Fast Algorithms for the Shortest Unique Palindromic Substring Problem 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(mlogσRLES+mlogm/loglogm) time so that all SUPSs for any subsequent query interval can be answered in O(logm/loglogm+α) time, where α is the number of outputs, and σRLES 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(loglogm/logloglogm+α) time and can be built in O(mlogσRLES+mlogm/loglogm) time, and the second one answers queries in O(log log m+ α) time and can be built in O(mlogσRLES) time. Both of these data structures require O(m) space.