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CORR
2012
Springer

A limit process for partial match queries in random quadtrees

7 years 5 months ago
A limit process for partial match queries in random quadtrees
We consider the problem of recovering items matching a partially specified pattern in multidimensional trees (quad trees 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 Cn(ξ) to visit in order to report the items matching a random query ξ, independent and uniformly distributed on [0, 1], satisfies E[Cn(ξ)] ∼ κnβ , where κ and β are explicit constants. We develop an approach based on the analysis of the cost Cn(s) of any fixed query s ∈ [0, 1], and give precise estimates for the variance and limit distribution. Moreover, a functional limit law for a rescaled version of the process (Cn(s))0≤s≤1 is derived in the space of c`adl`ag functions with the Skorokhod topology. For the worst case complexity maxs∈[0,1] Cn(s) the order of the expectation as well as a limit law are given. AMS 2010 s...
Nicolas Broutin, Ralph Neininger, Henning Sulzbach
Added 20 Apr 2012
Updated 20 Apr 2012
Type Journal
Year 2012
Where CORR
Authors Nicolas Broutin, Ralph Neininger, Henning Sulzbach
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