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 C-n(xi) to visit in order to report the items matching a random query xi, independent and uniformly distributed on [0, 1], satisfies E[C-n(xi)] similar to kappa n(beta), where kappa and beta are explicit constants. We develop an approach based on the analysis of the cost C-n(s) of any fixed query s is an element of [0, 1], and give precise estimates for the variance and limit distribution of the cost C-n(x). Our results permit us 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 is an element of[0,1]) C-n(x)] similar to gamma n(beta); this settles a question of Devroye [Pers. Comm., 2000].