Hopcroft’s Problem, Log-Star Shaving, 2D Fractional Cascading, and Decision Trees

Abstract

We revisit Hopcroft’s problem and related fundamental problems about geometric range searching. Givennpoints andnlines in the plane, we show how to count the number of point-line incidence pairs or the number of point-above-line pairs inO(n4/3) time, which matches the conjectured lower bound and improves the best previous time bound of\(n^{4/3}2^{O(\log ^*n)} \)obtained almost 30 years ago by Matoušek.We describe two interesting and different ways to achieve the result: the first is randomized and uses a new 2D version of fractional cascading for arrangements of lines; the second is deterministic and uses decision trees in a manner inspired by the sorting technique of Fredman (1976). The second approach extends to any constant dimension.Many consequences follow from these new ideas: for example, we obtain anO(n4/3)-time algorithm for line segment intersection counting in the plane,O(n4/3)-time randomized algorithms for distance selection in the plane and bichromatic closest pair and Euclidean minimum spanning tree in three or four dimensions, and a randomized data structure for halfplane range counting in the plane withO(n4/3) preprocessing time and space andO(n1/3) query time.