Abstract
We show that the size \(\gamma (t_n)\) of the smallest string attractor of the n-th Thue-Morse word \(t_n\) is 4 for any \(n\ge 4\), disproving the conjecture by Mantaci et al. [ICTCS 2019] that it is n. We also show that \(\delta (t_n) = \frac{10}{3+2^{4-n}}\) for \(n \ge 3\), where \(\delta (w)\) is the maximum over all \(k = 1,\ldots ,|w|\), the number of distinct substrings of length k in w divided by k, which is a measure of repetitiveness recently studied by Kociumaka et al. [LATIN 2020]. Furthermore, we show that the number \(z(t_n)\) of factors in the self-referencing Lempel-Ziv factorization of \(t_n\) is exactly 2n.
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Acknowledgments
This work was supported by JSPS KAKENHI Grant Numbers JP18K18002 (YN), JP17H01697 (SI), JP16H02783, JP20H04141 (HB), JP18H04098 (MT), and JST PRESTO Grant Number JPMJPR1922 (SI).
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Kutsukake, K., Matsumoto, T., Nakashima, Y., Inenaga, S., Bannai, H., Takeda, M. (2020). On Repetitiveness Measures of Thue-Morse Words. In: Boucher, C., Thankachan, S.V. (eds) String Processing and Information Retrieval. SPIRE 2020. Lecture Notes in Computer Science(), vol 12303. Springer, Cham. https://doi.org/10.1007/978-3-030-59212-7_15
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