Abstract
In this article we consider the following question: N words of length L are generated using a biased memoryless source, i.e. each letter is taken independently according to some fixed distribution on the alphabet, and collected in a set (duplicates are removed); what are the frequencies of the letters in a typical element of this random set? We prove that the typical frequency distribution of such a word can be characterized by considering the parameter \(\ell = L/\log N\). We exhibit two thresholds \(\ell _0<\ell _1\) that only depend on the source, such that if \(\ell \le \ell _0\), the distribution resembles the uniform distribution; if \(\ell \ge \ell _1\) it resembles the distribution of the source; and for \(\ell _0\le \ell \le \ell _1\) we characterize the distribution as an interpolation of the two extremal distributions.
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Notes
- 1.
It is a more natural view of the process to consider L as fixed and N as varying; this, however, leads to the somewhat artificial parameterization \(N=\exp (L/\ell )\).
- 2.
By “roughly” we mean up to some multiplicative power of L, with \(L=\varTheta (\log N)\) at our scale.
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Acknowledgments
The authors are grateful to Arnaud Carayol for his precious help when preparing this article, and an anonymous reviewer for suggesting the promising alternative \(\alpha \)-parametrization of the problem.
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Duchon, P., Nicaud, C. (2018). On the Biased Partial Word Collector Problem. In: Bender, M., Farach-Colton, M., Mosteiro, M. (eds) LATIN 2018: Theoretical Informatics. LATIN 2018. Lecture Notes in Computer Science(), vol 10807. Springer, Cham. https://doi.org/10.1007/978-3-319-77404-6_30
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DOI: https://doi.org/10.1007/978-3-319-77404-6_30
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