Building Sources of Zero Entropy: Rescaling and Inserting Delays (Invited Talk)

Authors Ali Akhavi, Fréderic Paccaut, Brigitte Vallée



PDF
Thumbnail PDF

File

LIPIcs.AofA.2022.1.pdf
  • Filesize: 0.89 MB
  • 28 pages

Document Identifiers

Author Details

Ali Akhavi
  • GREYC, Université de Caen-Normandie, France
  • LIPN, Université Sorbonne Paris Nord, France
Fréderic Paccaut
  • LAMFA CNRS UMR 7352, Université de Picardie Jules Verne, France
Brigitte Vallée
  • GREYC, CNRS and Université de Caen-Normandie, France

Cite As Get BibTex

Ali Akhavi, Fréderic Paccaut, and Brigitte Vallée. Building Sources of Zero Entropy: Rescaling and Inserting Delays (Invited Talk). In 33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 225, pp. 1:1-1:28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.AofA.2022.1

Abstract

Most of the natural sources that intervene in Information Theory have a positive entropy. They are well studied. The paper aims in building, in an explicit way, natural instances of sources with zero entropy. Such instances are obtained by slowing down sources of positive entropy, with processes which rescale sources or insert delays. These two processes - rescaling or inserting delays - are essentially the same; they do not change the fundamental intervals of the source, but only the "depth" at which they will be used, or the "speed" at which they are divided. However, they modify the entropy and lead to sources with zero entropy. The paper begins with a "starting" source of positive entropy, and uses a natural class of rescalings of sublinear type. In this way, it builds a class of sources of zero entropy that will be further analysed. As the starting sources possess well understood probabilistic properties, and as the process of rescaling does not change its fundamental intervals, the new sources keep the memory of some important probabilistic features of the initial source. Thus, these new sources may be thoroughly analysed, and their main probabilistic properties precisely described. We focus in particular on two important questions: exhibiting asymptotical normal behaviours à la Shannon-MacMillan-Breiman; analysing the depth of the tries built on the sources. In each case, we obtain a parameterized class of precise behaviours. The paper deals with the analytic combinatorics methodology and makes a great use of generating series.

Subject Classification

ACM Subject Classification
  • Theory of computation → Pattern matching
  • Theory of computation → Randomness, geometry and discrete structures
Keywords
  • Information Theory
  • Probabilistic analysis of sources
  • Sources with zero-entropy
  • Analytic combinatorics
  • Dirichlet generating functions
  • Transfer operator
  • Trie structure
  • Continued fraction expansion
  • Rice method
  • Quasi-power Theorem

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Viviane Baladi and Brigitte Vallée. Euclidean Algorithms are gaussian. Journal of Number Theory, Volume 110, Issue, pages 331-386, 2005. Google Scholar
  2. Valérie Berthé, Eda Cesaratto, Frédéric Paccaut, Pablo Rotondo, Martín D. Safe, and Brigitte Vallée. Two Arithmetical Sources and their associated Tries. In Michael Drmota and Clemens Heuberger, editors, 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms, AofA 2020, June 15-19, 2020, Klagenfurt, Austria (Virtual Conference), volume 159 of LIPIcs, pages 4:1-4:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.AofA.2020.4.
  3. Valérie Berthé, Eda Cesaratto, Pablo Rotondo, and Martin Dario Safe. Lochs-type Theorems beyond positive entropy. arXiv preprint, 2022. URL: https://doi.org/10.48550/arXiv.2202.04008v1.
  4. Eda Cesaratto and Brigitte Vallée. Gaussian Distribution of Trie depth for Strongly Tame sources. Comb. Probab. Comput., 24(1):54-103, 2015. URL: https://doi.org/10.1017/S0963548314000741.
  5. Julien Clément, James Allen Fill, Thu Hien Nguyen Thi, and Brigitte Vallée. Towards a realistic analysis of the Quickselect algorithm. Theory Comput. Syst., 58(4):528-578, 2016. URL: https://doi.org/10.1007/s00224-015-9633-5.
  6. Julien Clément, Philippe Flajolet, and Brigitte Vallée. Dynamical Sources in Information Theory: A General Analysis of Trie structures. Algorithmica, 29(1):307-369, 2001. URL: https://doi.org/10.1007/BF02679623.
  7. Julien Clément, Thu Hien Nguyen Thi, and Brigitte Vallée. Towards a realistic analysis of some popular sorting algorithms. Comb. Probab. Comput., 24(1):104-144, 2015. URL: https://doi.org/10.1017/S0963548314000649.
  8. Robert M Corless. Continued fractions and chaos. American Mathematical Monthly, 99:203-215, 1992. Google Scholar
  9. Karma Dajani and A. Howard Fieldsteel. Equipartition of interval partitions and an application to number theory. In Proceedings of the American Mathematical Society, volume 129, pages 3453-3460, 2001. Google Scholar
  10. Dmitry Dolgopyat. On decay of correlations in Anosov flows. Ann. of Math., 147(2):357-390, 1998. URL: https://doi.org/10.2307/121012.
  11. Philippe Flajolet. The Ubiquitous Digital Tree. In STACS 2006, 23rd Annual Symposium on Theoretical Aspects of Computer Science, Marseille, France, February 23-25, 2006, Proceedings, volume 3884 of Lecture Notes in Computer Science, pages 1-22. Springer, 2006. URL: https://doi.org/10.1007/11672142_1.
  12. Philippe Flajolet, Mathieu Roux, and Brigitte Vallée. Digital Trees and Memoryless sources: from Arithmetics to Analysis. Proceedings of AofA'10, DMTCS, proc AM, pages 231-258, 2010. URL: http://www.dmtcs.org/dmtcs-ojs/index.php/proceedings/article/view/dmAM0117/3289.
  13. Philippe Flajolet and Robert Sedgewick. Digital Search Trees revisited. SIAM J. Comput., 15(3):748-767, 1986. URL: https://doi.org/10.1137/0215054.
  14. Philippe Flajolet and Robert Sedgewick. Mellin Transforms and Asymptotics: Finite differences and Rice’s integrals. Theor. Comput. Sci., 144(1&2):101-124, 1995. URL: https://doi.org/10.1016/0304-3975(94)00281-M.
  15. Philippe Flajolet and Robert Sedgewick. Analytic Combinatorics. Cambridge University Press, 2009. URL: http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=9780521898065.
  16. Hsien-Kuei Hwang. On Convergence Rates in the central limit theorems for Combinatorial Structures. Eur. J. Comb., 19(3):329-343, 1998. URL: https://doi.org/10.1006/eujc.1997.0179.
  17. Hsien-Kuei Hwang. Limit Theorems for the Number of Summands in Integer Partitions. J. Comb. Theory, Ser. A, 96(1):89-126, 2001. URL: https://doi.org/10.1006/jcta.2000.3170.
  18. N. E. Nörlund. Leçons sur les équations linéaires aux différences finies. In Collection de monographies sur la théorie des fonctions. Gauthier-Villars, Paris, 1929. Google Scholar
  19. Niels Erik Nörlund. Vorlesungen über Differenzenrechnung. Chelsea Publishing Company, New York, 1954. Google Scholar
  20. Mathieu Roux and Brigitte Vallée. Information theory: Sources, Dirichlet series, and realistic analyses of data structures. In Petr Ambroz, Stepan Holub, and Zuzana Masáková, editors, Proceedings 8th International Conference Words 2011, Prague, Czech Republic, 12-16th September 2011, volume 63 of EPTCS, pages 199-214, 2011. URL: https://doi.org/10.4204/EPTCS.63.26.
  21. Wojtek Szpankowski. Average case analysis of algorithms on sequences. Interscience series in Discrete Mathematics and Optimization. Wiley, 2001. Google Scholar
  22. Brigitte Vallée. Dynamical Sources in Information Theory: Fundamental Intervals and Word Prefixes. Algorithmica, 29(1):262-306, 2001. URL: https://doi.org/10.1007/BF02679622.
  23. Brigitte Vallée. The Depoissonisation Quintet: Rice-Poisson-Mellin-Newton-Laplace. In James Allen Fill and Mark Daniel Ward, editors, 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms, AofA 2018, June 25-29, 2018, Uppsala, Sweden, volume 110 of LIPIcs, pages 35:1-35:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.AofA.2018.35.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail