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 ( S U P S ) 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 S U P S queries on run-length encoded strings. We show how to preprocess a given run-length encoded string R L E S of size m in O ( m ) space and O ( m log σ R L E S + m log m / log log m ) time so that all S U P S s for any subsequent query interval can be answered in O ( log m / log log m + α ) time, where α is the number of outputs, and σ R L E S is the number of distinct runs of R L E 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 + α ) time and can be built in O ( m log σ R L E S + m log m / log log m ) time, and the second one answers queries in O ( log log m + α ) time and can be built in O ( m log σ R L E S ) time. Both of these data structures require O ( m ) space.