Linear-time computation of generalized minimal absent words for multiple strings

TL;DR

This paper introduces an efficient algorithm for computing generalized minimal absent words across multiple strings, achieving optimal time complexity for specific cases and extending previous single-string methods.

Contribution

It generalizes the concept of minimal absent words to multiple strings and provides optimal algorithms for their computation in various scenarios.

Findings

Optimal $O(n + | ext{M}|)$ time for two strings

Extended algorithm for multiple strings with $O(n ext{ceil}(k / ext{log} n) + | ext{M}|)$ time

Efficient use of $O(n (k + ext{log} n))$ bits of space in the word RAM model

Abstract

A string $w$ is called a minimal absent word (MAW) for a string $S$ if $w$ does not occur as a substring in $S$ and all proper substrings of $w$ occur in $S$. MAWs are well-studied combinatorial string objects that have potential applications in areas including bioinformatics, musicology, and data compression. In this paper, we generalize the notion of MAWs to a set $\mathcal{S} = \{S_1, \ldots, S_k\}$ of multiple strings. We first describe our solution to the case of $k = 2$ strings, and show how to compute the set $\mathsf{M}$ of MAWs in optimal $O(n + |\mathsf{M}|)$ time and with $O(n)$ working space, where $n$ denotes the total length of the strings in $\mathcal{S}$. We then move on to the general case of $k > 2$ strings, and show how to compute the set $\mathsf{M}$ of MAWs in $O(n \lceil k / \log n \rceil + |\mathsf{M}|)$ time and with $O(n (k + \log n))$ bits of working space, in the word RAM model with machine word size $\omega = \log n$. The latter algorithm runs in optimal $O(n + |\mathsf{M}|)$ time for $k = O(\log n)$.