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

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Löffler, Maarten Mulzer, Wolfgang
Copyright Year	2012
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. 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.
Starting Page	941
Ending Page	974
Page Count	34
File Format	PDF
ISSN	00975397
DOI	10.1137/110825698
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	4
Volume Number	41
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2012-08-23
Access Restriction	Subscribed
Subject Keyword	Analysis of algorithms and problem complexity well-separated pair decomposition Delaunay triangulation Analysis of algorithms quadtree
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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