Fast Algorithms for the Shortest Unique Palindromic Substring Problem on Run-Length Encoded Strings

摘要

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 5. 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 RLEs of size m in O(m) space and O(m log σRLE_s + m-(log m/ log log m)~(1/2) time so that all SUPSs for any subsequent query interval can be answered in O (log m/ log log m)~(1/2)+ α) time, where α is the number of outputs, and σRLE_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 (log log m/ log log log m)~(1/2)+α) time and can be built in O (m log σRLE_s + m,(log m j log log m)~(1/2) time, and the second one answers queries in O(log log m + a) time and can be built in O(m log σRLE_s) time. Both of these data structures require O(m) space.