There are planar graphs almost as good as the complete graph
Journal of Computer and System Sciences
Delaunay graphs are almost as good as complete graphs
Discrete & Computational Geometry
Classes of graphs which approximate the complete Euclidean graph
Discrete & Computational Geometry
On sparse spanners of weighted graphs
Discrete & Computational Geometry
Optimally sparse spanners in 3-dimensional Euclidean space
SCG '93 Proceedings of the ninth annual symposium on Computational geometry
A fast algorithm for constructing sparse Euclidean spanners
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
Routing with guaranteed delivery in ad hoc wireless networks
Wireless Networks
Fast Greedy Algorithms for Constructing Sparse Geometric Spanners
SIAM Journal on Computing
Constructing Plane Spanners of Bounded Degree and Low Weight
ESA '02 Proceedings of the 10th Annual European Symposium on Algorithms
Localized construction of bounded degree and planar spanner for wireless ad hoc networks
Mobile Networks and Applications
Geometric Spanner Networks
Diamond triangulations contain spanners of bounded degree
ISAAC'06 Proceedings of the 17th international conference on Algorithms and Computation
Localized Delaunay triangulation with application in ad hoc wireless networks
IEEE Transactions on Parallel and Distributed Systems
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We consider the problem of computing spanners of Euclidean graphs embedded in the 2-dimensional Euclidean plane. We present an $O(n\lg{n})$ time algorithm that computes a spanner of a Euclidean graph that is of bounded degree and plane, where n is the number of points in the graph. Both upper bounds on the degree and the stretch factor significantly improve the previous bounds. We extend this algorithm to compute a bounded-degree plane lightweight spanner of a Euclidean graph. Our results rely on elegant structural and geometric results that we develop. Moreover, our results can be extended to Unit Disk graphs under the local distributed model of computation.