Shortest Unique Palindromic Substring Queries on Run-Length Encoded Strings

Abstract

For a string S, a palindromic substring S[i..j] is said to be a shortest unique palindromic substring ($SUPS$) for an interval [s, t] in S, if S[i..j] occurs exactly once in S, the interval [i, j] contains [s, t], and every palindromic substring containing [s, t] which is shorter than S[i..j] occurs at least twice in S. In this paper, we study the problem of answering $SUPS$ queries on run-length encoded strings. We show how to preprocess a given run-length encoded string $RLE _{S}$ of size m in O(m) space and $O(m \log \sigma _{ RLE _{S}} + m \sqrt{\log m / \log \log m})$ time so that all $SUPSs$ for any subsequent query interval can be answered in $O(\sqrt{\log m / \log \log m} + \alpha )$ time, where $\alpha$ is the number of outputs, and $\sigma _{ RLE _{S}}$ is the number of distinct runs of $RLE _{S}$.