On repetitiveness measures of Thue-Morse words
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 $n\geq 4$, 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 $n \geq 3$, 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 $2n$.
- Publication:
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arXiv e-prints
- Pub Date:
- May 2020
- DOI:
- 10.48550/arXiv.2005.09524
- arXiv:
- arXiv:2005.09524
- Bibcode:
- 2020arXiv200509524K
- Keywords:
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- Computer Science - Data Structures and Algorithms;
- Computer Science - Discrete Mathematics
- E-Print:
- accepted to SPIRE 2020