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 \( SUPS \) 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 \( RLE _{S}\).