Computing Covers Using Prefix Tables

TL;DR

This paper introduces a linear-time algorithm for computing the cover array of regular strings using prefix tables and extends the approach to indeterminate strings, improving efficiency over previous methods.

Contribution

It presents the first linear-time algorithm for cover array computation based on prefix tables and extends it to indeterminate strings, enhancing computational efficiency.

Findings

Linear-time algorithm for regular strings using prefix tables

Extension of the algorithm to indeterminate strings

Improved efficiency over previous border array-based methods

Abstract

An \emph{indeterminate string} $x = x[1..n]$ on an alphabet $\Sigma$ is a sequence of nonempty subsets of $\Sigma$; $x$ is said to be \emph{regular} if every subset is of size one. A proper substring $u$ of regular $x$ is said to be a \emph{cover} of $x$ iff for every $i \in 1..n$, an occurrence of $u$ in $x$ includes $x[i]$. The \emph{cover array} $\gamma = \gamma[1..n]$ of $x$ is an integer array such that $\gamma[i]$ is the longest cover of $x[1..i]$. Fifteen years ago a complex, though nevertheless linear-time, algorithm was proposed to compute the cover array of regular $x$ based on prior computation of the border array of $x$. In this paper we first describe a linear-time algorithm to compute the cover array of regular string $x$ based on the prefix table of $x$. We then extend this result to indeterminate strings.