Algorithms and combinatorial properties on shortest unique palindromic substrings

A palindrome is a string that reads the same forward and backward. A palindromic substring P of a string S is called a shortest unique palindromic substring (SUPS) for an interval [s,t] in S, if P occurs exactly once in S, this occurrence of P contains interval [s,t], and every palindromic substring of S which contains interval [s,t] and is shorter than P occurs at least twice in S. The SUPS problem is, given a string S, to preprocess S so that for any subsequent query interval [s,t] all the SUPSs for interval [s,t] can be answered quickly. We present an optimal solution to this problem. Namely, we show how to preprocess a given string S of length n in O(n) time and space so that all SUPSs for any subsequent query interval can be answered in O(α+1) time, where α is the number of outputs. We also discuss the number of SUPSs in a string.