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Testing Quasiperiodicity

Christine Awofeso, Ben Bals, Oded Lachish, Solon P. Pissis

TL;DR

This work studies testing quasiperiodicity in strings by focusing on covers of length at most $q$ and robustness to an $ abla$ distance $ abla n$. It introduces a randomized, query-based tester with input access via an oracle that runs in $O(q^3 \varepsilon^{-1} \log q)$ queries, independent of the string length $n$, and shows how the same ideas yield a simple $O(n)$-time streaming algorithm using $O(q)$ space for short covers. The results connect seeds, covers, and overlap-based constructions through gcd and conical-combination arguments, and provide rigorous correctness guarantees. Together, these contributions enable efficient verification of quasiperiodicity in both static and streaming settings, with practical implications for pattern matching on very long strings.

Abstract

A cover (or quasiperiod) of a string $S$ is a shorter string $C$ such that every position of $S$ is contained in some occurrence of $C$ as a substring. The notion of covers was introduced by Apostolico and Ehrenfeucht over 30 years ago [Theor. Comput. Sci. 1993] and it has received significant attention from the combinatorial pattern matching community. In this note, we show how to efficiently test whether $S$ admits a cover. Our tester can also be translated into a streaming algorithm.

Testing Quasiperiodicity

TL;DR

This work studies testing quasiperiodicity in strings by focusing on covers of length at most and robustness to an distance . It introduces a randomized, query-based tester with input access via an oracle that runs in queries, independent of the string length , and shows how the same ideas yield a simple -time streaming algorithm using space for short covers. The results connect seeds, covers, and overlap-based constructions through gcd and conical-combination arguments, and provide rigorous correctness guarantees. Together, these contributions enable efficient verification of quasiperiodicity in both static and streaming settings, with practical implications for pattern matching on very long strings.

Abstract

A cover (or quasiperiod) of a string is a shorter string such that every position of is contained in some occurrence of as a substring. The notion of covers was introduced by Apostolico and Ehrenfeucht over 30 years ago [Theor. Comput. Sci. 1993] and it has received significant attention from the combinatorial pattern matching community. In this note, we show how to efficiently test whether admits a cover. Our tester can also be translated into a streaming algorithm.

Paper Structure

This paper contains 4 sections, 11 theorems, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. An Efficient Tester for Covers and Seeds
  4. A Simple Streaming Algorithm for Covers via Seeds

Key Result

Theorem 5

An $\mathcal{O}(n)$-size representation of all seeds of a string $S\in\Sigma^n$ can be computed in $\mathcal{O}(n\log n)$ time and $\mathcal{O}(n)$ space. If $\Sigma=[\sigma]$, with $\sigma=n^{\mathcal{O}(1)}$, the same representation can be computed in $\mathcal{O}(n)$ time.

Figures (2)

  • Figure 1: Combining the seed occurrences of two fragments with a long overlap.
  • Figure 2: Editing a string that is not $\varepsilon$-far from $\mathop{\mathrm{\mathrm{QP}_\Sigma}}\nolimits(q)$ to be in $\mathop{\mathrm{\mathrm{QP}_\Sigma}}\nolimits(q)$.

Theorems & Definitions (26)

  • Definition 1: Cover and Quasiperiod
  • Example 2
  • Definition 3: Seed
  • Example 4
  • Theorem 5: DBLP:conf/soda/KociumakaKRRW12DBLP:conf/esa/Radoszewski23
  • Definition 6: Conical Combination
  • Theorem 7: Frobenius Number Bound, erdos1972linear
  • Corollary 8: erdos1972linear
  • Corollary 9
  • proof
  • ...and 16 more