Optimal Partition Trees

Abstract

We revisit one of the most fundamental classes of data structure problems in computational geometry: range searching. Matoušek (Discrete Comput. Geom. 10:157–182, 1993) gave a partition tree method for d-dimensional simplex range searching achieving O(n) space and O(n 1−1/d ) query time. Although this method is generally believed to be optimal, it is complicated and requires O(n 1+ε ) preprocessing time for any fixed ε>0. An earlier method by Matoušek (Discrete Comput. Geom. 8:315–334, 1992) requires O(nlogn) preprocessing time but O(n 1−1/d log O(1) n) query time. We give a new method that achieves simultaneously O(nlogn) preprocessing time, O(n) space, and O(n 1−1/d ) query time with high probability. Our method has several advantages: Numerous consequences follow (e.g., improved results for computing spanning trees with low crossing number, ray shooting among line segments, intersection searching, exact nearest neighbor search, linear programming queries, finding extreme points, …).