Absent Subsequences in Words

TL;DR

This paper investigates the properties and algorithms related to absent subsequences in words, focusing on minimal and shortest absent subsequences, with combinatorial characterizations and efficient computational methods.

Contribution

It introduces new characterizations, representations, and algorithms for identifying and computing minimal and shortest absent subsequences in strings.

Findings

Provided combinatorial characterizations of minimal and shortest absent subsequences.

Developed efficient algorithms for testing and computing these absent subsequences.

Designed data structures for quick queries on shortest absent subsequences within string factors.

Abstract

An absent factor of a string $w$ is a string $u$ which does not occur as a contiguous substring (a.k.a. factor) inside $w$. We extend this well-studied notion and define absent subsequences: a string $u$ is an absent subsequence of a string $w$ if $u$ does not occur as subsequence (a.k.a. scattered factor) inside $w$. Of particular interest to us are minimal absent subsequences, i.e., absent subsequences whose every subsequence is not absent, and shortest absent subsequences, i.e., absent subsequences of minimal length. We show a series of combinatorial and algorithmic results regarding these two notions. For instance: we give combinatorial characterisations of the sets of minimal and, respectively, shortest absent subsequences in a word, as well as compact representations of these sets; we show how we can test efficiently if a string is a shortest or minimal absent subsequence in a word, and we give efficient algorithms computing the lexicographically smallest absent subsequence of each kind; also, we show how a data structure for answering shortest absent subsequence-queries for the factors of a given string can be efficiently computed.