Many fundamental problems in computational geometry admit no algorithm running in $o(n \log n)$ time for $n$ planar input points, via classical reductions from sorting. Prominent examples include the computation of convex hulls, quadtrees, onion layer decompositions, Euclidean minimum spanning trees, KD-trees, Voronoi diagrams, and decremental closest-pair. A classical result shows that, given $n$ points sorted along a single direction, the convex hull can be constructed in linear time. Subsequent works established that for all of the other above problems, this information does not suffice. In 1989, Aggarwal, Guibas, Saxe, and Shor asked: Under which conditions can a Voronoi diagram be computed in $o(n \log n)$ time? Since then, the question of whether sorting along TWO directions enables a $o(n \log n)$-time algorithm for such problems has remained open and has been repeatedly mentioned in the literature. In this paper, we introduce the Presort Hierarchy: A problem is 1-Presortable if, given a sorting along one axis, it permits a (possibly randomised) $o(n \log n)$-time algorithm. It is 2-Presortable if sortings along both axes suffice. It is Presort-Hard otherwise. Our main result is that quadtrees, and by extension Delaunay triangulations, Voronoi diagrams, and Euclidean minimum spanning trees, are 2-Presortable: we present an algorithm with expected running time $O(n \sqrt{\log n})$. This addresses the longstanding open problem posed by Aggarwal, Guibas, Saxe, and Shor (albeit randomised). We complement this result by showing that some of the other above geometric problems are also 2-Presortable or Presort-Hard.

翻译：在计算几何中，许多基本问题通过经典的排序归约，无法在 $o(n \log n)$ 时间内处理 $n$ 个平面输入点。突出的例子包括凸包、四叉树、洋葱层分解、欧几里得最小生成树、KD-树、Voronoi 图以及递减最近对的计算。一个经典结果表明，给定沿单一方向排序的 $n$ 个点，凸包可以在线性时间内构建。后续工作证实，对于上述所有其他问题，仅此信息并不足够。1989年，Aggarwal、Guibas、Saxe 和 Shor 提出：在何种条件下可以在 $o(n \log n)$ 时间内计算 Voronoi 图？自此，沿两个方向排序是否能为此类问题实现 $o(n \log n)$ 时间算法的问题一直悬而未决，并在文献中被反复提及。本文中，我们引入了预排序层次结构：若给定沿一个轴的排序，一个问题允许（可能是随机的）$o(n \log n)$ 时间算法，则称其为 1-可预排序的。若沿两个轴的排序足够，则称其为 2-可预排序的。否则，称其为预排序困难的。我们的主要结果是：四叉树，以及由此延伸的 Delaunay 三角剖分、Voronoi 图和欧几里得最小生成树，是 2-可预排序的：我们提出了一种期望运行时间为 $O(n \sqrt{\log n})$ 的算法。这解决了 Aggarwal、Guibas、Saxe 和 Shor 提出的长期悬而未决的问题（尽管是随机算法）。我们通过证明上述其他一些几何问题也是 2-可预排序的或预排序困难的，来补充这一结果。