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.

翻译：我们重新审视Hopcroft的问题和与几何范围搜索相关的根本问题。 鉴于美元点数和飞机上的线条, 我们展示了如何用美元时间来计算点线事件对数或点线对数的数值, 以美元时间来计算点线事件对数或点线对数的数值, 与假设的较低约束值相对应, 并改进了以前由Matou\4/3}2 ⁇ O( log ⁇ n)} 近30年前由Matou\v{s}ek获得的最佳时间约束值。 我们描述了实现结果的两个有趣的不同方式: 第一个是随机化的, 并使用2D版的分解层卡片卡对数来安排线; 第二个是确定性, 以Fredman(1976年) 的排序技术所启发的方式使用决定性树。 第二个方法延伸到任何恒定的层面。 这些新想法产生了许多后果: 例如, 我们获得了一个$O(n_4/3} 美元时间算法, 在飞机上进行线段交叉计算, $O_4/3} 和在最接近的双方平面的轨道上, 4个时间随机算算和最接近的O 4 4美元。