Triangulating the Square and Squaring the Triangle: Quadtrees and Delaunay Triangulations are Equivalent

TL;DR

This paper proves the equivalence between Delaunay triangulations and compressed quadtrees, providing linear-time algorithms for converting between them and applying this to improve various Delaunay-related computations.

Contribution

It introduces two linear-time algorithms for converting between Delaunay triangulations and compressed quadtrees, extending previous work and enabling faster Delaunay computations.

Findings

Algorithms run in deterministic linear time

Equivalence enables new applications in Delaunay triangulation processing

Linear-time computation of Euclidean minimum spanning trees from quadtrees

Abstract

We show that Delaunay triangulations and compressed quadtrees are equivalent structures. More precisely, we give two algorithms: the first computes a compressed quadtree for a planar point set, given the Delaunay triangulation; the second finds the Delaunay triangulation, given a compressed quadtree. Both algorithms run in deterministic linear time on a pointer machine. Our work builds on and extends previous results by Krznaric and Levcopolous and Buchin and Mulzer. Our main tool for the second algorithm is the well-separated pair decomposition(WSPD), a structure that has been used previously to find Euclidean minimum spanning trees in higher dimensions (Eppstein). We show that knowing the WSPD (and a quadtree) suffices to compute a planar Euclidean minimum spanning tree (EMST) in linear time. With the EMST at hand, we can find the Delaunay triangulation in linear time. As a corollary, we obtain deterministic versions of many previous algorithms related to Delaunay triangulations, such as splitting planar Delaunay triangulations, preprocessing imprecise points for faster Delaunay computation, and transdichotomous Delaunay triangulations.