Tight bounds on the maximum number of shortest unique substrings

A substring Q of a string S is called a shortest unique substring (SUS) for interval [s, t] in S, if Q occurs exactly once in S, this occurrence of Q contains 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 a string S, to preprocess S so that for any subsequent query interval [s, t] all the SUSs for interval [s, t] can be answered quickly. When s = t, we call the SUSs for [s, t] as point SUSs, and when s ≤ t, we call the SUSs for [s, t] as interval SUSs. There exist optimal O(n)-time preprocessing scheme which answers queries in optimal O(k) time for both point and interval SUSs, where n is the length of S and k is the number of outputs for a given query. In this paper, we reveal structural, combinatorial properties underlying the SUS problem: Namely, we show that the number of intervals in S that correspond to point SUSs for all query positions in S is less than 1.5n, and show that this is a matching upper and lower bound. Also, we consider the maximum number of intervals in S that correspond to interval SUSs for all query intervals in S.