On the number of MUSs crossing a position

2025-08-22

AI Summary

This paper investigates the maximum number of minimal unique substrings (MUSs) crossing a specified position $i$ in a string. For a string $T$ of length $n$, we establish the first tight asymptotic bound $Theta(sqrt{n})$ on $|{ m MUS}(T,i)|$, resolving a long-standing gap between known upper and lower bounds. Our approach combines combinatorial string analysis with suffix array properties and a novel boundary construction technique. First, we prove that any position $i$ in any length-$n$ string is crossed by at most $O(sqrt{n})$ MUSs. Second, we explicitly construct an infinite family of strings in which some position is crossed by $Omega(sqrt{n})$ MUSs. The matching upper and lower bounds are both constructive and asymptotically tight, thereby determining the theoretical optimum for this problem.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Data Mining & Knowledge Management: Data Compression

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

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 $Θ(sqrt{n})$ upper and lower bounds for the number $|MUS(T,i)|$ of MUSs containing a position $i$ in a string $T$ of length $n$.

Problem

• Count MUSs crossing a specific string position
• Establish tight bounds for MUSs at position i
• Prove Theta(sqrt(n)) bounds for MUSs per index

Innovation

• Establishes tight bounds for MUS count
• Proves Theta(sqrt(n)) upper and lower bounds
• Analyzes minimal unique substrings crossing positions