Shortest unique palindromic substring queries in optimal time

TL;DR

This paper introduces an optimal preprocessing method for strings that allows quick retrieval of shortest unique palindromic substrings covering any query interval, significantly improving efficiency for such queries.

Contribution

It presents the first optimal solution to preprocess a string in linear time for fast retrieval of shortest unique palindromic substrings for any interval.

Findings

Preprocessing time and space are both O(n).

Query time for all SUPS is O(k+1), where k is the number of results.

The solution is optimal in both preprocessing and query efficiency.

Abstract

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 ($\mathit{SUPS}$) for an interval $[x, y]$ in $S$, if $P$ occurs exactly once in $S$, this occurrence of $P$ contains interval $[x, y]$, and every palindromic substring of $S$ which contains interval $[x, y]$ and is shorter than $P$ occurs at least twice in $S$. The $\mathit{SUPS}$ problem is, given a string $S$, to preprocess $S$ so that for any subsequent query interval $[x, y]$ all the $\mathit{SUPS}\mbox{s}$ for interval $[x, y]$ 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 $\mathit{SUPS}\mbox{s}$ for any subsequent query interval can be answered in $O(k+1)$ time, where $k$ is the number of outputs.