The Smallest String Attractors of Fibonacci and Period-Doubling Words

Abstract

A string attractor of a string $T[1..|T|]$ is a set of positions $Γ$ of $T$ such that any substring $w$ of $T$ has an occurrence that crosses a position in $Γ$, i.e., there is a position $i$ such that $w = T[i..i+|w|-1]$ and the intersection $[i,i+|w|-1]\cap Γ$ 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.

Figures (18)

• Figure 1: For $n=8$. Left: all smallest attractors of $F_n$, where each pair of horizontally aligned dots represents an attractor. Right: characterization of these attractors according to \ref{['thm:all_smallest_attractors_of_fib']}; the three dotted boxes (top to bottom) correspond to \ref{['lem:valid_att_from_flip', 'lem:valid_att_from_flip_offset', 'lem:valid_att_from_n-2_offset']}. (See \ref{['fig:att-example-n=9']} in \ref{['sec:additional-figures']} for the case $n=9$.)
• Figure 2: Illustration of \ref{['lem:s_parse_tree']} and \ref{['def:l_r_l_prime']}. Top: $F_9$. Bottom: the parse tree of $F_9$ based on singular words. For $2 \leq k \leq 7$: the gray blocks are descendants of the center child $S_{k-3}$ in the parse tree for $S_k$, i.e. $S_k = S_{k-2}S_{k-3}S_{k-2}$. The green and blue blocks correspond to positions in $L_k$ and $R_k$, respectively. (Notice that these positions are not crossed by any center child in the parse tree for $S_k$.) The positions corresponding to $L'_k$ are highlighted directly in orange.
• Figure 3: Illustration of the proof of \ref{['lem:invalidate_center']} for $n=11$. Circles and crosses at the top mark valid and invalid attractor positions, respectively. A cross highlighted in a given color indicates that the position is invalid because it fails to be crossed by the substring shown in the same color. (Note that some combinations of the positions marked by circles in $S_{n-3}$ and $S_{n-2}$ are still invalid; these are characterized in \ref{['lem:invalidate_second_half_of_R']}.)
• Figure 4: Illustration of the proof of \ref{['lem:invalidate_second_half_of_R']} for $n=9$, where $P = S_{n-3}[1..2]$ and $\alpha = S_{(n-1)\bmod 2}$. Circles (resp. squares) at the top mark positions in $U_n$ (resp. $V_n$) that are not ruled out by \ref{['lem:invalidate_center']}. A circle-square pair in $U_n \otimes V_n$ is invalid if they are both in a red region since they fail to be crossed by $S_{n-4} \ P$. (All other circle-square pairs are indeed valid, which we prove in \ref{['subsec:valid_attractors']}.)
• Figure 5: Illustration of \ref{['lem:valid_att_from_flip']} for $n=10$. White and black markers at the top denote positions in $\mathcal{A}(F_{n-1})\xspace$ and $\mathcal{A}(F_n)\xspace$, respectively. Highlighted markers, together with the highlighted occurrences of $F_{n-1}$ and $(F_{n-1})^R$, illustrate the mapping described in the proof. Pink and orange squiggly arrows illustrate Case \ref{['flip-case:in-g-n-1']} and Case \ref{['flip-case:in-f-n-2']}, respectively. (Note that $\Delta_{n-1} = \overline{\Delta_n}$ and $(\Delta_{n-1})^R=\Delta_n$.)

Theorems & Definitions (25)

• Definition 1: String Attractors DBLP:conf/stoc/KempaP18
• Definition 3: $F_n$ and $f_n$
• Definition 4: $G_n$, $\Delta_n$
• Definition 6: Singular Words
• Lemma 7: DBLP:journals/ejc/WenW94
• Lemma 8
• Corollary 9
• Lemma 10
• Definition 11: $U_n$ and $V_n$
• Proposition 12: DBLP:conf/ictcs/MantaciRRRS19