Finding maximal closed substrings

A string is closed if it has length 1 or has a nonempty border without internal occurrences. In this paper, we introduce the notion of a maximal closed substring (MCS), which is an occurrence of a closed substring that cannot be extended to the left nor to the right by one character into a longer cl...

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Bibliographic Details
Published in:Theoretical computer science Vol. 1060; p. 115628
Main Authors: Badkobeh, Golnaz, De Luca, Alessandro, Fici, Gabriele, Puglisi, Simon J.
Format: Journal Article
Language:English
Published: Elsevier B.V 18.01.2026
Subjects:
Closed string
Combinatorial algorithms
Combinatorics on words
Gapped repeat
Maximal repetition
Closed string
Maximal repetition
Combinatorial algorithms
Combinatorics on words
Gapped repeat
ISSN:0304-3975
Online Access:Get full text
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Summary:A string is closed if it has length 1 or has a nonempty border without internal occurrences. In this paper, we introduce the notion of a maximal closed substring (MCS), which is an occurrence of a closed substring that cannot be extended to the left nor to the right by one character into a longer closed substring. MCSs with exponent at least 2 are commonly called runs; those with exponent smaller than 2, instead, are particular cases of maximal gapped repeats. We provide an algorithm that, given a string of length n, locates all MCSs the string contains in O(nlogn) time.
ISSN:0304-3975
DOI:10.1016/j.tcs.2025.115628