On the construction of abstract voronoi diagrams
TLDR
It is shown that the abstract Voronoi diagram of n sites in the plane can be constructed in timeO(n logn) by a randomized algorithm based on Clarkson and Shor's randomized incremental construction technique.Abstract:
We show that the abstract Voronoi diagram ofn sites in the plane can be constructed in timeO(n logn) by a randomized algorithm. This yields an alternative, but simpler,O(n logn) algorithm in many previously considered cases and the firstO(n logn) algorithm in some cases, e.g., disjoint convex sites with the Euclidean distance function. Abstract Voronoi diagrams are given by a family of bisecting curves and were recently introduced by Klein [13]. Our algorithm is based on Clarkson and Shor's randomized incremental construction technique [7].read more
Citations
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References
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Journal ArticleDOI
A sweepline algorithm for Voronoi diagrams
TL;DR: A geometric transformation is introduced that allows Voronoi diagrams to be computed using a sweepline technique and is used to obtain simple algorithms for computing the Vor onoi diagram of point sites, of line segment sites, and of weighted point sites.
Proceedings ArticleDOI
Applications of random sampling in computational geometry, II
TL;DR: Asymptotically tight bounds for a combinatorial quantity of interest in discrete and computational geometry, related to halfspace partitions of point sets, are given.
Proceedings ArticleDOI
Closest-point problems
Michael Ian Shamos,Dan Hoey +1 more
TL;DR: The purpose of this paper is to introduce a single geometric structure, called the Voronoi diagram, which can be constructed rapidly and contains all of the relevant proximity information in only linear space, and is used to obtain O(N log N) algorithms for most of the problems considered.
Journal ArticleDOI
Power diagrams: properties, algorithms and applications
TL;DR: The close relationship to convex hulls and arrangements of hyperplanes is investigated and exploited, and efficient algorithms that compute the power diagram and its order-k modifications are obtained.
Book
Concrete and Abstract Voronoi Diagrams
TL;DR: Voronoi diagrams in nice metrics, acyclic partitions, and abstract Vor onoi diagrams.