Computing Covers under Substring Consistent Equivalence Relations

TL;DR

This paper generalizes the concept of string covers under substring consistent equivalence relations (SCERs) and shows that existing linear-time algorithms for shortest and longest cover arrays can be adapted to these relations.

Contribution

It introduces a generalized framework for covers under SCERs and proves the applicability of existing algorithms to this broader class of relations.

Findings

Existing algorithms for cover arrays extend to SCERs.

Generalized border arrays enable efficient computation under SCERs.

The framework broadens the understanding of quasiperiodicity in strings.

Abstract

Covers are a kind of quasiperiodicity in strings. A string $C$ is a cover of another string $T$ if any position of $T$ is inside some occurrence of $C$ in $T$. The shortest and longest cover arrays of $T$ have the lengths of the shortest and longest covers of each prefix of $T$, respectively. The literature has proposed linear-time algorithms computing longest and shortest cover arrays taking border arrays as input. An equivalence relation $\approx$ over strings is called a substring consistent equivalence relation (SCER) iff $X \approx Y$ implies (1) $|X| = |Y|$ and (2) $X[i:j] \approx Y[i:j]$ for all $1 \le i \le j \le |X|$. In this paper, we generalize the notion of covers for SCERs and prove that existing algorithms to compute the shortest cover array and the longest cover array of a string $T$ under the identity relation will work for any SCERs taking the accordingly generalized border arrays.