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 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 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 with O(n^4/3) preprocessing time and space and O(n^1/3) query time.

我们重新审视了 Hopcroft 问题和有关几何范围搜索的相关基本问题。给定平面中的n 个点和n条线，我们展示了如何在O ( n ^4/3 ) 时间内计算点-线重合对的数量或点-线上方对的数量，这与推测的下界和改进了 Matoušek 近 30 年前获得的 \(n^{4/3}2^{O(\log ^*n)} \) 的最佳先前时间界限。

我们描述了两种有趣且不同的方法来获得结果：第一种是随机的，并使用新的 2D 版本的分数级联来排列线条；第二种是确定性的，并以受 Fredman (1976) 排序技术启发的方式使用决策树。第二种方法扩展到任何常数维度。

这些新想法产生了许多结果：例如，我们获得了一个O ( n ^4/3 ) 时间的平面内线段相交计数算法，O ( n ^4/3 ) 时间的平面内距离选择随机算法三、四维双色最近对和欧几里得最小生成树，以及O ( n ^4/3 )预处理时间和空间、O ( n ^1/3 )查询时间的平面内半平面范围计数的随机数据结构。