[Paper Review] String Attractors and Combinatorics on Words

Sabrina Mantaci, Antonio Restivo+3 more|arXiv (Cornell University)|Jul 10, 2019

Algorithms and Data Compression23 references4 citations

TL;DR

This paper investigates string attractors as a combinatorial tool for analyzing finite and infinite words, showing that standard Sturmian words have a minimum string attractor of size 2, Thue-Morse words have a logarithmic-size attractor (conjectured minimal), and de Bruijin words exhibit an attractor size asymptotically growing as $ n / \log n $. The work establishes connections between attractor size and word complexity, proposing a new measure for combinatorial word analysis.

ABSTRACT

The notion of \emph{string attractor} has recently been introduced in [Prezza, 2017] and studied in [Kempa and Prezza, 2018] to provide a unifying framework for known dictionary-based compressors. A string attractor for a word $w=w[1]w[2]\cdots w[n]$ is a subset $Γ$ of the positions $\{1,\ldots,n\}$, such that all distinct factors of $w$ have an occurrence crossing at least one of the elements of $Γ$. While finding the smallest string attractor for a word is a NP-complete problem, it has been proved in [Kempa and Prezza, 2018] that dictionary compressors can be interpreted as algorithms approximating the smallest string attractor for a given word. In this paper we explore the notion of string attractor from a combinatorial point of view, by focusing on several families of finite words. The results presented in the paper suggest that the notion of string attractor can be used to define new tools to investigate combinatorial properties of the words.

Motivation & Objective

• To investigate the combinatorial properties of string attractors in well-known families of words.
• To determine the size of the smallest string attractor for standard Sturmian, Thue-Morse, and de Bruijin words.
• To explore whether the distribution and structure of attractor positions reflect deeper combinatorial features of the words.
• To propose string attractor size as a potential new measure of word complexity.
• To open new research directions on morphism-generated words and attractor distribution.

Proposed method

• The paper defines string attractors as sets of positions such that every distinct factor of a word has an occurrence crossing at least one position in the set.
• It applies known results from dictionary compression and the Burrows-Wheeler Transform to bound attractor size.
• For standard Sturmian words, the proof leverages their factorization properties and conjugacy structure to show a minimal attractor of size 2 with two consecutive positions.
• For Thue-Morse words, a construction of a string attractor of size $ \log n $ is provided, based on the recursive structure of the word.
• For de Bruijin words, the method combines LZ factorization bounds and the fact that all $ k $-length substrings occur exactly once to derive asymptotic bounds on attractor size.
• Theoretical bounds are derived using Theorem 7.1 and 7.2 on LZ parsing, leading to $ \frac{n}{\log n} \leq \gamma^* \leq \frac{n}{(1 - \epsilon_n)\log n} + 1 $ for de Bruijin words of length $ n + k - 1 $.

Experimental results

Research questions

• RQ1 What is the size of the smallest string attractor for standard Sturmian words, and can it be characterized combinatorially?
• RQ2 Is the $ \log n $-sized string attractor for Thue-Morse words minimal, and what structural properties of the word support this?
• RQ3 How does the size of the smallest string attractor grow for de Bruijin words, and what does this imply about their compressibility and structure?
• RQ4 Can the distribution of positions in a minimal string attractor reveal insights into the combinatorial complexity of a word?
• RQ5 What is the relationship between morphism generation and the size of the minimal string attractor in infinite words?

Key findings

• The smallest string attractor for any standard Sturmian word has size exactly 2, and it consists of two consecutive positions.
• A string attractor of size $ \log n $ exists for the Thue-Morse word of length $ n $, and the authors conjecture this is minimal.
• For de Bruijin words of order $ k $ over an alphabet of size $ \sigma $, the size of the smallest string attractor grows asymptotically as $ \frac{n}{\log n} $, where $ n = \sigma^k $.
• The lower bound $ \frac{n}{\log n} $ for de Bruijin words arises from the fact that all $ k $-length substrings occur exactly once, limiting the maximum distance between attractor positions.
• The upper bound for the smallest string attractor in de Bruijin words is $ \frac{n}{(1 - \epsilon_n)\log n} + 1 $, where $ \epsilon_n = 2\frac{1 + \log(\log(\sigma n))}{\log n} $.
• The results suggest that the distribution of attractor positions, not just their size, may carry combinatorial significance, opening new avenues for complexity analysis.