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Data structures and network algorithms
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Discrete & Computational Geometry
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Discrete & Computational Geometry
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Nordic Journal of Computing
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SFCS '75 Proceedings of the 16th Annual Symposium on Foundations of Computer Science
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Information Processing Letters
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SIAM Journal on Computing
Transdichotomous Results in Computational Geometry, I: Point Location in Sublogarithmic Time
SIAM Journal on Computing
Delaunay triangulations in O(sort(n)) time and more
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SIAM Journal on Computing
Delaunay triangulations in O(sort(n)) time and more
Journal of the ACM (JACM)
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Proceedings of the twenty-seventh annual symposium on Computational geometry
Convex hull of points lying on lines in o(nlogn) time after preprocessing
Computational Geometry: Theory and Applications
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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].