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Optimal Bounds for Computing \(\alpha \)-gapped Repeats

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Language and Automata Theory and Applications (LATA 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9618))

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Abstract

Following (Kolpakov et al., 2013; Gawrychowski and Manea, 2015), we continue the study of \(\alpha \) -gapped repeats in strings, defined as factors uvu with \(|uv|\le \alpha |u|\). Our main result is the \(O(\alpha n)\) bound on the number of maximal \(\alpha \)-gapped repeats in a string of length n, previously proved to be \(O(\alpha ^2 n)\) in (Kolpakov et al., 2013). For a closely related notion of maximal \(\delta \)-subrepetition (maximal factors of exponent between \(1+\delta \) and 2), our result implies the \(O(n/\delta )\) bound on their number, which improves the bound of (Kolpakov et al., 2010) by a \(\log n\) factor.

We also prove an algorithmic time bound \(O(\alpha n+S)\) (S size of the output) for computing all maximal \(\alpha \)-gapped repeats. Our solution, inspired by (Gawrychowski and Manea, 2015), is different from the recently published proof by (Tanimura et al., 2015) of the same bound. Together with our bound on S, this implies an \(O(\alpha n)\)-time algorithm for computing all maximal \(\alpha \)-gapped repeats.

R. Kolpakov—The author was partially supported by Russian Foundation for Fundamental Research (Grant 15-07-03102).

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Notes

  1. 1.

    Note that in [15], the number of maximal \(\alpha \)-gapped palindromes was conjectured to be \(O(\alpha ^2 n)\).

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Author information

Authors and Affiliations

  1. King’s College London, London, WC2R 2LS, UK

    Maxime Crochemore

  2. Lomonosov Moscow State University, Leninskie Gory, Moscow, 119992, Russia

    Roman Kolpakov

  3. LIGM/CNRS, Université Paris-Est, 77454, Marne-la-vallée, France

    Gregory Kucherov

Authors
  1. Maxime Crochemore
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  2. Roman Kolpakov
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  3. Gregory Kucherov
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Correspondence to Gregory Kucherov .

Editor information

Editors and Affiliations

  1. Rovira i Virgili University, Tarragona, Spain

    Adrian-Horia Dediu

  2. Czech Technical University, Prague, Czech Republic

    Jan Janoušek

  3. Rovira i Virgili University, Tarragona, Spain

    Carlos Martín-Vide

  4. Justus Liebig University Giessen, Gießen, Germany

    Bianca Truthe

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Crochemore, M., Kolpakov, R., Kucherov, G. (2016). Optimal Bounds for Computing \(\alpha \)-gapped Repeats. In: Dediu, AH., Janoušek, J., Martín-Vide, C., Truthe, B. (eds) Language and Automata Theory and Applications. LATA 2016. Lecture Notes in Computer Science(), vol 9618. Springer, Cham. https://doi.org/10.1007/978-3-319-30000-9_19

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  • DOI: https://doi.org/10.1007/978-3-319-30000-9_19

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