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The Smallest String Attractors of Fibonacci and Period-Doubling Words

Authors Mutsunori Banbara , Hideo Bannai , Peaker Guo , Dominik Köppl , Takuya Mieno , Yoshio Okamoto

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LIPIcs.CPM.2026.33.pdf
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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics on words
Keywords
  • String attractors
  • Fibonacci words
  • Period-doubling words
  • Combinatorics on words

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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]∩ 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^{⌈n/2⌉ - 2} distinct position pairs that are the smallest string attractors of the nth Fibonacci word for n ≥ 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 nth period-doubling word for n ≥ 2. Our results show that strings with the same smallest attractor size can have a drastically different number of distinct smallest attractors.

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Mutsunori Banbara, Hideo Bannai, Peaker Guo, Dominik Köppl, Takuya Mieno, and Yoshio Okamoto. The Smallest String Attractors of Fibonacci and Period-Doubling Words. In 37th Annual Symposium on Combinatorial Pattern Matching (CPM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 369, pp. 33:1-33:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.CPM.2026.33

Author Details

Mutsunori Banbara
  • Graduate School of Informatics, Nagoya University, Japan
Hideo Bannai
  • M&D Data Science Center, Institute of Integrated Research, Institute of Science Tokyo, Japan
Peaker Guo
  • M&D Data Science Center, Institute of Integrated Research, Institute of Science Tokyo, Japan
Dominik Köppl
  • Department of Computer Science and Engineering, University of Yamanashi, Kofu, Japan
Takuya Mieno
  • Graduate School of Informatics and Engineering, University of Electro-Communications, Chofu, Japan
Yoshio Okamoto
  • Graduate School of Informatics and Engineering, University of Electro-Communications, Chofu, Japan

Funding

  • Bannai, Hideo: JSPS KAKENHI Grant Number JP24K02899.
  • Köppl, Dominik: JSPS KAKENHI Grant Number JP25K21150.
  • Mieno, Takuya: JSPS KAKENHI Grant Number JP24K20734.
  • Okamoto, Yoshio: JSPS KAKENHI Grant Number JP23K10982.

References

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