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Bounds on Delaunay tessellations

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Abstract Motivated by applications in density estimation and data compression, this article considers the bounds on the number of tiles in a Delaunay tessellation as a function of both the number of tessellating points and the dimension. Results can also be interpreted for the dual of the Delaunay tessellation, the Voronoi diagram. Several theoretical lower and upper bounds are found in the combinatorics and computational geometry literatures and are brought together in this article. We make a comparison of these bounds with several empirically derived curves based on multivariate uniform and Gaussian generated random tessellating points. The upper bounds are found to be very conservative when compared with the empirically derived number of tiles, often off by many orders of magnitude. Copyright © 2010 John Wiley & Sons, Inc. This article is categorized under: Applications of Computational Statistics > Computational Mathematics

Comparison of uniform, normal, Cauchy, hypercube, and Seidel curves in three dimensions. The uniform, normal, and Cauchy curves are virtually identical. The Seidel upper bound is as much as 5 orders of magnitude larger than the empirical result.

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Comparison of uniform, normal, Cauchy, hypercube, and Seidel curves in two dimensions. The uniform, normal, Cauchy, and Seidel curves are virtually identical.

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Lower bounds on the number of tiles as a function of dimension.

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The tessellation in blue is the Voronoi Diagram, while the one in red is the Delaunay tessellation.

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Comparison of Uniform, Normal, Cauchy, Hypercube, and Seidel curves in six dimensions. The uniform, normal, and Cauchy curves are virtually identical. The Seidel upper bound is as much as 8 orders of magnitude larger than the empirical result.

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Comparison of uniform, normal, Cauchy, hypercube, and Seidel curves in four dimensions. The uniform, normal, and Cauchy curves are virtually identical. The Seidel upper bound is as much as 5 orders of magnitude larger than the empirical result.

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Applications of Computational Statistics > Computational Mathematics

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