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Efficiently Finding All Maximal alpha-gapped Repeats

Authors Pawel Gawrychowski, Tomohiro I, Shunsuke Inenaga, Dominik Köppl, Florin Manea

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Pawel Gawrychowski
Tomohiro I
Shunsuke Inenaga
Dominik Köppl
Florin Manea

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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) https://doi.org/10.4230/LIPIcs.STACS.2016.39

Abstract

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|.

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Keywords
  • combinatorics on words
  • counting algorithms

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