A limit field for orthogonal range searches in two-dimensional random point search trees

TL;DR

This paper analyzes the asymptotic behavior of orthogonal range query costs in random quadtrees and 2-d trees, showing convergence to a unique random field and resolving longstanding questions about worst-case versus typical query costs.

Contribution

It introduces a limit random field for orthogonal range search costs in 2D random trees, providing a unified asymptotic analysis for typical and worst-case queries.

Findings

Cost functions converge to a unique random field.

Worst-case query costs match typical query asymptotics.

Results apply to both quadtrees and 2-d trees.

Abstract

We consider the cost of general orthogonal range queries in random quadtrees. The cost of a given query is encoded into a (random) function of four variables which characterize the coordinates of two opposite corners of the query rectangle. We prove that, when suitably shifted and rescaled, the random cost function converges uniformly in probability towards a random field that is characterized as the unique solution to a distributional fixed-point equation. We also state similar results for $2$-d trees. Our results imply for instance that the worst case query satisfies the same asymptotic estimates as a typical query, and thereby resolve an old question of Chanzy, Devroye and Zamora-Cura [\emph{Acta Inf.}, 37:355--383, 2000]