Document Open Access Logo

Patricia’s Bad Distributions

Authors Louigi Addario-Berry , Pat Morin , Ralph Neininger

PDF
Thumbnail PDF

File

LIPIcs.AofA.2024.25.pdf
  • Filesize: 0.66 MB
  • 8 pages

Document Identifiers

Author Details

Louigi Addario-Berry
  • Department of Mathematics and Statistics, McGill University, Montréal, Canada
Pat Morin
  • School of Computer Science, Carleton University, Ottawa, Canada
Ralph Neininger
  • Institute of Mathematics, Goethe University Frankfurt, Germany

Acknowledgements

This research was mainly done during the Sixteenth Annual Workshop on Probability and Combinatorics at McGill University’s Bellairs Institute in Holetown, Barbados. We thank Bellairs Institute for its hospitality and support. We also thank the referees and Jasmin Straub for comments on a draft of this note.

Cite As Get BibTex

Louigi Addario-Berry, Pat Morin, and Ralph Neininger. Patricia’s Bad Distributions. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 25:1-25:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.AofA.2024.25

Abstract

The height of a random PATRICIA tree built from independent, identically distributed infinite binary strings with arbitrary diffuse probability distribution μ on {0,1}^ℕ is studied. We show that the expected height grows asymptotically sublinearly in the number of leaves for any such μ, but can be made to exceed any specific sublinear growth rate by choosing μ appropriately.

Subject Classification

ACM Subject Classification
  • Theory of computation → Sorting and searching
Keywords
  • PATRICIA tree
  • random tree
  • height of tree
  • analysis of algorithms

Metrics

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

Related Versions

References

  1. Ali Akhavi, Fréderic Paccaut, and Brigitte Vallée. Building Sources of Zero Entropy: Rescaling and Inserting Delays. In Mark Daniel Ward, editor, 33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022), volume 225 of Leibniz International Proceedings in Informatics (LIPIcs), pages 1:1-1:28, Dagstuhl, Germany, 2022. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.AofA.2022.1.
  2. Stéphane Boucheron, Gábor Lugosi, and Pascal Massart. A sharp concentration inequality with applications. Random Structures Algorithms, 16(3):277-292, 2000. URL: https://doi.org/10.1002/(SICI)1098-2418(200005)16:3<277::AID-RSA4>3.0.CO;2-1.
  3. Stéphane Boucheron, Gábor Lugosi, and Pascal Massart. Concentration inequalities. Oxford University Press, Oxford, 2013. A nonasymptotic theory of independence, With a foreword by Michel Ledoux. URL: https://doi.org/10.1093/acprof:oso/9780199535255.001.0001.
  4. Julien Clément, Philippe Flajolet, and Brigitte Vallée. Dynamical sources in information theory: a general analysis of trie structures. Algorithmica, 29(1-2):307-369, 2001. Average-case analysis of algorithms (Princeton, NJ, 1998). URL: https://doi.org/10.1007/BF02679623.
  5. Luc Devroye. A note on the probabilistic analysis of patricia trees. Random Structures & Algorithms, 3(2):203-214, 1992. URL: https://doi.org/10.1002/rsa.3240030209.
  6. Luc Devroye. A study of trie-like structures under the density model. Ann. Appl. Probab., 2(2):402-434, 1992. URL: http://links.jstor.org/sici?sici=1050-5164(199205)2:2<402:ASOTSU>2.0.CO;2-F&origin=MSN.
  7. Luc Devroye. Universal limit laws for depths in random trees. SIAM Journal on Computing, 28(2):409-432, 1998. URL: https://doi.org/10.1137/S0097539795283954.
  8. Luc Devroye. Universal asymptotics for random tries and PATRICIA trees. Algorithmica, 42(1):11-29, 2005. URL: https://doi.org/10.1007/s00453-004-1137-7.
  9. Steven N. Evans and Anton Wakolbinger. Radix sort trees in the large. Electronic Communications in Probability, 22(none):1-13, 2017. URL: https://doi.org/10.1214/17-ECP77.
  10. Steven N. Evans and Anton Wakolbinger. PATRICIA bridges. In Genealogies of interacting particle systems, volume 38 of Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., pages 231-265. World Sci. Publ., Hackensack, NJ, 2020. Google Scholar
  11. Michael Fuchs, Hsien-Kuei Hwang, and Vytas Zacharovas. An analytic approach to the asymptotic variance of trie statistics and related structures. ArXiv, abs/1303.4244, 2013. URL: https://api.semanticscholar.org/CorpusID:14429040.
  12. Gernot Gwehenberger. Anwendung einer binären Verweiskettenmethode beim Aufbau von Listen. Elektronische Rechenanlagen, 10:223-226, 1968. Google Scholar
  13. Jasper Ischebeck. Central limit theorems for fringe trees in patricia tries. Preprint arXiv:2305.14900 [math.PR], 2023. URL: https://doi.org/10.48550/arXiv.2305.14900.
  14. Svante Janson. Renewal theory in the analysis of tries and strings. Theor. Comput. Sci., 416:33-54, January 2012. URL: https://doi.org/10.1016/j.tcs.2011.10.007.
  15. Svante Janson. Central limit theorems for additive functionals and fringe trees in tries. Electronic Journal of Probability, 27(none):1-63, 2022. URL: https://doi.org/10.1214/22-EJP776.
  16. Charles Knessl and Wojciech Szpankowski. Limit laws for the height in PATRICIA tries. J. Algorithms, 44(1):63-97, 2002. Analysis of algorithms. URL: https://doi.org/10.1016/S0196-6774(02)00212-2.
  17. Donald E. Knuth. The art of computer programming. Volume 3. Addison-Wesley Series in Computer Science and Information Processing. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1973. Sorting and searching. Google Scholar
  18. Donald E. Knuth. The art of computer programming. Vol. 3. Addison-Wesley, Reading, MA, 1998. Sorting and searching, Second edition [of MR0445948]. Google Scholar
  19. Kevin Leckey, Ralph Neininger, and Wojciech Szpankowski. Towards More Realistic Probabilistic Models for Data Structures: The External Path Length in Tries under the Markov Model, pages 877-886. Society for Industrial and Applied Mathematics, 2013. URL: https://doi.org/10.1137/1.9781611973105.63.
  20. Hosam M. Mahmoud. Evolution of random search trees. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons, Inc., New York, 1992. A Wiley-Interscience Publication. Google Scholar
  21. Donald R. Morrison. Patricia—practical algorithm to retrieve information coded in alphanumeric. Journal of the ACM, 15:514-534, 1968. Google Scholar
  22. Masashi Okamoto. Some inequalities relating to the partial sum of binomial probabilities. Ann. Inst. Statist. Math., 10:29-35, 1958. URL: https://doi.org/10.1007/BF02883985.
  23. B. Pittel. Asymptotical growth of a class of random trees. Ann. Probab., 13(2):414-427, 1985. URL: https://doi.org/10.1214/aop/1176993000.
  24. Boris Pittel. Paths in a random digital tree: limiting distributions. Adv. in Appl. Probab., 18(1):139-155, 1986. URL: https://doi.org/10.2307/1427240.
  25. Wojciech Szpankowski. Average case analysis of algorithms on sequences. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley-Interscience, New York, 2001. With a foreword by Philippe Flajolet. URL: https://doi.org/10.1002/9781118032770.
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
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact