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)] ∼ κn , 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 càdlàg 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 subject classifications. Primary 60F17, 68Q25; secondary 60C05, 60G18.