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 definition 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 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 show that a string of length $n$ contains $\mathcal O(n^{1.5})$ MCSs. We also provide an output-sensitive algorithm that, given a string of length $n$ over a constant-size alphabet, locates all $m$ MCSs the string contains in $\mathcal O(n\log n + m)$ time.