Document Open Access Logo

Largest Clusters for Supercritical Percolation on Split Trees

Authors Gabriel Berzunza , Cecilia Holmgren

PDF
Thumbnail PDF

File

LIPIcs.AofA.2020.6.pdf
  • Filesize: 431 kB
  • 10 pages

Document Identifiers

Author Details

Gabriel Berzunza
  • Department of Mathematics, Uppsala University, Sweden
Cecilia Holmgren
  • Department of Mathematics, Uppsala University, Sweden

Funding

This work is supported by the Knut and Alice Wallenberg Foundation, the Swedish Research Council and The Swedish Foundations' starting grant from Ragnar Söderbergs Foundation.

Cite AsGet BibTex

Gabriel Berzunza and Cecilia Holmgren. Largest Clusters for Supercritical Percolation on Split Trees. In 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 159, pp. 6:1-6:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.AofA.2020.6

Abstract

We consider the model of random trees introduced by Devroye [Devroye, 1999], the so-called random split trees. The model encompasses many important randomized algorithms and data structures. We then perform supercritical Bernoulli bond-percolation on those trees and obtain a precise weak limit theorem for the sizes of the largest clusters. The approach we develop may be useful for studying percolation on other classes of trees with logarithmic height, for instance, we have also studied the case of complete d-regular trees.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Probabilistic algorithms
Keywords
  • Split trees
  • random trees
  • supercritical bond-percolation
  • cluster size
  • Poisson measures

Metrics

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

References

  1. Erich Baur. Percolation on random recursive trees. Random Structures Algorithms, 48(4):655-680, 2016. URL: https://doi.org/10.1002/rsa.20603.
  2. Jean Bertoin. Almost giant clusters for percolation on large trees with logarithmic heights. J. Appl. Probab., 50(3):603-611, 2013. URL: https://doi.org/10.1239/jap/1378401225.
  3. Jean Bertoin. Sizes of the largest clusters for supercritical percolation on random recursive trees. Random Structures Algorithms, 44(1):29-44, 2014. URL: https://doi.org/10.1002/rsa.20448.
  4. Jean Bertoin and Gerónimo Uribe Bravo. Supercritical percolation on large scale-free random trees. Ann. Appl. Probab., 25(1):81-103, 2015. URL: https://doi.org/10.1214/13-AAP988.
  5. Gabriel Berzunza. The existence of a giant cluster for percolation on large crump-mode-jagers trees. To appears in Advances in Applied Probability, arXiv preprint, 2020. URL: http://arxiv.org/abs/1806.10686.
  6. Gabriel Berzunza and Cecilia Holmgren. The asymptotic distribution of cluster sizes for supercritical percolation on random split trees. Preprint, 2020. Google Scholar
  7. Gabriel Berzunza, Xing Shi Cai, and Cecilia Holmgren. The asymptotic non-normality of the giant cluster for percolation on random split trees. arXiv e-prints, February 2019. URL: http://arxiv.org/abs/1902.08109.
  8. Patrick Billingsley. Convergence of probability measures. Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons, Inc., New York, second edition, 1999. A Wiley-Interscience Publication. URL: https://doi.org/10.1002/9780470316962.
  9. Béla Bollobás. Random graphs, volume 73 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, second edition, 2001. URL: https://doi.org/10.1017/CBO9780511814068.
  10. Nicolas Broutin and Cecilia Holmgren. The total path length of split trees. Ann. Appl. Probab., 22(5):1745-1777, 2012. URL: https://doi.org/10.1214/11-AAP812.
  11. Edward G Coffman Jr and James Eve. File structures using hashing functions. Communications of the ACM, 13(7):427-432, 1970. Google Scholar
  12. Luc Devroye. On the expected height of fringe-balanced trees. Acta Inform., 30(5):459-466, 1993. URL: https://doi.org/10.1007/BF01210596.
  13. Luc Devroye. Universal limit laws for depths in random trees. SIAM J. Comput., 28(2):409-432, 1999. URL: https://doi.org/10.1137/S0097539795283954.
  14. Michael Drmota. Random trees. SpringerWienNewYork, Vienna, 2009. An interplay between combinatorics and probability. URL: https://doi.org/10.1007/978-3-211-75357-6.
  15. Rick Durrett. Random graph dynamics, volume 20 of Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2010. Google Scholar
  16. Raphael A. Finkel and Jon Louis Bentley. Quad trees a data structure for retrieval on composite keys. Acta informatica, 4(1):1-9, 1974. Google Scholar
  17. Philippe Flajolet, Mathieu Roux, and Brigitte Vallée. Digital trees and memoryless sources: from arithmetics to analysis. In 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10), Discrete Math. Theor. Comput. Sci. Proc., AM, pages 233-260. Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2010. Google Scholar
  18. C. A. R. Hoare. Quicksort. Comput. J., 5:10-15, 1962. URL: https://doi.org/10.1093/comjnl/5.1.10.
  19. Cecilia Holmgren. Novel characteristic of split trees by use of renewal theory. Electron. J. Probab., 17:no. 5, 27, 2012. URL: https://doi.org/10.1214/EJP.v17-1723.
  20. Alex Iksanov and Martin Möhle. A probabilistic proof of a weak limit law for the number of cuts needed to isolate the root of a random recursive tree. Electron. Comm. Probab., 12:28-35, 2007. URL: https://doi.org/10.1214/ECP.v12-1253.
  21. Hosam M. Mahmoud and Boris Pittel. Analysis of the space of search trees under the random insertion algorithm. J. Algorithms, 10(1):52-75, 1989. URL: https://doi.org/10.1016/0196-6774(89)90023-0.
  22. A. Meir and J. W. Moon. Cutting down random trees. J. Austral. Math. Soc., 11:313-324, 1970. Google Scholar
  23. J. Pitman. Combinatorial stochastic processes, volume 1875 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2006. Lectures from the 32nd Summer School on Probability Theory held in Saint-Flour, July 7-24, 2002, With a foreword by Jean Picard. Google Scholar
  24. Jim Pitman. Coalescent random forests. J. Combin. Theory Ser. A, 85(2):165-193, 1999. URL: https://doi.org/10.1006/jcta.1998.2919.
  25. R. Pyke. Spacings. (With discussion.). J. Roy. Statist. Soc. Ser. B, 27:395-449, 1965. URL: http://links.jstor.org/sici?sici=0035-9246(1965)27:3<395:S>2.0.CO;2-C&origin=MSN.
  26. Sidney I. Resnick. Extreme values, regular variation, and point processes, volume 4 of Applied Probability. A Series of the Applied Probability Trust. Springer-Verlag, New York, 1987. URL: https://doi.org/10.1007/978-0-387-75953-1.
  27. A Walker and Derick Wood. Locally balanced binary trees. The Computer Journal, 19(4):322-325, 1976. Google Scholar
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 Imprint Privacy Contact