Tighter Bounds and Optimal Algorithms for All Maximal α-gapped Repeats and Palindromes: Finding All Maximal α-gapped Repeats and Palindromes in Optimal Worst Case Time on Integer Alphabets

Paweł Gawrychowski, Tomohiro I, Shunsuke Inenaga, Dominik Köppl, Florin Manea

Research output: Contribution to journalArticlepeer-review

14 Citations (Scopus)

Abstract

An α-gapped repeat (α ≥ 1) in a word w is a factor uvu of w such that |uv| ≤ α|u|; the two occurrences of u are called arms of this α-gapped repeat. An α-gapped repeat is called maximal if its arms cannot be extended simultaneously with the same character to the right nor to the left. We show that the number of all maximal α-gapped repeats occurring in words of length n is upper bounded by 18αn. In the case of α-gapped palindromes, i.e., factors uvu with |uv|≤ α|u|, we show that the number of all maximal α-gapped palindromes occurring in words of length n is upper bounded by 28αn + 7n. Both upper bounds allow us to construct algorithms finding all maximal α-gapped repeats and/or all maximal α-gapped palindromes of a word of length n on an integer alphabet of size nO ( 1 ) in O(αn) time. The presented running times are optimal since there are words that have Θ(αn) maximal α-gapped repeats/palindromes.

Original languageEnglish
Pages (from-to)162-191
Number of pages30
JournalTheory of Computing Systems
Volume62
Issue number1
DOIs
Publication statusPublished - Jan 1 2018

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computational Theory and Mathematics

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