A limit process for partial match queries in random quadtrees and 2-d trees
Résumé
We consider the problem of recovering items matching a partially specified pattern in multidimensional trees (quadtrees 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 $C_n(\xi)$ to visit in order to report the items matching a random query $\xi$, independent and uniformly distributed on $[0,1]$, satisfies $E{C_n(\xi)}\sim \kappa n^{\beta}$, 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 $s\in [0,1]$, and give precise estimates for the variance and limit distribution. Moreover, a functional limit law for a rescaled version of the process $(C_n(s))_{0\le s\le 1}$ is derived in the space of c\'{a}dl\'{a}g functions with the Skorokhod topology. For the worst case complexity $\max_{s\in [0,1]} C_n(s)$ the order of the expectation as well as a limit law are given.