Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-25T18:40:19.831Z Has data issue: false hasContentIssue false
  • English
  • Français

Strong Convergence of Partial Match Queries in Random Quadtrees

Published online by Cambridge University Press:  27 April 2012

NICOLAS CURIEN
NICOLAS CURIEN*
Affiliation:
Département de Mathématiques et Applications, École Normale Supérieure, 45 rue d'Ulm, 75230 Paris CEDEX 05, France (e-mail: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that the rescaled costs of partial match queries in a random two-dimensional quadtree converge almost surely towards a random limit which is identified as the terminal value of a martingale. Our approach shares many similarities with the theory of self-similar fragmentations.

Keywords

Primary 60F9960G4460G1868W40
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 21 , Issue 5 , September 2012 , pp. 683 - 694
Copyright
Copyright © Cambridge University Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Bertoin, J. (2006) Random Fragmentations and Coagulation Processes, Vol. 102 of Cambridge Studies in Advanced Mathematics, Cambridge University Press.Google Scholar
[2]Bertoin, J. (2008) Homogeneous multitype fragmentations. In In and Out of Equilibrium 2, Birkhäuser, Progr. Probab. 60 161–183.Google Scholar
[3]Bertoin, J. and Gnedin, A. (2004) Asymptotic laws for nonconservative self-similar fragmentations. Electron. J. Probab. 9 575–593.CrossRefGoogle Scholar
[4]Broutin, N., Neininger, R. and Sulzbach, H. (2012) A limit process for partial match queries in random quadtrees. Submitted. arXiv:abs/1202.1342Google Scholar
[5]Broutin, N., Neininger, R. and Sulzbach, H. (2012) Partial match queries in random quadtrees. In Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA) (Rabani, Y., ed.), pp. 1056–1065. arXiv:abs/1107.2231Google Scholar
[6]Curien, N. and Joseph, A. (2011) Partial match queries in random quadtrees: A probabilistic approach. Adv. Appl. Probab. 43 178–194.CrossRefGoogle Scholar
[7]Curien, N. and Le Gall, J.-F. (2011) Random recursive triangulations of the disk via fragmentation theory. Ann. Probab. 39 2224–2270.CrossRefGoogle Scholar
[8]Finkel, R. A. and Bentley, J. L. (1974) Quad trees: A data structure for retrieval on composite keys. Acta Informatica 4 1–9.CrossRefGoogle Scholar
[9]Flajolet, P., Gonnet, G., Puech, C. and Robson, J. M. (1993) Analytic variations on quadtrees. Algorithmica 10 473–500.CrossRefGoogle Scholar
[10]Flajolet, P. and Sedgewick, R. (2009) Analytic Combinatorics, Cambridge University Press.CrossRefGoogle Scholar