Document

**Published in:** LIPIcs, Volume 283, 34th International Symposium on Algorithms and Computation (ISAAC 2023)

A subsequence of a word w is a word u such that u = w[i₁] w[i₂] … w[i_k], for some set of indices 1 ≤ i₁ < i₂ < … < i_k ≤ |w|. A word w is k-subsequence universal over an alphabet Σ if every word in Σ^k appears in w as a subsequence. In this paper, we study the intersection between the set of k-subsequence universal words over some alphabet Σ and regular languages over Σ. We call a regular language L k-∃-subsequence universal if there exists a k-subsequence universal word in L, and k-∀-subsequence universal if every word of L is k-subsequence universal. We give algorithms solving the problems of deciding if a given regular language, represented by a finite automaton recognising it, is k-∃-subsequence universal and, respectively, if it is k-∀-subsequence universal, for a given k. The algorithms are FPT w.r.t. the size of the input alphabet, and their run-time does not depend on k; they run in polynomial time in the number n of states of the input automaton when the size of the input alphabet is O(log n). Moreover, we show that the problem of deciding if a given regular language is k-∃-subsequence universal is NP-complete, when the language is over a large alphabet. Further, we provide algorithms for counting the number of k-subsequence universal words (paths) accepted by a given deterministic (respectively, nondeterministic) finite automaton, and ranking an input word (path) within the set of k-subsequence universal words accepted by a given finite automaton.

Duncan Adamson, Pamela Fleischmann, Annika Huch, Tore Koß, Florin Manea, and Dirk Nowotka. k-Universality of Regular Languages. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 4:1-4:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{adamson_et_al:LIPIcs.ISAAC.2023.4, author = {Adamson, Duncan and Fleischmann, Pamela and Huch, Annika and Ko{\ss}, Tore and Manea, Florin and Nowotka, Dirk}, title = {{k-Universality of Regular Languages}}, booktitle = {34th International Symposium on Algorithms and Computation (ISAAC 2023)}, pages = {4:1--4:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-289-1}, ISSN = {1868-8969}, year = {2023}, volume = {283}, editor = {Iwata, Satoru and Kakimura, Naonori}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.4}, URN = {urn:nbn:de:0030-drops-193064}, doi = {10.4230/LIPIcs.ISAAC.2023.4}, annote = {Keywords: String Algorithms, Regular Languages, Finite Automata, Subsequences} }

Document

**Published in:** LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)

We consider subsequences with gap constraints, i. e., length-k subsequences p that can be embedded into a string w such that the induced gaps (i. e., the factors of w between the positions to which p is mapped to) satisfy given gap constraints gc = (C_1, C_2, …, C_{k-1}); we call p a gc-subsequence of w. In the case where the gap constraints gc are defined by lower and upper length bounds C_i = (L^-_i, L^+_i) ∈ ℕ² and/or regular languages C_i ∈ REG, we prove tight (conditional on the orthogonal vectors (OV) hypothesis) complexity bounds for checking whether a given p is a gc-subsequence of a string w. We also consider the whole set of all gc-subsequences of a string, and investigate the complexity of the universality, equivalence and containment problems for these sets of gc-subsequences.

Joel D. Day, Maria Kosche, Florin Manea, and Markus L. Schmid. Subsequences with Gap Constraints: Complexity Bounds for Matching and Analysis Problems. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 64:1-64:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{day_et_al:LIPIcs.ISAAC.2022.64, author = {Day, Joel D. and Kosche, Maria and Manea, Florin and Schmid, Markus L.}, title = {{Subsequences with Gap Constraints: Complexity Bounds for Matching and Analysis Problems}}, booktitle = {33rd International Symposium on Algorithms and Computation (ISAAC 2022)}, pages = {64:1--64:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-258-7}, ISSN = {1868-8969}, year = {2022}, volume = {248}, editor = {Bae, Sang Won and Park, Heejin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.64}, URN = {urn:nbn:de:0030-drops-173493}, doi = {10.4230/LIPIcs.ISAAC.2022.64}, annote = {Keywords: String algorithms, subsequences with gap constraints, pattern matching, fine-grained complexity, conditional lower bounds, parameterised complexity} }

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Complete Volume

**Published in:** LIPIcs, Volume 216, 30th EACSL Annual Conference on Computer Science Logic (CSL 2022)

LIPIcs, Volume 216, CSL 2022, Complete Volume

Florin Manea and Alex Simpson. LIPIcs, Volume 216, CSL 2022, Complete Volume. In 30th EACSL Annual Conference on Computer Science Logic (CSL 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 216, pp. 1-684, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@Proceedings{manea_et_al:LIPIcs.CSL.2022, title = {{LIPIcs, Volume 216, CSL 2022, Complete Volume}}, booktitle = {30th EACSL Annual Conference on Computer Science Logic (CSL 2022)}, pages = {1--684}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-218-1}, ISSN = {1868-8969}, year = {2022}, volume = {216}, editor = {Manea, Florin and Simpson, Alex}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2022}, URN = {urn:nbn:de:0030-drops-157199}, doi = {10.4230/LIPIcs.CSL.2022}, annote = {Keywords: LIPIcs, Volume 216, CSL 2022, Complete Volume} }

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Front Matter

**Published in:** LIPIcs, Volume 216, 30th EACSL Annual Conference on Computer Science Logic (CSL 2022)

Front Matter, Table of Contents, Preface, Conference Organization

Florin Manea and Alex Simpson. Front Matter, Table of Contents, Preface, Conference Organization. In 30th EACSL Annual Conference on Computer Science Logic (CSL 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 216, pp. 0:i-0:xx, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{manea_et_al:LIPIcs.CSL.2022.0, author = {Manea, Florin and Simpson, Alex}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {30th EACSL Annual Conference on Computer Science Logic (CSL 2022)}, pages = {0:i--0:xx}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-218-1}, ISSN = {1868-8969}, year = {2022}, volume = {216}, editor = {Manea, Florin and Simpson, Alex}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2022.0}, URN = {urn:nbn:de:0030-drops-157202}, doi = {10.4230/LIPIcs.CSL.2022.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

Document

**Published in:** LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

A pattern α is a string of variables and terminal letters. We say that α matches a word w, consisting only of terminal letters, if w can be obtained by replacing the variables of α by terminal words. The matching problem, i.e., deciding whether a given pattern matches a given word, was heavily investigated: it is NP-complete in general, but can be solved efficiently for classes of patterns with restricted structure. In this paper, we approach this problem in a generalized setting, by considering approximate pattern matching under Hamming distance. More precisely, we are interested in what is the minimum Hamming distance between w and any word u obtained by replacing the variables of α by terminal words. Firstly, we address the class of regular patterns (in which no variable occurs twice) and propose efficient algorithms for this problem, as well as matching conditional lower bounds. We show that the problem can still be solved efficiently if we allow repeated variables, but restrict the way the different variables can be interleaved according to a locality parameter. However, as soon as we allow a variable to occur more than once and its occurrences can be interleaved arbitrarily with those of other variables, even if none of them occurs more than once, the problem becomes intractable.

Paweł Gawrychowski, Florin Manea, and Stefan Siemer. Matching Patterns with Variables Under Hamming Distance. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 48:1-48:24, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{gawrychowski_et_al:LIPIcs.MFCS.2021.48, author = {Gawrychowski, Pawe{\l} and Manea, Florin and Siemer, Stefan}, title = {{Matching Patterns with Variables Under Hamming Distance}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {48:1--48:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.48}, URN = {urn:nbn:de:0030-drops-144886}, doi = {10.4230/LIPIcs.MFCS.2021.48}, annote = {Keywords: Pattern with variables, Matching algorithms, Hamming distance, Conditional lower bounds, Patterns with structural restrictions} }

Document

**Published in:** LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

A word u is a subsequence of another word w if u can be obtained from w by deleting some of its letters. In the early 1970s, Imre Simon defined the relation ∼_k (called now Simon-Congruence) as follows: two words having exactly the same set of subsequences of length at most k are ∼_k-congruent. This relation was central in defining and analysing piecewise testable languages, but has found many applications in areas such as algorithmic learning theory, databases theory, or computational linguistics. Recently, it was shown that testing whether two words are ∼_k-congruent can be done in optimal linear time. Thus, it is a natural next step to ask, for two words w and u which are not ∼_k-equivalent, what is the minimal number of edit operations that we need to perform on w in order to obtain a word which is ∼_k-equivalent to u.
In this paper, we consider this problem in a setting which seems interesting: when u is a k-subsequence universal word. A word u with alph(u) = Σ is called k-subsequence universal if the set of subsequences of length k of u contains all possible words of length k over Σ. As such, our results are a series of efficient algorithms computing the edit distance from w to the language of k-subsequence universal words.

Joel D. Day, Pamela Fleischmann, Maria Kosche, Tore Koß, Florin Manea, and Stefan Siemer. The Edit Distance to k-Subsequence Universality. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 25:1-25:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{day_et_al:LIPIcs.STACS.2021.25, author = {Day, Joel D. and Fleischmann, Pamela and Kosche, Maria and Ko{\ss}, Tore and Manea, Florin and Siemer, Stefan}, title = {{The Edit Distance to k-Subsequence Universality}}, booktitle = {38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)}, pages = {25:1--25:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-180-1}, ISSN = {1868-8969}, year = {2021}, volume = {187}, editor = {Bl\"{a}ser, Markus and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.25}, URN = {urn:nbn:de:0030-drops-136705}, doi = {10.4230/LIPIcs.STACS.2021.25}, annote = {Keywords: Subsequence, Scattered factor, Subword, Universality, k-subsequence universality, Edit distance, Efficient algorithms} }

Document

**Published in:** LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

Simon’s congruence ∼_k is a relation on words defined by Imre Simon in the 1970s and intensely studied since then. This congruence was initially used in connection to piecewise testable languages, but also found many applications in, e.g., learning theory, databases theory, or linguistics. The ∼_k-relation is defined as follows: two words are ∼_k-congruent if they have the same set of subsequences of length at most k. A long standing open problem, stated already by Simon in his initial works on this topic, was to design an algorithm which computes, given two words s and t, the largest k for which s∼_k t. We propose the first algorithm solving this problem in linear time O(|s|+|t|) when the input words are over the integer alphabet {1,…,|s|+|t|} (or other alphabets which can be sorted in linear time). Our approach can be extended to an optimal algorithm in the case of general alphabets as well.
To achieve these results, we introduce a novel data-structure, called Simon-Tree, which allows us to construct a natural representation of the equivalence classes induced by ∼_k on the set of suffixes of a word, for all k ≥ 1. We show that such a tree can be constructed for an input word in linear time. Then, when working with two words s and t, we compute their respective Simon-Trees and efficiently build a correspondence between the nodes of these trees. This correspondence, which can also be constructed in linear time O(|s|+|t|), allows us to retrieve the largest k for which s∼_k t.

Paweł Gawrychowski, Maria Kosche, Tore Koß, Florin Manea, and Stefan Siemer. Efficiently Testing Simon’s Congruence. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 34:1-34:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{gawrychowski_et_al:LIPIcs.STACS.2021.34, author = {Gawrychowski, Pawe{\l} and Kosche, Maria and Ko{\ss}, Tore and Manea, Florin and Siemer, Stefan}, title = {{Efficiently Testing Simon’s Congruence}}, booktitle = {38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)}, pages = {34:1--34:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-180-1}, ISSN = {1868-8969}, year = {2021}, volume = {187}, editor = {Bl\"{a}ser, Markus and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.34}, URN = {urn:nbn:de:0030-drops-136796}, doi = {10.4230/LIPIcs.STACS.2021.34}, annote = {Keywords: Simon’s congruence, Subsequence, Scattered factor, Efficient algorithms} }

Document

Track B: Automata, Logic, Semantics, and Theory of Programming

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

For quadratic word equations, there exists an algorithm based on rewriting rules which generates a directed graph describing all solutions to the equation. For regular word equations - those for which each variable occurs at most once on each side of the equation - we investigate the properties of this graph, such as bounds on its diameter, size, and DAG-width, as well as providing some insights into symmetries in its structure. As a consequence, we obtain a combinatorial proof that the problem of deciding whether a regular word equation has a solution is in NP.

Joel D. Day and Florin Manea. On the Structure of Solution Sets to Regular Word Equations. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 124:1-124:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{day_et_al:LIPIcs.ICALP.2020.124, author = {Day, Joel D. and Manea, Florin}, title = {{On the Structure of Solution Sets to Regular Word Equations}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {124:1--124:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.124}, URN = {urn:nbn:de:0030-drops-125314}, doi = {10.4230/LIPIcs.ICALP.2020.124}, annote = {Keywords: Quadratic Word Equations, Regular Word Equations, String Solving, NP} }

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**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

It is a long standing conjecture that the problem of deciding whether a quadratic word equation has a solution is in NP. It has also been conjectured that the length of a minimal solution to a quadratic equation is at most exponential in the length of the equation, with the latter conjecture implying the former. We show that both conjectures hold for some natural subclasses of quadratic equations, namely the classes of regular-reversed, k-ordered, and variable-sparse quadratic equations. We also discuss a connection of our techniques to the topic of unavoidable patterns, and the possibility of exploiting this connection to produce further similar results.

Joel D. Day, Florin Manea, and Dirk Nowotka. Upper Bounds on the Length of Minimal Solutions to Certain Quadratic Word Equations. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 44:1-44:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{day_et_al:LIPIcs.MFCS.2019.44, author = {Day, Joel D. and Manea, Florin and Nowotka, Dirk}, title = {{Upper Bounds on the Length of Minimal Solutions to Certain Quadratic Word Equations}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {44:1--44:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.44}, URN = {urn:nbn:de:0030-drops-109889}, doi = {10.4230/LIPIcs.MFCS.2019.44}, annote = {Keywords: Quadratic Word Equations, Length Upper Bounds, NP, Unavoidable Patterns} }

Document

Track B: Automata, Logic, Semantics, and Theory of Programming

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

We investigate the locality number, a recently introduced structural parameter for strings (with applications in pattern matching with variables), and its connection to two important graph-parameters, cutwidth and pathwidth. These connections allow us to show that computing the locality number is NP-hard but fixed-parameter tractable (when the locality number or the alphabet size is treated as a parameter), and can be approximated with ratio O(sqrt{log{opt}} log n). As a by-product, we also relate cutwidth via the locality number to pathwidth, which is of independent interest, since it improves the best currently known approximation algorithm for cutwidth. In addition to these main results, we also consider the possibility of greedy-based approximation algorithms for the locality number.

Katrin Casel, Joel D. Day, Pamela Fleischmann, Tomasz Kociumaka, Florin Manea, and Markus L. Schmid. Graph and String Parameters: Connections Between Pathwidth, Cutwidth and the Locality Number (Track B: Automata, Logic, Semantics, and Theory of Programming). In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 109:1-109:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{casel_et_al:LIPIcs.ICALP.2019.109, author = {Casel, Katrin and Day, Joel D. and Fleischmann, Pamela and Kociumaka, Tomasz and Manea, Florin and Schmid, Markus L.}, title = {{Graph and String Parameters: Connections Between Pathwidth, Cutwidth and the Locality Number}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {109:1--109:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.109}, URN = {urn:nbn:de:0030-drops-106858}, doi = {10.4230/LIPIcs.ICALP.2019.109}, annote = {Keywords: Graph and String Parameters, NP-Completeness, Approximation Algorithms} }

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**Published in:** LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

For k >= 3, a k-rollercoaster is a sequence of numbers whose every maximal contiguous subsequence, that is increasing or decreasing, has length at least k; 3-rollercoasters are called simply rollercoasters. Given a sequence of distinct real numbers, we are interested in computing its maximum-length (not necessarily contiguous) subsequence that is a k-rollercoaster. Biedl et al. (2018) have shown that each sequence of n distinct real numbers contains a rollercoaster of length at least ceil[n/2] for n>7, and that a longest rollercoaster contained in such a sequence can be computed in O(n log n)-time (or faster, in O(n log log n) time, when the input sequence is a permutation of {1,...,n}). They have also shown that every sequence of n >=slant (k-1)^2+1 distinct real numbers contains a k-rollercoaster of length at least n/(2(k-1)) - 3k/2, and gave an O(nk log n)-time (respectively, O(n k log log n)-time) algorithm computing a longest k-rollercoaster in a sequence of length n (respectively, a permutation of {1,...,n}).
In this paper, we give an O(nk^2)-time algorithm computing the length of a longest k-rollercoaster contained in a sequence of n distinct real numbers; hence, for constant k, our algorithm computes the length of a longest k-rollercoaster in optimal linear time. The algorithm can be easily adapted to output the respective k-rollercoaster. In particular, this improves the results of Biedl et al. (2018), by showing that a longest rollercoaster can be computed in optimal linear time. We also present an algorithm computing the length of a longest k-rollercoaster in O(n log^2 n)-time, that is, subquadratic even for large values of k <= n. Again, the rollercoaster can be easily retrieved. Finally, we show an Omega(n log k) lower bound for the number of comparisons in any comparison-based algorithm computing the length of a longest k-rollercoaster.

Paweł Gawrychowski, Florin Manea, and Radosław Serafin. Fast and Longest Rollercoasters. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 30:1-30:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{gawrychowski_et_al:LIPIcs.STACS.2019.30, author = {Gawrychowski, Pawe{\l} and Manea, Florin and Serafin, Rados{\l}aw}, title = {{Fast and Longest Rollercoasters}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, pages = {30:1--30:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-100-9}, ISSN = {1868-8969}, year = {2019}, volume = {126}, editor = {Niedermeier, Rolf and Paul, Christophe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.30}, URN = {urn:nbn:de:0030-drops-102694}, doi = {10.4230/LIPIcs.STACS.2019.30}, annote = {Keywords: sequences, alternating runs, patterns in permutations} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

A rollercoaster is a sequence of real numbers for which every maximal contiguous subsequence - increasing or decreasing - has length at least three. By translating this sequence to a set of points in the plane, a rollercoaster can be defined as an x-monotone polygonal path for which every maximal sub-path, with positive- or negative-slope edges, has at least three vertices. Given a sequence of distinct real numbers, the rollercoaster problem asks for a maximum-length (not necessarily contiguous) subsequence that is a rollercoaster. It was conjectured that every sequence of n distinct real numbers contains a rollercoaster of length at least ceil[n/2] for n>7, while the best known lower bound is Omega(n/log n). In this paper we prove this conjecture. Our proof is constructive and implies a linear-time algorithm for computing a rollercoaster of this length. Extending the O(n log n)-time algorithm for computing a longest increasing subsequence, we show how to compute a maximum-length rollercoaster within the same time bound. A maximum-length rollercoaster in a permutation of {1,...,n} can be computed in O(n log log n) time.
The search for rollercoasters was motivated by orthogeodesic point-set embedding of caterpillars. A caterpillar is a tree such that deleting the leaves gives a path, called the spine. A top-view caterpillar is one of maximum degree 4 such that the two leaves adjacent to each vertex lie on opposite sides of the spine. As an application of our result on rollercoasters, we are able to find a planar drawing of every n-vertex top-view caterpillar on every set of 25/3(n+4) points in the plane, such that each edge is an orthogonal path with one bend. This improves the previous best known upper bound on the number of required points, which is O(n log n). We also show that such a drawing can be obtained in linear time, when the points are given in sorted order.

Therese Biedl, Ahmad Biniaz, Robert Cummings, Anna Lubiw, Florin Manea, Dirk Nowotka, and Jeffrey Shallit. Rollercoasters and Caterpillars. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 18:1-18:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{biedl_et_al:LIPIcs.ICALP.2018.18, author = {Biedl, Therese and Biniaz, Ahmad and Cummings, Robert and Lubiw, Anna and Manea, Florin and Nowotka, Dirk and Shallit, Jeffrey}, title = {{Rollercoasters and Caterpillars}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {18:1--18:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.18}, URN = {urn:nbn:de:0030-drops-90224}, doi = {10.4230/LIPIcs.ICALP.2018.18}, annote = {Keywords: sequences, alternating runs, patterns in permutations, caterpillars} }

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**Published in:** LIPIcs, Volume 93, 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)

A pattern is a word consisting of constants from an alphabet Sigma of terminal symbols and variables from a set X. Given a pattern alpha, the decision-problem whether a given word w may be obtained by substituting the variables in \alpha for words over Sigma is called the matching problem. While this problem is, in general, NP-complete, several classes of patterns for which it can be efficiently solved are already known. We present two new classes of patterns, called k-local, and strongly-nested, and show that the respective matching problems, as well as membership can be solved efficiently for any fixed k.

Joel D. Day, Pamela Fleischmann, Florin Manea, and Dirk Nowotka. Local Patterns. In 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 93, pp. 24:1-24:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{day_et_al:LIPIcs.FSTTCS.2017.24, author = {Day, Joel D. and Fleischmann, Pamela and Manea, Florin and Nowotka, Dirk}, title = {{Local Patterns}}, booktitle = {37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)}, pages = {24:1--24:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-055-2}, ISSN = {1868-8969}, year = {2018}, volume = {93}, editor = {Lokam, Satya and Ramanujam, R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2017.24}, URN = {urn:nbn:de:0030-drops-84013}, doi = {10.4230/LIPIcs.FSTTCS.2017.24}, annote = {Keywords: Patterns with Variables, Local Patterns, Combinatorial Pattern Matching, Descriptive Patterns} }

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

We investigate the class of regular-ordered word equations. In such equations, each variable occurs at most once in each side and the order of the variables occurring in both left and right hand sides is preserved (the variables can be, however, separated by potentially distinct constant factors). Surprisingly, we obtain that solving such simple equations, even when the sides contain exactly the same variables, is NP-hard. By considerations regarding the combinatorial structure of the minimal solutions of the more general quadratic equations we obtain that the satisfiability problem for regular-ordered equations is in NP. The complexity of solving such word equations under regular constraints is also settled. Finally, we show that a related class of simple word equations, that generalises one-variable equations, is in P.

Joel D. Day, Florin Manea, and Dirk Nowotka. The Hardness of Solving Simple Word Equations. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 18:1-18:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{day_et_al:LIPIcs.MFCS.2017.18, author = {Day, Joel D. and Manea, Florin and Nowotka, Dirk}, title = {{The Hardness of Solving Simple Word Equations}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {18:1--18:14}, 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.2017.18}, URN = {urn:nbn:de:0030-drops-81233}, doi = {10.4230/LIPIcs.MFCS.2017.18}, annote = {Keywords: Word Equations, Regular Patterns, Regular Constraints} }

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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 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

For alpha >=1, an alpha-gapped repeat in a word w is a factor uvu of w such that |uv| <= alpha * |u|; the two occurrences of a factor u in such a repeat are called arms. Such a repeat is called maximal if its arms cannot be extended simultaneously with the same symbol to the right nor to the left. We show that the number of all maximal alpha-gapped repeats occurring in words of length n is upper bounded by 18 * alpha * n, allowing us to construct an algorithm finding all maximal alpha-gapped repeats of a word on an integer alphabet of size n^{O}(1)} in {O}(alpha * n) time. This result is optimal as there are words that have Theta(alpha * n) maximal alpha-gapped repeats. Our techniques can be extended to get comparable results in the case of alpha-gapped palindromes, i.e., factors uvu^{T} with |uv| <= alpha |u|.

Pawel Gawrychowski, Tomohiro I, Shunsuke Inenaga, Dominik Köppl, and Florin Manea. Efficiently Finding All Maximal alpha-gapped Repeats. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 39:1-39:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{gawrychowski_et_al:LIPIcs.STACS.2016.39, author = {Gawrychowski, Pawel and I, Tomohiro and Inenaga, Shunsuke and K\"{o}ppl, Dominik and Manea, Florin}, title = {{Efficiently Finding All Maximal alpha-gapped Repeats}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {39:1--39:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-001-9}, ISSN = {1868-8969}, year = {2016}, volume = {47}, editor = {Ollinger, Nicolas and Vollmer, Heribert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.39}, URN = {urn:nbn:de:0030-drops-57408}, doi = {10.4230/LIPIcs.STACS.2016.39}, annote = {Keywords: combinatorics on words, counting algorithms} }

Document

**Published in:** LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

A pattern (i. e., a string of variables and terminals) maps to a word, if this is obtained by uniformly replacing the variables by terminal words; deciding this is NP-complete. We present efficient
algorithms\footnote{The computational model we use is the standard unit-cost RAM with logarithmic word size. Also, all logarithms appearing in our time complexity evaluations are in base 2.} that solve this problem for restricted classes of patterns. Furthermore, we show that it is NP-complete to decide, for a given number k and a word w, whether w can be factorised into k distinct factors; this shows that the injective version (i.e., different variables are replaced by different words) of the above matching problem is NP-complete even for very restricted cases.

Henning Fernau, Florin Manea, Robert Mercas, and Markus L. Schmid. Pattern Matching with Variables: Fast Algorithms and New Hardness Results. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 302-315, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015)

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@InProceedings{fernau_et_al:LIPIcs.STACS.2015.302, author = {Fernau, Henning and Manea, Florin and Mercas, Robert and Schmid, Markus L.}, title = {{Pattern Matching with Variables: Fast Algorithms and New Hardness Results}}, booktitle = {32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)}, pages = {302--315}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-78-1}, ISSN = {1868-8969}, year = {2015}, volume = {30}, editor = {Mayr, Ernst W. and Ollinger, Nicolas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.302}, URN = {urn:nbn:de:0030-drops-49220}, doi = {10.4230/LIPIcs.STACS.2015.302}, annote = {Keywords: combinatorial pattern matching, combinatorics on words, patterns with variables, \$\{cal NP\}\$-complete string problems} }

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

Pseudo-repetitions are a natural generalisation of the classical notion of repetitions in sequences: they are the repeated concatenation of a word and its encoding under a certain morphism or antimorphism (anti-/morphism, for short). We approach the problem of deciding efficiently, for a word w and a literal anti-/morphism f, whether w contains an instance of a given pattern involving a variable x and its image under f, i.e., f(x). Our results generalise both the problem of finding fixed repetitive structures (e.g., squares, cubes) inside a word and the problem of finding palindromic structures inside a word. For instance, we can detect efficiently a factor of the form xx^Rxxx^R, or any other pattern of such type. We also address the problem of testing efficiently, in the same setting, whether the word w contains an arbitrary pseudo-repetition of a given exponent.

Pawel Gawrychowski, Florin Manea, and Dirk Nowotka. Testing Generalised Freeness of Words. In 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 25, pp. 337-349, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2014)

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@InProceedings{gawrychowski_et_al:LIPIcs.STACS.2014.337, author = {Gawrychowski, Pawel and Manea, Florin and Nowotka, Dirk}, title = {{Testing Generalised Freeness of Words}}, booktitle = {31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)}, pages = {337--349}, 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.2014.337}, URN = {urn:nbn:de:0030-drops-44694}, doi = {10.4230/LIPIcs.STACS.2014.337}, annote = {Keywords: Stringology, Pattern matching, Repetition, Pseudo-repetition} }

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**Published in:** LIPIcs, Volume 24, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)

We investigate the solution set of the pseudoperiodic extension of the classical Lyndon and Sch\"utzenberger word equations. Consider u_1 ... u_l = v_1 ... v_m w_1 ... w_n, where u_i is in {u, theta(u)} for all 1 <= i <= l, v_j is in {v, theta(v)} for all 1 <= j <= m, w_k is in {w, theta(w)} for all 1 <= k <= n and u, v and w are variables, and theta is an antimorphic involution. A solution is called pseudoperiodic, if u,v,w are in {t, theta(t)}^+ for a word t. [Czeizler et al./I&C/2011] established that for small values of l, m, and n non-periodic solutions exist, and that for large enough values all solutions are pseudoperiodic. However, they leave a gap between those bounds which we close for a number of cases. Namely, we show that for l = 3 and either m,n >= 12 or m,n >= 5 and either m and n are not both even or not all u_i's are equal, all solutions are pseudoperiodic.

Florin Manea, Mike Müller, and Dirk Nowotka. On the Pseudoperiodic Extension of u^l = v^m w^n. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 24, pp. 475-486, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2013)

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@InProceedings{manea_et_al:LIPIcs.FSTTCS.2013.475, author = {Manea, Florin and M\"{u}ller, Mike and Nowotka, Dirk}, title = {{On the Pseudoperiodic Extension of u^l = v^m w^n}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)}, pages = {475--486}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-64-4}, ISSN = {1868-8969}, year = {2013}, volume = {24}, editor = {Seth, Anil and Vishnoi, Nisheeth K.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2013.475}, URN = {urn:nbn:de:0030-drops-43948}, doi = {10.4230/LIPIcs.FSTTCS.2013.475}, annote = {Keywords: Word equations, Pseudoperiodicity, Lyndon-Sch\"{u}tzenberger equation} }

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**Published in:** LIPIcs, Volume 20, 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)

Pseudo-repetitions are a natural generalization of the classical notion of repetitions in sequences. We solve fundamental algorithmic questions on pseudo-repetitions by application of insightful combinatorial results on words. More precisely, we efficiently decide whether a word is a pseudo-repetition and find all the pseudo-repetitive factors of a word.

Pawel Gawrychowski, Florin Manea, Robert Mercas, Dirk Nowotka, and Catalin Tiseanu. Finding Pseudo-repetitions. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 257-268, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2013)

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@InProceedings{gawrychowski_et_al:LIPIcs.STACS.2013.257, author = {Gawrychowski, Pawel and Manea, Florin and Mercas, Robert and Nowotka, Dirk and Tiseanu, Catalin}, title = {{Finding Pseudo-repetitions}}, booktitle = {30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)}, pages = {257--268}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-50-7}, ISSN = {1868-8969}, year = {2013}, volume = {20}, editor = {Portier, Natacha and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2013.257}, URN = {urn:nbn:de:0030-drops-39394}, doi = {10.4230/LIPIcs.STACS.2013.257}, annote = {Keywords: Stringology, Pattern matching, Repetition, Pseudo-repetition} }

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