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Voronoi diagrams and their duals, Delaunay triangulations, are used in many areas of computing and the sciences. Starting in 3dimensions, there is a substantial (i.e. polynomial) difference between the best case and the worst case complexity of these objects when starting with n points. This motivates the search for algorithms that are outputsenstiive rather than relying only on worstcase guarantees. In this talk, I will describe a simple, new algorithm for computing Voronoi diagrams in ddimensions that runs in O(f log n log spread) time, where f is the output size and the spread of the input points is the ratio of the diameter to the closest pair distance. For a wide range of inputs, this is the best known algorithm. The algorithm is novel in the that it turns the classic algorithm of Delaunay refinement for mesh generation on its head, working backwards from a quality mesh to the Delaunay triangulation of the input. Along the way, we will see instances of several other classic problems for which no higherdimensional results are known, including kinetic convex hulls and splitting Delaunay triangulations.
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