Triangulating the Square and Squaring the Triangle : 1 Quadtrees and Delaunay Triangulations are Equivalent

4 We show that Delaunay triangulations and compressed quadtrees are equivalent structures. More precisely, 5 we give two algorithms: the first computes a compressed quadtree for a planar point set, given the Delaunay 6 triangulation; the second finds the Delaunay triangulation, given a compressed quadtree. Both algorithms 7 run in deterministic linear time on a pointer machine. Our work builds on and extends previous results by 8 Krznaric and Levcopolous [38] and Buchin and Mulzer [9]. Our main tool for the second algorithm is the 9 well-separated pair decomposition (WSPD) [12], a structure that has been used previously to find Euclidean 10 minimum spanning trees in higher dimensions [26]. We show that knowing the WSPD (and a quadtree) 11 suffices to compute a planar Euclidean minimum spanning tree (EMST) in linear time. With the EMST at 12 hand, we can find the Delaunay triangulation in linear time [20]. 13 As a corollary, we obtain deterministic versions of many previous algorithms related to Delaunay trian14 gulations, such as splitting planar Delaunay triangulations [18,19], preprocessing imprecise points for faster 15 Delaunay computation [8, 40], and transdichotomous Delaunay triangulations [9, 14,15]. 16