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...

Let G = (V

Given a set S of n points in the plane, we give an O(n log n)-time algorithm that constructs a plane t-spanner for S, for t 10:02, such that the degree of each point of S is bounded from above by 27, and the total edge length is proportional to the weight of a minimum spanning tree of S. These constants are all worst case constants that are artifacts of our proofs. In practice, we believe them to be much smaller. Previously, no algorithms were known for constructing plane t-spanners of bounded degree.

A k\Gammaspanner of a connected graph G = (V; E) is a subgraph G 0 consisting of all the vertices of V and a subset of the edges, with the additional property that the distance between any two vertices in G 0 is larger than the distance in G by no more than a factor of k. This paper concerns the hardness of finding spanners with a number of edges close to the optimum. It is proved that for every fixed k, approximating the spanner problem is at least as hard as approximating the set cover problem We also consider a weighted version of the spanner problem, and prove an essential difference between the approximability of the case k = 2, and the case k 5. Department of Computer Science, The Open University, 16 Klauzner st., Ramat Aviv, Israel, guyk@shaked.openu.ac.il. 1 Introduction The concept of graph spanners has been studied in several recent papers in the context of communication networks, distributed computing, robotics and computational geometry [ADDJ-90, C-94, CK-94,...

Abstract This paper settles the following two longstanding open problems: 1. What is the worst-case approximation ratio between the greedy and the minimum weight triangulation? 2. Is there a polynomial time algorithm that always produces a triangulation whose length is within a constant factor from the minimum? The answer to the first question is that the known \Omega (pn) lower bound is tight. The second question is answered in the affirmative by using a slight modification of an O(n log n) algorithm for the greedy triangulation. We also derive some other interesting results. For example, we show that a constant-factor approximation of the minimum weight convex partition can be obtained within the same time bounds. 1 Introduction Let S be any set of n points in the plane. A triangulation of S is a maximal straight-line graph whose vertices are the points in S. Any triangulation of S partitions the convex hull of S into empty triangles. A triangulation that has received special attention is the minimum weight triangulation, in which the optimization criteria is to minimize the total edge length. This triangulation has some good properties [2] and is e.g. useful for numerical approximation of bivariate data [20].

A triangulation of a planar point set S is a maximal plane straight-line graph with vertex set S. In the minimum weight triangulation (MWT) problem, we are looking for a triangulation of a given point set that minimizes the sum of the edge lengths. We prove that the decision version of this problem is NP-hard. We use a reduction from PLANAR-1-IN-3-SAT. The correct working of the gadgets is established with computer assistance, using geometric inclusion and exclusion criteria for MWT edges, such as the diamond test and the LMT-Skeleton heuristic, as well as dynamic programming on polygonal faces.

A k-spanner of a connected graph G = (V; E) is a subgraph G 0 consisting of all the vertices of V and a subset of the edges, with the additional property that the distance between any two vertices in G 0 is larger than that distance in G by no more than a factor of k. This note concerns the problem of finding the sparsest 2-spanner in a given graph, and presents an approximation algorithm for this problem with approximation ratio log(|E|/|V|).

We consider online routing algorithms for finding paths between the vertices of plane graphs.

Let V be a set of n points in 3-dimensional Euclidean space. A subgraph of the complete Euclidean graph is a t-spanner if for any u and v in V, the length of the shortest path from u to v in the spanner is at most t times d(u, v). We show that for any t> 1, a greedy algorithm produces a t-spanner with O(n) edges, and total edge weight O(1). tot(it4ST), where MST is a minimum spanning tree of V. 1