## On computing Voronoi diagrams by divide-prune-and-conquer (1996)

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Venue: | IN PROC. 12TH ANNUAL ACM SYMPOS. COMPUT. GEOM |

Citations: | 14 - 3 self |

### BibTeX

@INPROCEEDINGS{Amato96oncomputing,

author = {Nancy M. Amato and Edgar A. Ramos},

title = {On computing Voronoi diagrams by divide-prune-and-conquer},

booktitle = {IN PROC. 12TH ANNUAL ACM SYMPOS. COMPUT. GEOM},

year = {1996},

publisher = {}

}

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### Abstract

Using a divide, prune, and conquer approach based on geometric partitioning, we obtain: (1) An output sensitive algorithm for computing a weighted Voronoi diagram in R 4 (the projection of certain polyhedra in R 5) that runs in time O((n+f) log³ f) where n is the number of sites and f is the number of output cells; and (2) a deterministic parallel algorithm in the EREW PRAM model for computing an algebraic planar Voronoi diagram (in which bisectors between sites are simple curves consisting of a constant number of algebraic pieces of constant degree) that runs in time O(log² n) using optimal O(n log n) work. The first result implies an algorithm for the problems of computing the convex hull of a point set and the intersection of a set of halfspaces in R 5, and computing the Euclidean Voronoi diagram in R 4. The second result implies both sequential and parallel work-optimal deterministic algorithms for a number of Voronoi diagram problems (including line segments in the plane), and other non-Voronoi diagram problems that can fit in the framework (including the intersection of equal radius balls in R³ and some lower envelope problems in R³).