Hostname: page-component-74656f9884-8r8nv Total loading time: 0 Render date: 2026-11-10T12:20:44.376Z Has data issue: false hasContentIssue false
  • English
  • Français

Partial match queries in two-dimensional quadtrees: a probabilistic approach

Part of: Limit theorems Markov processes Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Nicolas Curien  and
Adrien Joseph
Nicolas Curien*
Affiliation:
École Normale Supérieure
Adrien Joseph*
Affiliation:
Université Pierre et Marie Curie
*
∗ Postal address: Département de Mathématiques et Applications, École Normale Supérieure, 45 rue d'Ulm, 75005 Paris, France. Email address: nicolas.curien@ens.fr
∗∗ Postal address: Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie, 4 place Jussieu Tour 16-26, 75005 Paris, France. Email address: adrien.joseph@upmc.fr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We analyze the mean cost of the partial match queries in random two-dimensional quadtrees. The method is based on fragmentation theory. The convergence is guaranteed by a coupling argument of Markov chains, whereas the value of the limit is computed as the fixed point of an integral equation.

Keywords

Quadtreepartial match queryfragmentation theoryMarkov chaincouplingintegral equation

MSC classification

Primary: 60F99: None of the above, but in this section
Secondary: 60G18: Self-similar processes 60J05: Discrete-time Markov processes on general state spaces

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 43 , Issue 1 , March 2011 , pp. 178 - 194
Copyright
Copyright © Applied Probability Trust 2011 

References

Bertoin, J. (2006). Random Fragmentations and Coagulation Processes (Camb. Stud. Adv. Math. 102). Cambridge University Press.CrossRefGoogle Scholar
Bertoin, J. and Gnedin, A. V. (2004). Asymptotic laws for nonconservative self-similar fragmentations. Electron. J. Prob. 9, 575–593.CrossRefGoogle Scholar
Chern, H.-H. and Hwang, H.-K. (2003). Partial match queries in random quadtrees. SIAM J. Comput. 32, 904–915.CrossRefGoogle Scholar
Curien, N. and Le Gall, J.-F. (2011). Random recursive triangulations of the disk via fragmentation theory. To appear in Ann. Prob. CrossRefGoogle Scholar
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
Flajolet, P. and Sedgewick, R. (2009). Analytic Combinatorics. Cambridge University Press.CrossRefGoogle Scholar
Flajolet, P., Gonnet, G., Puech, C. and Robson, J. M. (1993). Analytic variations on quadtrees. Algorithmica 10, 473–500.CrossRefGoogle Scholar
Flajolet, P., Labelle, G., Laforest, L. and Salvy, B. (1995). Hypergeometrics and the cost structure of quadtrees. Random Structures Algorithms 7, 117–144.CrossRefGoogle Scholar
Iserles, A. and Liu, Y. (1997). Integro-differential equations and generalized hypergeometric functions. J. Math. Anal. Appl. 208, 404–424.CrossRefGoogle Scholar
Liu, Q. (1997). Sur une équation fonctionnelle et ses applications: une extension du théorème de Kesten-Stigum concernant des processus de branchement. Adv. Appl. Prob. 29, 353–373.CrossRefGoogle Scholar
Liu, Q. (2001). Asymptotic properties and absolute continuity of laws stable by random weighted mean. Stoch. Process. Appl. 95, 83–107.CrossRefGoogle Scholar
Meyn, S. and Tweedie, R. L. (2009). Markov Chains and Stochastic Stability, 2nd edn. Cambridge University Press.CrossRefGoogle Scholar