There are planar graphs almost as good as the complete graph
Journal of Computer and System Sciences
Euclidean spanners: short, thin, and lanky
STOC '95 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing
The rectilinear Steiner arborescence problem is NP-complete
SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
An Improved Algorithm for the Minimum Manhattan Network Problem
ISAAC '02 Proceedings of the 13th International Symposium on Algorithms and Computation
Approximating a minimum Manhattan network
Nordic Journal of Computing
A rounding algorithm for approximating minimum manhattan networks
APPROX'05/RANDOM'05 Proceedings of the 8th international workshop on Approximation, Randomization and Combinatorial Optimization Problems, and Proceedings of the 9th international conference on Randamization and Computation: algorithms and techniques
A rounding algorithm for approximating minimum Manhattan networks
Theoretical Computer Science
Light orthogonal networks with constant geometric dilation
Journal of Discrete Algorithms
Minimum Manhattan network is NP-complete
Proceedings of the twenty-fifth annual symposium on Computational geometry
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Given a set of points in the plane and a constant t ≥ 1, a Euclidean t-spanner is a network in which, for any pair of points, the ratio of the network distance and the Euclidean distance of the two points is atmost t. Such networks have applications in transportation or communication network design and have been studied extensively.In this paper we study 1-spanners under the Manhattan (or L1-) metric. Such networks are called Manhattan networks. A Manhattan network for a set of points is a set of axis-parallel line segments whose union contains an x- and y-monotone path for each pair of points. It is not known whether it is NP-hard to compute minimum Manhattan networks (MMN), i.e., Manhattan networks of minimum total length. In this paper we present an approximation algorithm for this problem. Given a set P of n points, our algorithm computes in O(n logn) time and linear space a Manhattan network for P whose length is at most 3 times the length of an MMN of P.We also establish a mixed-integer programming formulation for the MMN problem. With its help we extensively investigate the performance of our factor-3 approximation algorithm on random point sets.