Minimal unique palindromic substrings after single-character substitution

Abstract

A palindrome is a string that reads the same forward and backward. A palindromic substring w of a string T is called a minimal unique palindromic substring (MUPS ) of T if w occurs only once in T and any proper palindromic substring of w occurs at least twice in T . MUPSs are utilized for answering the shortest unique palindromic substring problem, which is motivated by molecular biology [Inoue et al., 2018]. Given a string T of length n, all MUPSs of T can be computed in O(n) time. In this paper, we study the problem of updating the set of MUPSs when a character in the input string T is substituted by another character. We first analyze the number d of changes of MUPSs when a character is substituted, and show that d is in O(log n). Further, we present an algorithm that uses O(n) time and space for preprocessing, and updates the set of MUPSs in O(log σ + (log log n) 2 + d) time where σ is the alphabet size. We also propose a variant of the algorithm, which runs in optimal O(d) time when the alphabet size is constant.