Shortest unique palindromic substring queries in optimal time

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 \( SUPS \text {s}\) 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 \( SUPS \text {s}\) for any subsequent query interval can be answered in \(O(\alpha + 1)\) time, where \(\alpha \) is the number of outputs.