Shortest Unique Substring Queries on Run-Length Encoded Strings

We consider the problem of answering shortest unique substring (SUS) queries on run-length encoded strings. For a string S, a unique substring u = S[i..j] is said to be a shortest unique substring (SUS) of S containing an interval [s, t] (i j\u27-i\u27, S[i\u27..j\u27] occurs at least twice in S. Given a run-length encoding of size m of a string of length N, we show that we can construct a data structure of size O(m+pi_s(N, m)) in O(m log m + pi_c(N, m)) time such that queries can be answered in O(pi_q(N, m) + k) time, where k is the size of the output (the number of SUSs), and pi_s(N,m), pi_c(N,m), pi_q(N,m) are, respectively, the size, construction time, and query time for a predecessor/successor query data structure of m elements for the universe of [1,N]. Using the data structure by Beam and Fich (JCSS 2002), this results in a data structure of O(m) space that is constructed in O(m log m) time, and answers queries in O(sqrt(log m/loglog m)+k) time