On the number of MUSs crossing a position

TL;DR

This paper establishes tight bounds on the maximum number of minimal unique substrings (MUSs) that can contain a specific position in a string, enhancing understanding of MUS distribution.

Contribution

It provides matching  bounds for the number of MUSs crossing a position, a significant theoretical advancement in string analysis.

Findings

Upper and lower bounds are both   for the number of MUSs crossing a position.

The bounds are tight, confirming the maximum number of crossing MUSs is .

The results improve understanding of MUS distribution in strings.

Abstract

A string $w$ is said to be a minimal unique substring (MUS) of a string $T$ if $w$ occurs exactly once in $T$, and any proper substring of $w$ occurs at least twice in $T$. It is known that the number of MUSs in a string $T$ of length $n$ is at most $n$, and that the set $MUS(T)$ of all MUSs in $T$ can be computed in $O(n)$ time [Ilie and Smyth, 2011]. Let $MUS(T,i)$ denote the set of MUSs that contain a position $i$ in a string $T$. In this short paper, we present matching $\Theta(\sqrt{n})$ upper and lower bounds for the number $|MUS(T,i)|$ of MUSs containing a position $i$ in a string $T$ of length $n$.