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December 2013 A limit process for partial match queries in random quadtrees and $2$-d trees
Nicolas Broutin, Ralph Neininger, Henning Sulzbach
Ann. Appl. Probab. 23(6): 2560-2603 (December 2013). DOI: 10.1214/12-AAP912


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].


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Nicolas Broutin. Ralph Neininger. Henning Sulzbach. "A limit process for partial match queries in random quadtrees and $2$-d trees." Ann. Appl. Probab. 23 (6) 2560 - 2603, December 2013.


Published: December 2013
First available in Project Euclid: 22 October 2013

zbMATH: 1358.68080
MathSciNet: MR3127945
Digital Object Identifier: 10.1214/12-AAP912

Primary: 05C05, 60C05, 60F17
Secondary: 05A15, 05A16, 11Y16

Rights: Copyright © 2013 Institute of Mathematical Statistics


Vol.23 • No. 6 • December 2013
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