Document Open Access Logo

Inversions in Split Trees and Conditional Galton-Watson Trees

Authors Xing Shi Cai , Cecilia Holmgren , Svante Janson , Tony Johansson, Fiona Skerman

PDF
Thumbnail PDF

File

LIPIcs.AofA.2018.15.pdf
  • Filesize: 486 kB
  • 12 pages

Document Identifiers

Author Details

Xing Shi Cai
  • Department of Mathematics, Uppsala University, Uppsala, Sweden
Cecilia Holmgren
  • Department of Mathematics, Uppsala University, Uppsala, Sweden
Svante Janson
  • Department of Mathematics, Uppsala University, Uppsala, Sweden
Tony Johansson
  • Department of Mathematics, Uppsala University, Uppsala, Sweden
Fiona Skerman
  • Department of Mathematics, Uppsala University, Uppsala, Sweden

Funding

This work was partially supported by the Knut and Alice Wallenberg Foundation, the Swedish Research Council and the Ragnar Söderbergs Foundation.

Cite AsGet BibTex

Xing Shi Cai, Cecilia Holmgren, Svante Janson, Tony Johansson, and Fiona Skerman. Inversions in Split Trees and Conditional Galton-Watson Trees. In 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 110, pp. 15:1-15:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.AofA.2018.15

Abstract

We study I(T), the number of inversions in a tree T with its vertices labeled uniformly at random. We first show that the cumulants of I(T) have explicit formulas. Then we consider X_n, the normalized version of I(T_n), for a sequence of trees T_n. For fixed T_n's, we prove a sufficient condition for X_n to converge in distribution. For T_n being split trees [Devroye, 1999], we show that X_n converges to the unique solution of a distributional equation. Finally, when T_n's are conditional Galton-Watson trees, we show that X_n converges to a random variable defined in terms of Brownian excursions. Our results generalize and extend previous work by Panholzer and Seitz [Panholzer and Seitz, 2012].

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Probabilistic algorithms
Keywords
  • inversions
  • random trees
  • split trees
  • Galton-Watson trees
  • permutation
  • cumulant

Metrics

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

References

  1. David Aldous. The continuum random tree. II. An overview. In Stochastic analysis (Durham, 1990), volume 167 of London Math. Soc. Lecture Note Ser., pages 23-70. Cambridge Univ. Press, Cambridge, 1991. Google Scholar
  2. David Aldous. The continuum random tree. III. Ann. Probab., 21(1):248-289, 1993. Google Scholar
  3. Nicolas Broutin and Cecilia Holmgren. The total path length of split trees. Ann. Appl. Probab., 22(5):1745-1777, 2012. Google Scholar
  4. Xing Shi Cai, C. Holmgren, S. Janson, T. Johansson, and F. Skerman. Inversions in split trees and conditional Galton-Watson trees. ArXiv e-prints, 2017. URL: http://arxiv.org/abs/1709.00216.
  5. G. Cramer. Introduction à l'analyse des lignes courbes algébriques. Cramer, 1750. Google Scholar
  6. Luc Devroye. Universal limit laws for depths in random trees. SIAM J. Comput., 28(2):409-432, 1999. Google Scholar
  7. William Feller. An introduction to probability theory and its applications. Vol. I. John Wiley &Sons, Inc., New York-London-Sydney, 3rd edition, 1968. Google Scholar
  8. P. Flajolet, P. Poblete, and A. Viola. On the analysis of linear probing hashing. Algorithmica, 22(4):490-515, 1998. Google Scholar
  9. Ira M. Gessel, Bruce E. Sagan, and Yeong Nan Yeh. Enumeration of trees by inversions. J. Graph Theory, 19(4):435-459, 1995. Google Scholar
  10. Bernhard Gittenberger. A note on: "State spaces of the snake and its tour - convergence of the discrete snake". J. Theoret. Probab., 16(4):1063-1067 (2004), 2003. Google Scholar
  11. Svante Janson. The Wiener index of simply generated random trees. Random Structures Algorithms, 22(4):337-358, 2003. Google Scholar
  12. Svante Janson. Simply generated trees, conditioned Galton-Watson trees, random allocations and condensation: extended abstract. In 23rd Intern. Meeting on Probabilistic, Combinatorial, and Asymptotic Methods for the Analysis of Algorithms (AofA'12), Discrete Math. Theor. Comput. Sci. Proc., AQ, pages 479-490. Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2012. Google Scholar
  13. Svante Janson and Philippe Chassaing. The center of mass of the ISE and the Wiener index of trees. Electron. Comm. Probab., 9:178-187, 2004. Google Scholar
  14. Svante Janson and Jean-François Marckert. Convergence of discrete snakes. J. Theoret. Probab., 18(3):615-647, 2005. Google Scholar
  15. Donald E. Knuth. The art of computer programming. Vol. 3. Addison-Wesley, Reading, MA, 1998. Google Scholar
  16. C. L. Mallows and John Riordan. The inversion enumerator for labeled trees. Bull. Amer. Math. Soc., 74:92-94, 1968. Google Scholar
  17. Barbara H. Margolius. Permutations with inversions. J. Integer Seq., 4(2):Article 01.2.4, 13, 2001. Google Scholar
  18. Ralph Neininger. On a multivariate contraction method for random recursive structures with applications to Quicksort. Random Structures Algorithms, 19(3-4):498-524, 2001. Analysis of algorithms (Krynica Morska, 2000). Google Scholar
  19. Ralph Neininger and Ludger Rüschendorf. On the internal path length of d-dimensional quad trees. Random Structures Algorithms, 15(1):25-41, 1999. Google Scholar
  20. Alois Panholzer and Georg Seitz. Limiting distributions for the number of inversions in labelled tree families. Ann. Comb., 16(4):847-870, 2012. Google Scholar
  21. Uwe Rösler. A limit theorem for "Quicksort". RAIRO Inform. Théor. Appl., 25(1):85-100, 1991. Google Scholar
  22. Vladimir N. Sachkov. Probabilistic methods in combinatorial analysis, volume 56 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1997. Translated from the Russian, Revised by the author. 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