The Presort Hierarchy for Geometric Problems

TL;DR

The paper proposes a Presort Hierarchy that classifies planar geometric problems by their sensitivity to presorting along axes, defining 1- and 2-Presortable versus Presort-Hard. It resolves a long-standing open problem by showing quadtrees (and by reductions Delaunay/Voronoi diagrams and Euclidean MSTs) are 2-Presortable with a randomized algorithm running in $O(n \sqrt{\log n})$ time, leveraging a 2-presorted input and orthogonal range successor data structures. It further places several other problems in the hierarchy (e.g., KD-trees, orthogonal segment intersection, maximum empty circle) as 1- or 2-Presortable or Presort-Hard, and proves Presort-Hardness for onion layers and decremental closest-pair problems, delineating fundamental limits of presorting. Together, these results map the landscape of proximity structures under presorted inputs and offer avenues for faster geometric computation in presorted regimes, while highlighting key derandomisation and dimensional-extension challenges.

Abstract

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.

Figures (4)

• Figure 1: Depiction of type i. and type ii. quadtree splits. (b) Our variable names.
• Figure 2: Visualization of our main algorithm. The auxiliary data structures $D(R_{P_i})$ for $i = 0,1,2,\dots$ are used to implicitly represent a skip list.
• Figure 3: The construction for computing some triangulation, given $A_x$. We take the union of the convex hull (left) and an $x$-monotone polygonal chain through all the vertices (middle) to arrive at a connected plane graph whose vertices are exactly $P$ (right).
• Figure 4: Example of a family of point sets that admits $\Omega((n/4)!)$ different onion layer decompositions, but has a fixed $x$- and $y$-sorted order.

Theorems & Definitions (17)

• Theorem 1: Buchin and Mulzer BuchinMulzer2011Delaunay
• Definition 2: Figure \ref{['fig:quadtree-split']}
• Theorem 3
• Lemma 4
• Theorem 5
• Corollary 6: Equivalence from Loffler2012Triangulating
• Theorem 6
• Theorem 7
• Theorem 8
• Theorem 9