The Presort Hierarchy for Geometric Problems

2026-02-09

AI Summary

This study investigates which classical computational geometry problems can surpass the $O(n \log n)$ time lower bound when input points are pre-sorted along one- or two-dimensional coordinate axes. To this end, the paper introduces the “Presort Hierarchy” framework, offering the first systematic characterization of how presorted information influences the complexity of geometric problems. Leveraging randomized algorithms and complexity reductions, the authors establish that constructing quadtrees, Voronoi diagrams, Delaunay triangulations, and Euclidean minimum spanning trees belongs to the 2-Presortable class. They present expected-time algorithms with complexity $O(n\sqrt{\log n})$, substantially improving upon the traditional lower bound and resolving a long-standing open problem dating back to 1989.

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• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

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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.

Problem

• computational geometry
• Voronoi diagram
• presorting
• time complexity
• lower bound

Innovation

• Presort Hierarchy
• 2-Presortable
• Voronoi diagram
• computational geometry
• sub-n log n algorithms