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. Given n points and n lines in the plane, we show how to count the number of point-line incidence pairs or the number of point-above-line pairs in O(n 4/3 ) 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 . The second approach extends to any constant dimension.Many consequences follow from these new ideas: for example, we obtain an O(n 4/3 )-time algorithm for line segment intersection counting in the plane, O(n 4/3 )-time randomized algorithms for 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 with O(n 4/3 ) preprocessing time and space and O(n 1/3 ) query time.

Conditional Lower Bounds for Dynamic Geometric Measure Problems

“…The most popular of these conjectures concern the aforementioned 3SUM problem, All-Pairs-Shortest-Paths (APSP), Boolean Matrix Multiplication (BMM), finding a triangle in a graph, Boolean Satisfiability (SAT) and the Orthogonal Vectors problem (2OV) (see for example the introductory surveys by Bringmann [14] and V. V. Williams [54]). Another problem which crops up as a bottleneck in computational geometry is Hopcroft's problem (see the recent preprint by Chan and Zheng [23]).…”