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

We consider the problem of recovering items matching a partially specified pattern in multidimensional trees (quadtrees 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(.) to visit in order to report the items matching a random query ., independent and uniformly distributed on [0,1], satisfies E[Cn(.)]..n., where . and . 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 of the cost Cn(x). Our results permit us 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)]..n.; this settles a question of Devroye [Pers. Comm., 2000].