The Edit Distance to $k$-Subsequence Universality

TL;DR

This paper introduces algorithms to determine the minimum number of edits required to transform a given word into a $k$-subsequence universal word, which contains all possible length-$k$ subsequences over its alphabet.

Contribution

It presents efficient algorithms for computing the minimal edit distance to achieve $k$-subsequence universality from any given word.

Findings

Algorithms effectively compute minimal edits for $k$-subsequence universality.

The methods optimize the process of transforming words into universal forms.

Results demonstrate practical efficiency for relevant applications.

Abstract

A word $u$ is a subsequence of another word $w$ if $u$ can be obtained from $w$ by deleting some of its letters. The word $w$ with alph$(w)=\Sigma$ is called $k$-subsequence universal if the set of subsequences of length $k$ of $w$ contains all possible words of length $k$ over $\Sigma$. We propose a series of efficient algorithms computing the minimal number of edit operations (insertion, deletion, substitution) one needs to apply to a given word in order to reach the set of $k$-subsequence universal words.