Computing String Covers in Sublinear Time

TL;DR

This paper introduces sublinear time algorithms and data structures for computing string covers, including the shortest cover, using space-efficient representations, and analyzes their complexity in various computational models.

Contribution

It presents the first sublinear time algorithms for computing all covers of a string and designs a constant-time cover array query data structure.

Findings

All covers can be computed in optimal $O(n/ ext{log}_\sigma n)$ time.

A data structure of size $O(n( ext{log}\sigma + ext{log} ext log n)/ ext log n)$ supports constant-time cover array queries.

The structure of cover arrays for Fibonacci strings is characterized.

Abstract

Let $T$ be a string of length $n$ over an integer alphabet of size $\sigma$. In the word RAM model, $T$ can be represented in $O(n /\log_\sigma n)$ space. We show that a representation of all covers of $T$ can be computed in the optimal $O(n/\log_\sigma n)$ time; in particular, the shortest cover can be computed within this time. We also design an $O(n(\log\sigma + \log \log n)/\log n)$-sized data structure that computes in $O(1)$ time any element of the so-called (shortest) cover array of $T$, that is, the length of the shortest cover of any given prefix of $T$. As a by-product, we describe the structure of cover arrays of Fibonacci strings. On the negative side, we show that the shortest cover of a length-$n$ string cannot be computed using $o(n/\log n)$ operations in the PILLAR model of Charalampopoulos, Kociumaka, and Wellnitz (FOCS 2020).