String Attractors: Verification and Optimization

TL;DR

This paper advances the theory of string attractors by providing efficient algorithms for verification, minimality, and approximation, leveraging new reductions based on the truncated suffix tree, and explores the complexity of related problems.

Contribution

It introduces a novel reduction using truncated suffix trees, enabling faster algorithms for verifying and approximating string attractors, and analyzes the complexity of the sharp-$k$-attractor problem.

Findings

Minimum 3-attractor can be found in O(n) time under certain conditions.

A 2.45-approximation for 3-attractor is achievable in O(n) time.

Sharp-$k$-attractor problem is NP-complete for k ≥ 3.

Abstract

String attractors [STOC 2018] are combinatorial objects recently introduced to unify all known dictionary compression techniques in a single theory. A set $\Gamma\subseteq [1..n]$ is a $k$-attractor for a string $S\in[1..\sigma]^n$ if and only if every distinct substring of $S$ of length at most $k$ has an occurrence straddling at least one of the positions in $\Gamma$. Finding the smallest $k$-attractor is NP-hard for $k\geq3$, but polylogarithmic approximations can be found using reductions from dictionary compressors. It is easy to reduce the $k$-attractor problem to a set-cover instance where string's positions are interpreted as sets of substrings. The main result of this paper is a much more powerful reduction based on the truncated suffix tree. Our new characterization of the problem leads to more efficient algorithms for string attractors: we show how to check the validity and minimality of a $k$-attractor in near-optimal time and how to quickly compute exact and approximate solutions. For example, we prove that a minimum $3$-attractor can be found in optimal $O(n)$ time when $\sigma\in O(\sqrt[3+\epsilon]{\log n})$ for any constant $\epsilon>0$, and $2.45$-approximation can be computed in $O(n)$ time on general alphabets. To conclude, we introduce and study the complexity of the closely-related sharp-$k$-attractor problem: to find the smallest set of positions capturing all distinct substrings of length exactly $k$. We show that the problem is in P for $k=1,2$ and is NP-complete for constant $k\geq 3$.