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 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 Cn(ξ) to visit in order to report the items matching an independent and uniformly on [0, 1] random query ξ satisfies E[Cn(ξ)] ∼ κnβ , where κ and β are explicit constants. We develop an approach based on the analysis of the cost Cn(x) of any fixed query x ∈ [0, 1], and give precise estimates for the variance and limit distribution of the cost Cn(x). Our results permit to describe a limit process for the costs Cn(x) as x varies in [0, 1]; one of the consequences is that E[maxx∈[0,1] Cn(x)] ∼ γnβ ; this settles a question of Devroye [Pers. Comm., 2000].

• Type: Journal
• Year: 2012
• Where: SODA
• Authors: Nicolas Broutin, Ralph Neininger, Henning Sulzbach
• Keywords: Algorithms, Explicit Constants, Pers Comm, Precise Estimates, SODA 2012