License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.AofA.2020.4
URN: urn:nbn:de:0030-drops-120345
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12034/
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Berthé, Valérie ; Cesaratto, Eda ; Paccaut, Frédéric ; Rotondo, Pablo ; Safe, Martín D. ; Vallée, Brigitte

Two Arithmetical Sources and Their Associated Tries

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Abstract

This article is devoted to the study of two arithmetical sources associated with classical partitions, that are both defined through the mediant of two fractions. The Stern-Brocot source is associated with the sequence of all the mediants, while the Sturm source only keeps mediants whose denominator is "not too large". Even though these sources are both of zero Shannon entropy, with very similar Renyi entropies, their probabilistic features yet appear to be quite different. We then study how they influence the behaviour of tries built on words they emit, and we notably focus on the trie depth.
The paper deals with Analytic Combinatorics methods, and Dirichlet generating functions, that are usually used and studied in the case of good sources with positive entropy. To the best of our knowledge, the present study is the first one where these powerful methods are applied to a zero-entropy context. In our context, the generating function associated with each source is explicit and related to classical functions in Number Theory, as the ζ function, the double ζ function or the transfer operator associated with the Gauss map. We obtain precise asymptotic estimates for the mean value of the trie depth that prove moreover to be quite different for each source. Then, these sources provide explicit and natural instances which lead to two unusual and different trie behaviours.

BibTeX - Entry

@InProceedings{berth_et_al:LIPIcs:2020:12034,
  author =	{Val{\'e}rie Berth{\'e} and Eda Cesaratto and Fr{\'e}d{\'e}ric Paccaut and Pablo Rotondo and Mart{\'\i}n D. Safe and Brigitte Vall{\'e}e},
  title =	{{Two Arithmetical Sources and Their Associated Tries}},
  booktitle =	{31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020)},
  pages =	{4:1--4:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-147-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{159},
  editor =	{Michael Drmota and Clemens Heuberger},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12034},
  URN =		{urn:nbn:de:0030-drops-120345},
  doi =		{10.4230/LIPIcs.AofA.2020.4},
  annote =	{Keywords: Combinatorics of words, Information Theory, Probabilistic analysis, Analytic combinatorics, Dirichlet generating functions, Sources, Partitions, Trie structure, Continued fraction expansion, Farey map, Sturm words, Transfer operator}
}

Keywords: Combinatorics of words, Information Theory, Probabilistic analysis, Analytic combinatorics, Dirichlet generating functions, Sources, Partitions, Trie structure, Continued fraction expansion, Farey map, Sturm words, Transfer operator
Collection: 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020)
Issue Date: 2020
Date of publication: 10.06.2020

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