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On the Complexity of Determining the Period of a String

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Combinatorial Pattern Matching (CPM 2000)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1848))

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Abstract

We study the complexity of a classical combinatorial problem of computing the period of a string. We investigate both the average- and the worst-case complexity of the problem.

We deliver almost tight bounds for the average-case. We show that every algorithm computing the period must examine \( \Omega (\sqrt m ) \) symbols of an input string of length m. On the other hand we present an algorithm that computes the period by examining on average \( {\rm O}\left( {\sqrt {m \cdot \log _{|\sum |} m} } \right) \) symbols, where \( \left| \sum \right| \geqslant 2 \) stands for the input alphabet.

We also present a deterministic algorithm that computes the period of a string using m + O(m 3/4) comparisons. This is the first algorithm that have the worst-case complexity m + o(m).

Work partly done while the author was with the Heinz Nixdorf Institute and Department of Mathematics & Computer Science at the University of Paderborn, D-33095 Paderborn, Germany.

Supported in part by Nuffield Foundation Award for newly appointed lecturers NUF-NAL.

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Author information

Authors and Affiliations

  1. Department of Computer and Information Science, New Jersey Institute of Technology, Newark, NJ, 07102-1982, USA

    Artur Czumaj

  2. Department of Computer Science, University of Liverpool, Peach Str., L69 7ZF, Liverpool, UK

    Leszek Gçasieniec

Authors
  1. Artur Czumaj
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  2. Leszek Gçasieniec
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Editor information

Editors and Affiliations

  1. Dipartimento di Matematica ed Applicazioni, Universitá die Palermo, Via Archirafi 34, 90123, Palermo, Italy

    Raffaele Giancarlo

  2. Centre de recherches mathématiques, Université de Montréal, CP 6128, succursale Centre-Ville, Montréal, Québec, Canada, H3C 3J7

    David Sankoff

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Czumaj, A., Gçasieniec, L. (2000). On the Complexity of Determining the Period of a String. In: Giancarlo, R., Sankoff, D. (eds) Combinatorial Pattern Matching. CPM 2000. Lecture Notes in Computer Science, vol 1848. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45123-4_34

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  • DOI: https://doi.org/10.1007/3-540-45123-4_34

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  • Print ISBN: 978-3-540-67633-1

  • Online ISBN: 978-3-540-45123-5

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