Counting Palindromes in Substrings

We propose a data structure and an online algorithm to report the number of distinct palindromes in any substring of an input string. Assume that the string S of length n arrives symbol-by-symbol and every symbol is followed by zero or more queries of the form “report the number of distinct palindromes in S[i..j]”. We use \(O(n\log n)\) total time to process the string plus \(O(\log n)\) time per query. The required space is \(O(n\log n)\) in general and O(n) in a natural particular case. As a simple application, we describe an algorithm reporting all palindromic rich substrings of an input string in \(O(n\log n)\) time and O(n) space.