Partial match queries in random quadtrees

Abstract : We consider the problem of recovering items matching a partially specified pattern in multidimensional trees (quad trees 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 an independent and uniformly on $[0,1]$ random query $\xi$ satisfies $\Ec{C_n(\xi)}\sim \kappa n^{\beta}$, where $\kappa$ and $\beta$ are explicit constants. We develop an approach based on the analysis of the cost $C_n(x)$ of any fixed query $x\in [0,1]$, and give precise estimates for the variance and limit distribution of the cost $C_n(x)$. Our results permit to describe a limit process for the costs $C_n(x)$ as $x$ varies in $[0,1]$; one of the consequences is that $E{\max_{x\in [0,1]} C_n(x)} \sim \gamma n^\beta$.
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Contributor : Nicolas Broutin <>
Submitted on : Sunday, January 13, 2013 - 4:05:32 PM
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Nicolas Broutin, Ralph Neininger, Henning Sulzbach. Partial match queries in random quadtrees. SoDA 2012 - Symposium on Discrete Algorithms, ACM; SIAM, Jan 2012, Kyoto, Japan. pp.1056-1065, ⟨10.1137/1.9781611973099.83⟩. ⟨hal-00773365⟩



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