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 \emph{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 \emph{runs}; those with exponent smaller than $2$, instead, are particular cases of \emph{maximal gapped repeats}. We provide an algorithm that, given a string of length $n$ locates all MCSs the string contains in $\mathcal O(n\log n)$ time.