Combinatorics of minimal absent words for a sliding window

TL;DR

This paper investigates how the set of minimal absent words (MAWs) changes as a fixed-length sliding window moves over a string, providing tight bounds on the number of changes for different alphabet sizes.

Contribution

It introduces tight bounds on the maximum changes in MAWs during sliding window shifts, improving previous bounds for both general and binary alphabets.

Findings

Derived tight upper bounds for MAW changes

Established tight lower bounds matching the upper bounds

Improved upon previous bounds in the literature

Abstract

A string $w$ is called a minimal absent word (MAW) for another string $T$ if $w$ does not occur in $T$ but the proper substrings of $w$ occur in $T$. For example, let $\Sigma = \{\mathtt{a, b, c}\}$ be the alphabet. Then, the set of MAWs for string $w = \mathtt{abaab}$ is $\{\mathtt{aaa, aaba, bab, bb, c}\}$. In this paper, we study combinatorial properties of MAWs in the sliding window model, namely, how the set of MAWs changes when a sliding window of fixed length $d$ is shifted over the input string $T$ of length $n$, where $1 \leq d < n$. We present \emph{tight} upper and lower bounds on the maximum number of changes in the set of MAWs for a sliding window over $T$, both in the cases of general alphabets and binary alphabets. Our bounds improve on the previously known best bounds [Crochemore et al., 2020].