Fast Algorithm for Partial Covers in Words

TL;DR

This paper introduces a new concept of partial covers in words, along with an efficient data structure and algorithms to compute shortest partial covers for various parameters, extending the classical notion of word covers.

Contribution

It proposes a novel $oldsymbol{ ext{ extit{$oldsymbol{ ext{ extalpha}}}$-partial cover}}$ concept and develops an $O(n)$-sized data structure with $O(n extlog n)$ construction time to efficiently compute shortest partial covers.

Findings

Data structure of size $O(n)$ constructed in $O(n extlog n)$ time.

Algorithm computes shortest $ extalpha$-partial covers for all $ extalpha=1, ext2, extots,n$.

Extension of cover concept to partial covers with efficient computation methods.

Abstract

A factor $u$ of a word $w$ is a cover of $w$ if every position in $w$ lies within some occurrence of $u$ in $w$. A word $w$ covered by $u$ thus generalizes the idea of a repetition, that is, a word composed of exact concatenations of $u$. In this article we introduce a new notion of $\alpha$-partial cover, which can be viewed as a relaxed variant of cover, that is, a factor covering at least $\alpha$ positions in $w$. We develop a data structure of $O(n)$ size (where $n=|w|$) that can be constructed in $O(n\log n)$ time which we apply to compute all shortest $\alpha$-partial covers for a given $\alpha$. We also employ it for an $O(n\log n)$-time algorithm computing a shortest $\alpha$-partial cover for each $\alpha=1,2,\ldots,n$.