Discrete & Computational Geometry

, Volume 31, Issue 1, pp 83–124

Polyhedral Voronoi Diagrams of Polyhedra in Three Dimensions


DOI: 10.1007/s00454-003-2950-5

Cite this article as:
Koltun, V. & Sharir, M. Discrete Comput Geom (2004) 31: 83. doi:10.1007/s00454-003-2950-5


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.

Copyright information

© Springer-Verlag 2003

Authors and Affiliations

  1. 1.School of Computer Science, Tel Aviv University, Tel Aviv 69978Israel
  2. 2.Courant Institute of Mathematical Sciences, New York University, New York, NY 10012USA