We introduce a geometric transformation that allows Voronoi diagrams to be computed using a sweepline technique. The transformation is used to obtain simple algorithms for computing the Voronoi diagram of point sites, of line segment sites, and of weighted point sites. All algorithms have O(n log n) worst-case running time and use O(n) space.
A model of computation based on random access machines operating in parallel and sharing a common memory is presented.The computational power of this model is related to that of traditional models.In particular, deterministic parallel RAM's can accept in polynomial time exactly the sets accepted by polynomial tape bounded Turing machines; nondeterministic RAM's can accept in polynomial time exactly the sets accepted by nondeterministic exponential time bounded Turing machines.Similar results hold for other classes.The effect of limiting the size of the common memory is also considered.
Introduction The Voronoi diagram of a set of sites partitions space into regions, one per site the region for a site s consists of all points closer to s than to any other site. The dual of the Voronoi diagram, the Delaunay triangulation, is the unique triangulation so that the circumsphere of every triangle contains no sites in its interior. Voronoi diagrams and Delaunay triangulations have been rediscovered or applied in many areas of mathematics and the natural sciences they are central topics in computational geometry, with hundreds of papers discussing algorithms and extensions. Section 2 discusses the de nition and basic properties in the usual case of point sites in R d with the Euclidean metric, while section 3 gives basic algorithms. Some of the many extensions obtained by v arying metric, sites, environment, and constraints are discussed in section 4. Section 5 nishes with some interesting and nonobvious structural properties of Voronoi diagrams and Delaunay triangulations. Glossary site: A de ning object for a Voronoi diagram or Delaunay triangulation. Also generator, source, Voronoi point. Voronoi face: The set of points for which a single site is closest or more generally a set of sites is closest. Also Voronoi region, Voronoi cell. Voronoi diagram: The set of all Voronoi faces. Also Thiessen diagram, Wigner-Seitz diagram, Blum transform, Dirichlet tesselation. Delaunay triangulation: The unique triangulation of a set of sites so that the circumsphere of each triangle has no sites in its interior.
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