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

Abstract : We consider the problem of recovering items matching a partially specified pattern in multidimensional trees (quadtrees and k-d trees). We assume the classical 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 complexity, measured as 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 $E{C_n(\xi)}\sim \kappa n^{\beta}$, 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. Moreover, a functional limit law for a rescaled version of the process $(C_n(s))_{0\le s\le 1}$ is derived in the space of c\'{a}dl\'{a}g functions with the Skorokhod topology. For the worst case complexity $\max_{s\in [0,1]} C_n(s)$ the order of the expectation as well as a limit law are given.
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https://hal.inria.fr/hal-00773363
Contributor : Nicolas Broutin <>
Submitted on : Sunday, January 13, 2013 - 4:03:47 PM
Last modification on : Monday, February 18, 2019 - 7:52:04 PM

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Nicolas Broutin, Ralph Neininger, Henning Sulzbach. A limit process for partial match queries in random quadtrees and 2-d trees. Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2013, 23 (6), pp.2560-2603. ⟨10.1214/12-AAP912⟩. ⟨hal-00773363⟩

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