Tight bounds on the maximum number of shortest unique substrings

TL;DR

This paper investigates the structural properties of shortest unique substrings (SUS) in strings, establishing tight bounds on their maximum number for point and interval SUSs, and providing insights into their combinatorial characteristics.

Contribution

It reveals tight bounds on the maximum number of SUSs in strings and analyzes their structural and combinatorial properties, advancing understanding of SUS complexity.

Findings

Number of point SUS intervals is less than 1.5n, matching the upper and lower bounds.

Provides structural and combinatorial insights into SUS properties.

Establishes tight bounds for maximum interval SUSs in strings.

Abstract

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 \leq 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.