Voronoi diagrams and arrangements
Discrete & Computational Geometry
Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
Near-quadratic bounds for the L1 Voronoi diagram of moving points
Computational Geometry: Theory and Applications - Special issue: computational geometry, theory and applications
The Union of Convex Polyhedra in Three Dimensions
SIAM Journal on Computing
On Translational Motion Planning of a Convex Polyhedron in 3-Space
SIAM Journal on Computing
Voronoi diagrams and Delaunay triangulations
Handbook of discrete and computational geometry
Voronoi diagrams of lines in 3-space under polyhedral convex distance functions
Journal of Algorithms - Special issue on SODA '95 papers
A tight bound for the complexity of voroni diagrams under polyhedral convex distance functions in 3D
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Three dimensional euclidean Voronoi diagrams of lines with a fixed number of orientations
Proceedings of the eighteenth annual symposium on Computational geometry
Almost Tight Upper Bounds for Vertical Decompositions in Four Dimensions
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
A Replacement for Voronoi Diagrams of Near Linear Size
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Three dimensional euclidean Voronoi diagrams of lines with a fixed number of orientations
Proceedings of the eighteenth annual symposium on Computational geometry
Efficient max-norm distance computation and reliable voxelization
Proceedings of the 2003 Eurographics/ACM SIGGRAPH symposium on Geometry processing
Approximating nearest neighbor among triangles in convex position
Information Processing Letters
3D hyperbolic Voronoi diagrams
Computer-Aided Design
ICCSA'06 Proceedings of the 2006 international conference on Computational Science and Its Applications - Volume Part V
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We show that that the complexity of the Voronoi diagram of a collection of disjoint polyhedra in 3-space that have n vertices overall, under a convex distance function induced by a polyhedron with O(1) facets, is O(n 2+&egr;), for any &egr;0. We also show that when the sites are n segments in 3-space, this complexity is O(n 2 &agr;(n) log n). This generalizes previous results by Chew et al. [9] and by Aronov and Sharir [4], and solves an open problem put forward by Agarwal and Sharir [2]. Specific distance functions for which our results hold are the L 1 and the L ∞ 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 δ0.