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A substring Q of a string S is called a shortest unique substring (SUS) forinterval [s,t] in S, if Q occurs exactly once in S, this occurrence of Qcontains interval [s,t], and every substring of S which contains interval [s,t]and is shorter than Q occurs at least twice in S. The SUS problem is, given astring S, to preprocess S so that for any subsequent query interval [s,t] allthe SUSs for interval [s,t] can be answered quickly. When s = t, we call theSUSs for [s,t] as point SUSs, and when s \leq t, we call the SUSs for [s,t] asinterval SUSs. There exist optimal O(n)-time preprocessing scheme which answersqueries in optimal O(k) time for both point and interval SUSs, where n is thelength of S and k is the number of outputs for a given query. In this paper, wereveal structural, combinatorial properties underlying the SUS problem: Namely,we show that the number of intervals in S that correspond to point SUSs for allquery positions in S is less than 1.5n, and show that this is a matching upperand lower bound. Also, we consider the maximum number of intervals in S thatcorrespond to interval SUSs for all query intervals in S.

Tight bounds on the maximum number of shortest unique substrings