Polyhedral Voronoi Diagrams of Polyhedra in Three Dimensions
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We show that the complexity of the Voronoi diagram of a collection of disjoint polyhedra in general position in 3-space that have n vertices overall, under a convex distance function induced by a polyhedron with O(1) facets, is O(n2+ε), for any ε > 0. We also show that when the sites are n segments in 3-space, this complexity is O(n2 α(n) log n). This generalizes previous results by Chew et al. and by Aronov and Sharir, and solves an open problem put forward by Agarwal and Sharir. Specific distance functions for which our results hold are the L1 and L\infty metrics. These results imply that we can preprocess a collection of polyhedra as above into a near-quadratic data structure that can answer δ-approximate Euclidean nearest-neighbor queries amidst the polyhedra in time O(log (n/δ) ), for an arbitrarily small δ > 0.
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