The Annals of Applied Probability

A limit process for partial match queries in random quadtrees and $2$-d trees

Nicolas Broutin, Ralph Neininger, and Henning Sulzbach

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We consider the problem of recovering items matching a partially specified pattern in multidimensional trees (quadtrees and $k$-d trees). We assume the traditional model where the data consist of independent and uniform points in the unit square. For this model, in a structure on $n$ points, it is known that the number of nodes $C_{n}(\xi)$ to visit in order to report the items matching a random query $\xi$, independent and uniformly distributed on $[0,1]$, satisfies $\mathbf{E} [{C_{n}(\xi)}]\sim\kappa n^{\beta}$, where $\kappa$ and $\beta$ are explicit constants. We develop an approach based on the analysis of the cost $C_{n}(s)$ of any fixed query $s\in[0,1]$, and give precise estimates for the variance and limit distribution of the cost $C_{n}(x)$. Our results permit us to describe a limit process for the costs $C_{n}(x)$ as $x$ varies in $[0,1]$; one of the consequences is that $\mathbf{E} [{\max_{x\in[0,1]}C_{n}(x)}]\sim\gamma n^{\beta}$; this settles a question of Devroye [Pers. Comm., 2000].

Article information

Ann. Appl. Probab., Volume 23, Number 6 (2013), 2560-2603.

First available in Project Euclid: 22 October 2013

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Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability 60F17: Functional limit theorems; invariance principles 05C05: Trees
Secondary: 11Y16: Algorithms; complexity [See also 68Q25] 05A15: Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx] 05A16: Asymptotic enumeration

Analysis of algorithms quadtree limit distribution contraction method


Broutin, Nicolas; Neininger, Ralph; Sulzbach, Henning. A limit process for partial match queries in random quadtrees and $2$-d trees. Ann. Appl. Probab. 23 (2013), no. 6, 2560--2603. doi:10.1214/12-AAP912.

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