Compact Data Structures for Shortest Unique Substring Queries

Abstract

Given a string T of length n, a substring $u = T[i..j]$ of T is called a shortest unique substring (SUS) for an interval [s, t] if (a) u occurs exactly once in T, (b) u contains the interval [s, t] (i.e. $i \le s \le$$t \le j$), and (c) every substring v of T with $|v| < |u|$ containing [s, t] occurs at least twice in T. Given a query interval $[s, t] \subset [1, n]$, the interval SUS problem is to output all the SUSs for the interval [s, t]. In this article, we propose a $4n + o(n)$ bits data structure answering an interval SUS query in output-sensitive $O(occ)$ time, where $occ$ is the number of returned SUSs. Additionally, we focus on the point SUS problem, which is the interval SUS problem for $s = t$. Here, we propose a $\lceil (\log _2{3} + 1)n \rceil + o(n)$ bits data structure answering a point SUS query in the same output-sensitive time.