Optimal bounds for computing α-gapped repeats

Maxime Crochemore, Roman Kolpakov, Gregory Kucherov

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)


Following (Kolpakov et al., 2013; Gawrychowski and Manea, 2015), we continue the study of α-gapped repeats in strings, defined as factors of the form uvu with |uv|=|u|+|v|≤α|u|. Our main result is the O(αn) bound on the number of maximal α-gapped repeats in a string of length n, previously proved to be O(α2n) in (Kolpakov et al., 2013). For a closely related notion of maximal δ-subrepetition (maximal factors of exponent between 1+δ and 2), our result implies the O(n/δ) bound on their number, which improves the bound of (Kolpakov et al., 2010) by a log⁡n factor. We also prove an algorithmic time bound O(αn+S) (S size of the output) for computing all maximal α-gapped repeats. Our solution, inspired by (Gawrychowski and Manea, 2015), is different from the recently published proof by (Tanimura et al., 2015) of the same bound. Together with our bound on S, this implies an O(αn)-time algorithm for computing all maximal α-gapped repeats.

Original languageEnglish
Article number104434
JournalInformation and Computation
Publication statusPublished - Oct 2019
Externally publishedYes


  • Algorithms on strings
  • Combinatorial algorithms
  • Combinatorics on words
  • Gapped repeats
  • Repeats
  • Subrepetitions
  • Time complexity


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