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

TL;DR

This paper improves the computational complexity of fundamental geometric problems like point-line incidences and range searching to match conjectured lower bounds, introducing new randomized and deterministic methods inspired by fractional cascading and decision trees.

Contribution

It presents the first $O(n^{4/3})$-time algorithms for key geometric problems, using novel randomized and deterministic techniques that extend to higher dimensions.

Findings

Achieved $O(n^{4/3})$ time for counting point-line incidences.

Developed a randomized algorithm for line segment intersection counting.

Created a data structure with $O(n^{4/3})$ preprocessing and $O(n^{1/3})$ query time for halfplane range counting.

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 Matou\v{s}ek. 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 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.