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How Much Different Are Two Words with Different Shortest Periods

  • Mai Alzamel
  • Maxime Crochemore
  • Costas S. Iliopoulos
  • Tomasz Kociumaka
  • Ritu Kundu
  • Jakub RadoszewskiEmail author
  • Wojciech Rytter
  • Tomasz Waleń
Conference paper
Part of the IFIP Advances in Information and Communication Technology book series (IFIPAICT, volume 520)

Abstract

Sometimes the difference between two distinct words of the same length cannot be smaller than a certain minimal amount. In particular if two distinct words of the same length are both periodic or quasiperiodic, then their Hamming distance is at least 2. We study here how the minimum Hamming distance \( dist (x,y)\) between two words xy of the same length n depends on their periods. Similar problems were considered in [1] in the context of quasiperiodicities. We say that a period p of a word x is primitive if x does not have any smaller period \(p'\) which divides p. For integers pn (\(p\le n\)) we define \(\mathcal {P}_{p}(n)\) as the set of words of length n with primitive period p. We show several results related to the following functions introduced in this paper for \(p\ne q\) and \(n \ge \max (p,q)\).
$$\begin{aligned} {\mathcal D}_{p,q}(n) = \min \,\{\, dist (x,y)\,:\; x\in \mathcal {P}_{p}(n), \,y\in \mathcal {P}_{q}(n)\,\}, \\ N_{p,q}(h) = \max \,\{\, n \,:\; {\mathcal D}_{p,q}(n)\le h\,\}. \qquad \qquad \end{aligned}$$

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Copyright information

© IFIP International Federation for Information Processing 2018

Authors and Affiliations

  • Mai Alzamel
    • 1
  • Maxime Crochemore
    • 1
    • 2
  • Costas S. Iliopoulos
    • 1
  • Tomasz Kociumaka
    • 3
  • Ritu Kundu
    • 1
  • Jakub Radoszewski
    • 1
    • 3
    Email author
  • Wojciech Rytter
    • 3
  • Tomasz Waleń
    • 3
  1. 1.Department of InformaticsKing’s College LondonLondonUK
  2. 2.Université Paris-EstMarne-la-ValléeFrance
  3. 3.Faculty of Mathematics, Informatics and MechanicsUniversity of WarsawWarsawPoland

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