Fast Algorithms for the Shortest Unique Palindromic Substring Problem 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 $O(m \log \sigma _{\mathit {RLE}_{S}} + m \sqrt {\log m / \log \log m})$ time so that all SUPSs for any subsequent query interval can be answered in $O(\sqrt {\log m / \log \log m} + \alpha )$ time, where α is the number of outputs, and $\sigma _{\mathit {RLE}_{S}}$ is the number of distinct runs of RLE_S. Additionaly, we consider a variant of the SUPS problem where a query interval is also given in a run-length encoded form. For this variant of the problem, we present two alternative algorithms with faster queries. The first one answers queries in $O(\sqrt {\log \log m /\log \log \log m} + \alpha )$ time and can be built in $O(m \log \sigma _{\mathit {RLE}_{S}} + m \sqrt {\log m / \log \log m})$ time, and the second one answers queries in $O(\log \log m + \alpha )$ time and can be built in $O(m \log \sigma _{\mathit {RLE}_{S}})$ time. Both of these data structures require O(m) space.