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(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)]∼κnβ, 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 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)]∼γnβ; this settles a question of
Devroye [Pers. Comm., 2000].Comment: Published in at http://dx.doi.org/10.1214/12-AAP912 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org). arXiv admin note: text
overlap with arXiv:1107.223