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A Separation of \(\gamma \) and b via Thue–Morse Words

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String Processing and Information Retrieval (SPIRE 2021)

Abstract

We prove that for \(n\ge 2\), the size \(b(t_n)\) of the smallest bidirectional scheme for the nth Thue–Morse word \(t_n\) is \(n+2\). Since Kutsukake et al. [SPIRE 2020] show that the size \(\gamma (t_n)\) of the smallest string attractor for \(t_n\) is 4 for \(n \ge 4\), this shows for the first time that there is a separation between the size of the smallest string attractor \(\gamma \) and the size of the smallest bidirectional scheme b, i.e., there exist string families such that \(\gamma = o(b)\).

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Acknowledgements

This work was supported by JSPS KAKENHI Grant Numbers JP20H04141 (HB), JP20J21147 (MF), JP19K20213 (TI), JP21K17701 (DK), JP20J11983 (TM).

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Correspondence to Hideo Bannai .

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Bannai, H., Funakoshi, M., I, T., Köppl, D., Mieno, T., Nishimoto, T. (2021). A Separation of \(\gamma \) and b via Thue–Morse Words. In: Lecroq, T., Touzet, H. (eds) String Processing and Information Retrieval. SPIRE 2021. Lecture Notes in Computer Science(), vol 12944. Springer, Cham. https://doi.org/10.1007/978-3-030-86692-1_14

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  • DOI: https://doi.org/10.1007/978-3-030-86692-1_14

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