Voronoi diagrams can also be thought of as lower envelopes, in the sense mentioned at the beginning of this subsection. Namely, for each point x not situated on a bisecting curve, the relation p x q defines a total ordering on S. If we construct a set of surfaces H p , p S,in3-space such that H p is below H q i# p x q holds, then the projection of their lower envelope equals the abstract Voronoi diagram.

We survey results in geometric network design theory, including algorithms for constructing minimum spanning trees and low-dilation graphs. 1 Introduction This survey covers topics in geometric network design theory. The problem is easy to state: connect a collection of sites by a "good" network. For instance, one may wish to connect components of a VLSI circuit by networks of wires, in a way that uses little surface area on the chip, draws little power, and propagates signals quickly. Similar problems come up in other applications such as telecommunications, road network design, and medical imaging [1]. One network design problem, the Traveling Salesman problem, is sufficiently important to have whole books devoted to it [79]. Problems involving some form of geometric minimum or maximum spanning tree also arise in the solution of other geometric problems such as clustering [12], mesh generation [56], and robot motion planning [93]. One can vary the network design problem in many w...

An edge-flipping operation in a triangulation T of a set of points in the plane is a local restructuring that changes T into a triangulation that differs from T in exactly one edge. The edge-flipping distance between two triangulations of the same set of points is the minimum number of edge-flipping operations needed to convert one into the other. In the context of computing the rotation distance of binary trees Sleator, Tarjan, and Thurston [7] show an upper bound of 2n \Gamma 10 on the maximum edge-flipping distance between triangulations of convex polygons with n nodes, n ? 12. Using volumetric arguments in hyperbolic 3-space they prove that the bound is tight. In this paper we establish an upper bound on the edgeflipping distance between triangulations of a general set of points in the plane by showing that not more edge-flipping operations than the number of intersections between the edges of two triangulations are needed to transform these triangulations into another, and we pre...

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Given a set S of n points in the plane, a triangulation is a maximal set of nonintersecting edges connecting the points in S. The weight of the triangulation is the sum of the lengths of the edges. In this paper, we show that for fi ? 1= sin , the β-skeleton of S is a subgraph of a minimum weight triangulation of S, where = tan \Gamma1 (3= q 2 p 3) =3:1. There exists a four point example such that the β-skeleton for β ! 1= sin(=3) is not a subgraph of the minimum weight triangulation. Keywords: Minimum weight triangulation, beta skeleton, computational geometry.

Voronoi Diagrams. Lecture Notes in Computer Science, vol. 400, Springer-Verlag, USA. Klein, R. and Wood, D. 1988. "Voronoi diagrams and mixed metrics," Lecture Notes in Computer Science, Springer-Verlag, vol. 294, pp. 281-291. Klein, R., Mehlhorn, K. and Meiser, S. 1993. "Randomized incremental construction of abstract Voronoi diagrams", Computational Geometry Theory and Applications, vol. 3, pp. 157-184. Knuth, D. 1968. The Art of Computer Programming. Volume1: Fundamental Algorithms. Addison-Wesley, Reading, MA. Knuth, D. 1973. The Art of Computer Programming. Volume3: Sorting and Searching. Addison-Wesley, Reading, MA. Kokubo, I. 1985. "Computational methods of generalized Voronoi diagrams," M. Sc. Thesis, Department of Mathematical Engineering and Information Physics, University of Tokyo, Japan. Krantz, A. 1996. "Analysis of an efficient algorithms for the hard-sphere problem," ACM Transaction on Modeling and Computer Simulation, vol. 6, no. 3, pp. 185-209. Lambert, T. 1993....

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Minimum-weight triangulation (MWT) is NP-hard. It has a polynomial-time constant-factor approximation algorithm, and a variety of effective polynomialtime heuristics that, for many instances, can find the exact MWT. Linear programs (LPs) for MWT are well-studied, but previously no connection was known between any LP and any approximation algorithm or heuristic for MWT. Here we show the first such connections: for an LP formulation due to Dantzig et al. (1985): (i) the integrality gap is bounded by a constant; (ii) given any instance, if the aforementioned heuristics find the MWT, then so does the LP. 1