The Smallest String Attractors of Fibonacci and Period-Doubling Words

TL;DR

This paper characterizes all smallest string attractors for Fibonacci and period-doubling words, revealing the complexity and variety of minimal attractors beyond their size, with explicit formulas for their counts.

Contribution

It provides a complete characterization and recursive formulas for all smallest string attractors of Fibonacci and period-doubling words, highlighting their structural diversity.

Findings

Fibonacci words have a known smallest attractor size of 2.

The set of all smallest attractors for Fibonacci words is fully characterized.

The number of smallest attractors varies drastically among strings with the same size.

Abstract

A string attractor of a string $T[1..|T|]$ is a set of positions $\Gamma$ of $T$ such that any substring $w$ of $T$ has an occurrence that crosses a position in $\Gamma$, i.e., there is a position $i$ such that $w = T[i..i+|w|-1]$ and the intersection $[i,i+|w|-1]\cap \Gamma$ is nonempty. The size of the smallest string attractor of Fibonacci words is known to be $2$. We completely characterize the set of all smallest string attractors of Fibonacci words, and show a recursive formula describing the $2^{n-4} + 2^{\lceil n/2 \rceil - 2}$ distinct position pairs that are the smallest string attractors of the $n$th Fibonacci word for $n \geq 7$. Similarly, the size of the smallest string attractor of period-doubling words is known to be $2$. We also completely characterize the set of all smallest string attractors of period-doubling words, and show a formula describing the two distinct position pairs that are the smallest string attractors of the $n$th period-doubling word for $n\geq 2$. Our results show that strings with the same smallest attractor size can have a drastically different number of distinct smallest attractors.