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

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 [40] and Buchin and Mulzer [10]. Our main tool for the second algorithm is the well-separated pair decomposition (WSPD) [13], a structure that has been used previously to find Euclidean minimum spanning trees in higher dimensions [27]. We show that knowing the WSPD (and a quadtree) suffices to compute a planar EMST in linear time. With the EMST at hand, we can find the Delaunay triangulation in linear time [21]. As a corollary, we obtain deterministic versions of many previous algorithms related to Delaunay triangulations, such as splitting planar Delaunay triangulations [19, 20], preprocessing imprecise points for faster Delaunay computation [9, 42], and transdichotomous Delaunay triangulations [10, 15, 16].