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Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2023.100

Convergence of the Number of Period Sets in Strings

Authors: Eric Rivals, Michelle Sweering, and Pengfei Wang

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Abstract

Consider words of length n. The set of all periods of a word of length n is a subset of {0,1,2,…,n-1}. However, any subset of {0,1,2,…,n-1} is not necessarily a valid set of periods. In a seminal paper in 1981, Guibas and Odlyzko proposed to encode the set of periods of a word into an n long binary string, called an autocorrelation, where a one at position i denotes the period i. They considered the question of recognizing a valid period set, and also studied the number of valid period sets for strings of length n, denoted κ_n. They conjectured that ln(κ_n) asymptotically converges to a constant times ln²(n). Although improved lower bounds for ln(κ_n)/ln²(n) were proposed in 2001, the question of a tight upper bound has remained open since Guibas and Odlyzko’s paper. Here, we exhibit an upper bound for this fraction, which implies its convergence and closes this longstanding conjecture. Moreover, we extend our result to find similar bounds for the number of correlations: a generalization of autocorrelations which encodes the overlaps between two strings.

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Eric Rivals, Michelle Sweering, and Pengfei Wang. Convergence of the Number of Period Sets in Strings. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 100:1-100:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{rivals_et_al:LIPIcs.ICALP.2023.100,
  author =	{Rivals, Eric and Sweering, Michelle and Wang, Pengfei},
  title =	{{Convergence of the Number of Period Sets in Strings}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{100:1--100:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.100},
  URN =		{urn:nbn:de:0030-drops-181527},
  doi =		{10.4230/LIPIcs.ICALP.2023.100},
  annote =	{Keywords: Autocorrelation, period, border, combinatorics, correlation, periodicity, upper bound, asymptotic convergence}
}

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Convergence of the Number of Period Sets in Strings

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