Minimal Unique Substrings and Minimal Absent Words in a Sliding Window

TL;DR

This paper investigates algorithms for efficiently computing minimal unique substrings and minimal absent words within a sliding window over a string, providing new bounds and an improved algorithm for MUSs.

Contribution

It introduces an $O(n ext{log}\sigma)$-time algorithm for MUSs in sliding windows and tight bounds on MAWs changes, improving previous results.

Findings

Developed an $O(n ext{log}\sigma)$-time algorithm for MUSs in sliding windows.

Established tight bounds on the maximum changes in MAWs in sliding windows.

Improved upon previous bounds for MAWs changes in sliding window scenarios.

Abstract

A substring $u$ of a string $T$ is called a minimal unique substring (MUS) of $T$ if $u$ occurs exactly once in $T$ and any proper substring of $u$ occurs at least twice in $T$. A string $w$ is called a minimal absent word (MAW) of $T$ if $w$ does not occur in $T$ and any proper substring of $w$ occurs in $T$. In this paper, we study the problems of computing MUSs and MAWs in a sliding window over a given string $T$. We first show how the set of MUSs can change in a sliding window over $T$, and present an $O(n\log\sigma)$-time and $O(d)$-space algorithm to compute MUSs in a sliding window of width $d$ over $T$, where $\sigma$ is the maximum number of distinct characters in every window. We then give tight upper and lower bounds on the maximum number of changes in the set of MAWs in a sliding window over $T$. Our bounds improve on the previous results in [Crochemore et al., 2017].