Shortest Unique Palindromic Substring Queries on Run-Length Encoded Strings

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 $\mathit{SUPS}$ 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 $O(m (\log \sigma_{RLE_{S}} + \sqrt{\log m / \log\log m}))$ time so that all $\mathit{SUPSs}$ 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 $RLE_{S}$.