Efficient partition trees

Abstract

We prove a theorem on partitioning point sets in Ed (d fixed) and give an efficient construction of partition trees based on it. This yields a simplex range searching structure with linear space, O(n log n) deterministic preprocessing time, and O(n1-1/d(log n)O(1)) query time. With O(nlog n) preprocessing time, where δ is an arbitrary positive constant, a more complicated data structure yields query time O(n1-1/d(log log n)O(1)). This attains the lower bounds due to Chazelle [C1] up to polylogarithmic factors, improving and simplifying previous results of Chazelle et al. [CSW]. The partition result implies that, for rd≤n1-δ, a (1/r)-approximation of size O(rd) with respect to simplices for an n-point set in Ed can be computed in O(n log r) deterministic time. A (1/r)-cutting of size O(rd) for a collection of n hyperplanes in Ed can be computed in O(n log r) deterministic time, provided that r≤n1/(2 d-1). © 1992 Springer-Verlag New York Inc.