String attractors and bi-infinite words

Abstract

String attractors are a combinatorial tool coming from the field of data compression. It is a set of positions within a word which intersects an occurrence of every factor. While one-sided infinite words admitting a finite string attractor are eventually periodic, the situation is different for two-sided infinite words. In this paper, we characterise the bi-infinite words admitting a finite string attractor as the characteristic Sturmian words and their morphic images. For words that do not admit finite string attractors, we study the structure and properties of their infinite string attractors.

Figures (5)

• Figure 1: Proof that if $f$ is a factor of $u$, then $f$ has an occurrence starting in $w'$.
• Figure 2: Proof of the case where the left and right period words are conjugates and $j \geq i$. The grey portion represents the positions in $\Gamma'$.
• Figure 3: The graph $\mathcal{G}_{n_1}(x)$.
• Figure 4: Proof that if $u$ is not a factor of $w$ and has an occurrence starting in $\varphi(\Gamma) \setminus \Gamma'$, then it has another occurrence crossing $\Gamma'$. The grey portion represents the positions in $\Gamma'$.
• Figure 5: The case where no occurrence of $w$ begins or ends in $\Gamma'$. The grey portion represents the positions in $\Gamma'$.

Theorems & Definitions (69)

• Remark 1
• Definition 2: String attractor
• Example 3
• Proposition 4
• Definition 5
• Definition 6: Span
• Proposition 7
• proof
• Definition 8
• Proposition 9