.24}, URN = {urn:nbn:de:0030-drops-86855}, doi = {10.4230/LIPIcs.CPM.2018.24}, annote = {Keywords: Lyndon word, Lyndon factorization, Straight line program} }

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**Published in:** LIPIcs, Volume 92, 28th International Symposium on Algorithms and Computation (ISAAC 2017)

We consider the problem of computing all maximal repetitions contained in a string that is given in run-length encoding.
Given a run-length encoding of a string, we show that the maximum number of maximal repetitions contained in the string is at most m+k-1, where m is the size of the run-length encoding, and k is the number of run-length factors whose exponent is at least 2.
We also show an algorithm for computing all maximal repetitions in O(m \alpha(m)) time and O(m) space, where \alpha denotes the inverse Ackermann function.

Yuta Fujishige, Yuto Nakashima, Shunsuke Inenaga, Hideo Bannai, and Masayuki Takeda. Almost Linear Time Computation of Maximal Repetitions in Run Length Encoded Strings. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 33:1-33:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{fujishige_et_al:LIPIcs.ISAAC.2017.33, author = {Fujishige, Yuta and Nakashima, Yuto and Inenaga, Shunsuke and Bannai, Hideo and Takeda, Masayuki}, title = {{Almost Linear Time Computation of Maximal Repetitions in Run Length Encoded Strings}}, booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)}, pages = {33:1--33:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-054-5}, ISSN = {1868-8969}, year = {2017}, volume = {92}, editor = {Okamoto, Yoshio and Tokuyama, Takeshi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.33}, URN = {urn:nbn:de:0030-drops-82610}, doi = {10.4230/LIPIcs.ISAAC.2017.33}, annote = {Keywords: maximal repetitions,run length encoding} }

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**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

The longest common extension (LCE) problem is to preprocess a given string w of length n so that the length of the longest common prefix between suffixes of w that start at any two given positions is answered quickly. In this paper, we present a data structure of O(z \tau^2 + \frac{n}{\tau}) words of space which answers LCE queries in O(1) time and can be built in O(n \log \sigma) time, where 1 \leq \tau \leq \sqrt{n} is a parameter, z is the size of the Lempel-Ziv 77 factorization of w and \sigma is the alphabet size. The proposed LCE data structure not access the input string w when answering queries, and thus w can be deleted after preprocessing. On top of this main result, we obtain further results using (variants of) our LCE data structure, which include the following:
- For highly repetitive strings where the z\tau^2 term is dominated by \frac{n}{\tau}, we obtain a constant-time and sub-linear space LCE query data structure.
- Even when the input string is not well compressible via Lempel-Ziv 77 factorization, we still can obtain a constant-time and sub-linear space LCE data structure for suitable \tau and for \sigma \leq 2^{o(\log n)}.
- The time-space trade-off lower bounds for the LCE problem by Bille et al. [J. Discrete Algorithms, 25:42-50, 2014] and by Kosolobov [CoRR, abs/1611.02891, 2016] do not apply in some cases with our LCE data structure.

Yuka Tanimura, Takaaki Nishimoto, Hideo Bannai, Shunsuke Inenaga, and Masayuki Takeda. Small-Space LCE Data Structure with Constant-Time Queries. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 10:1-10:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{tanimura_et_al:LIPIcs.MFCS.2017.10, author = {Tanimura, Yuka and Nishimoto, Takaaki and Bannai, Hideo and Inenaga, Shunsuke and Takeda, Masayuki}, title = {{Small-Space LCE Data Structure with Constant-Time Queries}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {10:1--10:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.10}, URN = {urn:nbn:de:0030-drops-81021}, doi = {10.4230/LIPIcs.MFCS.2017.10}, annote = {Keywords: longest common extension, truncated suffix trees, t-covers} }

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**Published in:** LIPIcs, Volume 78, 28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017)

The constrained LCS problem asks one to find a longest common subsequence of two input strings A and B with some constraints. The STR-IC-LCS problem is a variant of the constrained LCS problem, where the solution must include a given constraint string C as a substring. Given two strings A and B of respective lengths M and N, and a constraint string C of length at most min{M, N}, the best known algorithm for the STR-IC-LCS problem, proposed by Deorowicz (Inf. Process. Lett., 11:423-426, 2012), runs in O(MN) time. In this work, we present an O(mN + nM)-time solution to the STR-IC-LCS problem, where m and n denote the sizes of the run-length encodings of A and B, respectively. Since m <= M and n <= N always hold, our algorithm is always as fast as Deorowicz's algorithm, and is faster when input strings are compressible via RLE.

Keita Kuboi, Yuta Fujishige, Shunsuke Inenaga, Hideo Bannai, and Masayuki Takeda. Faster STR-IC-LCS Computation via RLE. In 28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 78, pp. 20:1-20:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{kuboi_et_al:LIPIcs.CPM.2017.20, author = {Kuboi, Keita and Fujishige, Yuta and Inenaga, Shunsuke and Bannai, Hideo and Takeda, Masayuki}, title = {{Faster STR-IC-LCS Computation via RLE}}, booktitle = {28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017)}, pages = {20:1--20:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-039-2}, ISSN = {1868-8969}, year = {2017}, volume = {78}, editor = {K\"{a}rkk\"{a}inen, Juha and Radoszewski, Jakub and Rytter, Wojciech}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2017.20}, URN = {urn:nbn:de:0030-drops-73335}, doi = {10.4230/LIPIcs.CPM.2017.20}, annote = {Keywords: longest common subsequence, STR-IC-LCS, run-length encoding} }

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**Published in:** LIPIcs, Volume 78, 28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017)

Given a string on an integer alphabet, we present an algorithm that computes the set of all distinct squares belonging to this string in time linear to the string length. As an application, we show how to compute the tree topology of the minimal augmented suffix tree in linear time. Asides from that, we elaborate an algorithm computing the longest previous table in a succinct representation using compressed working space.

Hideo Bannai, Shunsuke Inenaga, and Dominik Köppl. Computing All Distinct Squares in Linear Time for Integer Alphabets. In 28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 78, pp. 22:1-22:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bannai_et_al:LIPIcs.CPM.2017.22, author = {Bannai, Hideo and Inenaga, Shunsuke and K\"{o}ppl, Dominik}, title = {{Computing All Distinct Squares in Linear Time for Integer Alphabets}}, booktitle = {28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017)}, pages = {22:1--22:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-039-2}, ISSN = {1868-8969}, year = {2017}, volume = {78}, editor = {K\"{a}rkk\"{a}inen, Juha and Radoszewski, Jakub and Rytter, Wojciech}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2017.22}, URN = {urn:nbn:de:0030-drops-73224}, doi = {10.4230/LIPIcs.CPM.2017.22}, annote = {Keywords: tandem repeats, distinct squares, counting algorithms} }

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**Published in:** LIPIcs, Volume 78, 28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017)

A substring Q of a string S is called a shortest unique substring (SUS) for interval [s,t] in S, if Q occurs exactly once in S, this occurrence of Q contains interval [s,t], and every substring of S which contains interval [s,t] and is shorter than Q occurs at least twice in S. The SUS problem is, given a string S, to preprocess S so that for any subsequent query interval [s,t] all the SUSs for interval [s,t] can be answered quickly. When s = t, we call the SUSs for [s, t] as point SUSs, and when s <= t, we call the SUSs for [s, t] as interval SUSs. There exist optimal O(n)-time preprocessing scheme which answers queries in optimal O(k) time for both point and interval SUSs, where n is the length of S and k is the number of outputs for a given query. In this paper, we reveal structural, combinatorial properties underlying the SUS problem: Namely, we show that the number of intervals in S that correspond to point SUSs for all query positions in S is less than 1.5n, and show that this is a matching upper and lower bound. Also, we consider the maximum number of intervals in S that correspond to interval SUSs for all query intervals in S.

Takuya Mieno, Shunsuke Inenaga, Hideo Bannai, and Masayuki Takeda. Tight Bounds on the Maximum Number of Shortest Unique Substrings. In 28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 78, pp. 24:1-24:11, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{mieno_et_al:LIPIcs.CPM.2017.24, author = {Mieno, Takuya and Inenaga, Shunsuke and Bannai, Hideo and Takeda, Masayuki}, title = {{Tight Bounds on the Maximum Number of Shortest Unique Substrings}}, booktitle = {28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017)}, pages = {24:1--24:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-039-2}, ISSN = {1868-8969}, year = {2017}, volume = {78}, editor = {K\"{a}rkk\"{a}inen, Juha and Radoszewski, Jakub and Rytter, Wojciech}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2017.24}, URN = {urn:nbn:de:0030-drops-73460}, doi = {10.4230/LIPIcs.CPM.2017.24}, annote = {Keywords: shortest unique substrings, maximal unique substrings} }

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**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

The directed acyclic word graph (DAWG) of a string y is the smallest (partial) DFA which recognizes all suffixes of y and has only O(n) nodes and edges. We present the first O(n)-time algorithm for computing the DAWG of a given string y of length n over an integer alphabet of polynomial size in n. We also show that a straightforward modification to our DAWG construction algorithm leads to the first O(n)-time algorithm for constructing the affix tree of a given string y over an integer alphabet. Affix trees are a text indexing structure supporting bidirectional pattern searches. As an application to our O(n)-time DAWG construction algorithm, we show that the set MAW(y) of all minimal absent words of y can be computed in optimal O(n + |MAW(y)|) time and O(n) working space for integer alphabets.

Yuta Fujishige, Yuki Tsujimaru, Shunsuke Inenaga, Hideo Bannai, and Masayuki Takeda. Computing DAWGs and Minimal Absent Words in Linear Time for Integer Alphabets. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 38:1-38:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{fujishige_et_al:LIPIcs.MFCS.2016.38, author = {Fujishige, Yuta and Tsujimaru, Yuki and Inenaga, Shunsuke and Bannai, Hideo and Takeda, Masayuki}, title = {{Computing DAWGs and Minimal Absent Words in Linear Time for Integer Alphabets}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {38:1--38:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.38}, URN = {urn:nbn:de:0030-drops-64528}, doi = {10.4230/LIPIcs.MFCS.2016.38}, annote = {Keywords: string algorithms, DAWGs, suffix trees, minimal absent words} }

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**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

We consider the problem of answering shortest unique substring (SUS) queries on run-length encoded strings. For a string S, a unique substring u = S[i..j] is said to be a shortest unique substring (SUS) of S containing an interval [s, t] (i <= s <= t <= j) if for any i' <= s <= t <= j' with j-i > j'-i', S[i'..j'] occurs at least twice in S.
Given a run-length encoding of size m of a string of length N, we show that we can construct a data structure of size O(m+pi_s(N, m)) in O(m log m + pi_c(N, m)) time such that queries can be answered in
O(pi_q(N, m) + k) time, where k is the size of the output (the number of SUSs), and pi_s(N,m), pi_c(N,m), pi_q(N,m) are, respectively, the size, construction time, and query time for a predecessor/successor query data structure of m elements for the universe of [1,N]. Using the data structure by Beam and Fich (JCSS 2002), this results in a data structure of O(m) space that is constructed in O(m log m) time, and answers queries in O(sqrt(log m/loglog m)+k) time.

Takuya Mieno, Shunsuke Inenaga, Hideo Bannai, and Masayuki Takeda. Shortest Unique Substring Queries on Run-Length Encoded Strings. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 69:1-69:11, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{mieno_et_al:LIPIcs.MFCS.2016.69, author = {Mieno, Takuya and Inenaga, Shunsuke and Bannai, Hideo and Takeda, Masayuki}, title = {{Shortest Unique Substring Queries on Run-Length Encoded Strings}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {69:1--69:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.69}, URN = {urn:nbn:de:0030-drops-65033}, doi = {10.4230/LIPIcs.MFCS.2016.69}, annote = {Keywords: string algorithms, shortest unique substring, run-length encoding} }

Document

**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

A Longest Common Extension (LCE) query on a text T of length N asks for the length of the longest common prefix of suffixes starting at given two positions. We show that the signature encoding G of size w = O(min(z log N log^* M, N)) [Mehlhorn et al., Algorithmica 17(2):183-198, 1997] of T, which can be seen as a compressed representation of T, has a capability to support LCE queries in O(log N + log ell log^* M) time, where ell is the answer to the query, z is the size of the Lempel-Ziv77 (LZ77) factorization of T, and M >= 4N is an integer that can be handled in constant time under word RAM model. In compressed space, this is the fastest deterministic LCE data structure in many cases. Moreover, G can be enhanced to support efficient update operations: After processing G in O(w f_A) time, we can insert/delete any (sub)string of length y into/from an arbitrary position of T in O((y + log Nlog^* M) f_A) time, where f_A = O(min{ (loglog M loglog w)/(logloglog M), sqrt(log w/loglog w)}). This yields the first fully dynamic LCE data structure working in compressed space. We also present efficient construction algorithms from various types of inputs: We can construct G in O(N f_A) time from uncompressed string T; in O(n loglog (n log^* M) log N log^* M) time from grammar-compressed string T represented by a straight-line program of size n; and in O(z f_A log N log^* M) time from LZ77-compressed string T with z factors. On top of the above contributions, we show several applications of our data structures which improve previous best known results on grammar-compressed string processing.

Takaaki Nishimoto, Tomohiro I, Shunsuke Inenaga, Hideo Bannai, and Masayuki Takeda. Fully Dynamic Data Structure for LCE Queries in Compressed Space. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 72:1-72:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{nishimoto_et_al:LIPIcs.MFCS.2016.72, author = {Nishimoto, Takaaki and I, Tomohiro and Inenaga, Shunsuke and Bannai, Hideo and Takeda, Masayuki}, title = {{Fully Dynamic Data Structure for LCE Queries in Compressed Space}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {72:1--72:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.72}, URN = {urn:nbn:de:0030-drops-65045}, doi = {10.4230/LIPIcs.MFCS.2016.72}, annote = {Keywords: dynamic texts, longest common extension (LCE) queries, straight-line program} }

Document

**Published in:** LIPIcs, Volume 54, 27th Annual Symposium on Combinatorial Pattern Matching (CPM 2016)

Given a string S of n symbols, a longest common extension query LCE(i,j) asks for the length of the longest common prefix of the $i$th and $j$th suffixes of S. LCE queries have several important applications in string processing, perhaps most notably to suffix sorting. Recently, Bille et al. (J. Discrete Algorithms 25:42-50, 2014, Proc. CPM 2015:65-76) described several data structures for answering LCE queries that offers a space-time trade-off between data structure size and query time. In particular, for a parameter 1 <= tau <= n, their best deterministic solution is a data structure of size O(n/tau) which allows LCE queries to be answered in O(tau) time. However, the construction time for all deterministic versions of their data structure is quadratic in n. In this paper, we propose a deterministic solution that achieves a similar space-time trade-off of O(tau * min(log(tau),log(n/tau)) query time using O(n/tau) space, but significantly improve the construction time to O(n*tau).

Yuka Tanimura, Tomohiro I, Hideo Bannai, Shunsuke Inenaga, Simon J. Puglisi, and Masayuki Takeda. Deterministic Sub-Linear Space LCE Data Structures With Efficient Construction. In 27th Annual Symposium on Combinatorial Pattern Matching (CPM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 54, pp. 1:1-1:10, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{tanimura_et_al:LIPIcs.CPM.2016.1, author = {Tanimura, Yuka and I, Tomohiro and Bannai, Hideo and Inenaga, Shunsuke and Puglisi, Simon J. and Takeda, Masayuki}, title = {{Deterministic Sub-Linear Space LCE Data Structures With Efficient Construction}}, booktitle = {27th Annual Symposium on Combinatorial Pattern Matching (CPM 2016)}, pages = {1:1--1:10}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-012-5}, ISSN = {1868-8969}, year = {2016}, volume = {54}, editor = {Grossi, Roberto and Lewenstein, Moshe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2016.1}, URN = {urn:nbn:de:0030-drops-60655}, doi = {10.4230/LIPIcs.CPM.2016.1}, annote = {Keywords: longest common extension, longest common prefix, sparse suffix array} }

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**Published in:** LIPIcs, Volume 54, 27th Annual Symposium on Combinatorial Pattern Matching (CPM 2016)

A square factorization of a string w is a factorization of w in which each factor is a square. Dumitran et al. [SPIRE 2015, pp. 54-66] showed how to find a square factorization of a given string of length n in O(n log n) time, and they posed a question whether it can be done in O(n) time. In this paper, we answer their question positively, showing an O(n)-time algorithm for square factorization in the standard word RAM model with machine word size omega = Omega(log n). We also show an O(n + (n log^2 n) / omega)-time (respectively, O(n log n)-time) algorithm to find a square factorization which contains the maximum (respectively, minimum) number of squares.

Yoshiaki Matsuoka, Shunsuke Inenaga, Hideo Bannai, Masayuki Takeda, and Florin Manea. Factorizing a String into Squares in Linear Time. In 27th Annual Symposium on Combinatorial Pattern Matching (CPM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 54, pp. 27:1-27:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{matsuoka_et_al:LIPIcs.CPM.2016.27, author = {Matsuoka, Yoshiaki and Inenaga, Shunsuke and Bannai, Hideo and Takeda, Masayuki and Manea, Florin}, title = {{Factorizing a String into Squares in Linear Time}}, booktitle = {27th Annual Symposium on Combinatorial Pattern Matching (CPM 2016)}, pages = {27:1--27:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-012-5}, ISSN = {1868-8969}, year = {2016}, volume = {54}, editor = {Grossi, Roberto and Lewenstein, Moshe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2016.27}, URN = {urn:nbn:de:0030-drops-60645}, doi = {10.4230/LIPIcs.CPM.2016.27}, annote = {Keywords: Squares, Runs, Factorization of Strings} }

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**Published in:** LIPIcs, Volume 25, 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)

We present a new on-line algorithm for computing the Lempel-Ziv factorization of a string that runs in O(N.log(N)) time and uses only O(N.log(s)) bits of working space, where N is the length of the string and s is the size of the alphabet. This is a notable improvement compared to the performance of previous on-line algorithms using the same order of working space but running in either O(N.log^3(N)) time [Okanohara and Sadakane, 2009] or O(N.log^2(N)) time [Starikovskaya, 2012]. The key to our new algorithm is in the utilization of an elegant but less popular index structure called Directed Acyclic Word Graphs, or DAWGs [Blumer et al., 1985]. We also present an opportunistic variant of our algorithm, which, given the run length encoding of size m of a string of length N, computes the Lempel-Ziv factorization of the string on-line, in O(m.min{log(log(m)).log(log(N))/(log(log(log(N)))), (log(m))^{1/2}/(log(log(m)))^{1/2})}) time and O(m.log(N)) bits of space.

Jun'ichi Yamamoto, Tomohiro I, Hideo Bannai, Shunsuke Inenaga, and Masayuki Takeda. Faster Compact On-Line Lempel-Ziv Factorization. In 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 25, pp. 675-686, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2014)

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@InProceedings{yamamoto_et_al:LIPIcs.STACS.2014.675, author = {Yamamoto, Jun'ichi and I, Tomohiro and Bannai, Hideo and Inenaga, Shunsuke and Takeda, Masayuki}, title = {{Faster Compact On-Line Lempel-Ziv Factorization}}, booktitle = {31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)}, pages = {675--686}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-65-1}, ISSN = {1868-8969}, year = {2014}, volume = {25}, editor = {Mayr, Ernst W. and Portier, Natacha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2014.675}, URN = {urn:nbn:de:0030-drops-44976}, doi = {10.4230/LIPIcs.STACS.2014.675}, annote = {Keywords: Lempel-Ziv Factorization, String Index} }

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