Delaunay graphs are almost as good as complete graphs
Discrete & Computational Geometry
Locality in distributed graph algorithms
SIAM Journal on Computing
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
A new way to weigh Malnourished Euclidean graphs
Proceedings of the sixth annual ACM-SIAM symposium on Discrete algorithms
Distributed computing: a locality-sensitive approach
Distributed computing: a locality-sensitive approach
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
Approximating geometric bottleneck shortest paths
Computational Geometry: Theory and Applications
Local approximation schemes for topology control
Proceedings of the twenty-fifth annual ACM symposium on Principles of distributed computing
Localized construction of bounded degree and planar spanner for wireless ad hoc networks
Mobile Networks and Applications
Geometric Spanner Networks
Computational Geometry: Algorithms and Applications
Computational Geometry: Algorithms and Applications
Sensor networks: distributed algorithms reloaded – or revolutions?
SIROCCO'06 Proceedings of the 13th international conference on Structural Information and Communication Complexity
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 and unit disk graphs embedded in the two-dimensional Euclidean plane. We are particularly interested in spanners that possess useful properties such as planarity, bounded degree, and/or light weight. Such spanners have been extensively studied in the area of computational geometry and have been used as the building block for constructing efficient and reliable wireless network communication topologies. We study the above problem under two computational models: the centralized and the distributed model. In the distributed model we focus on algorithms that are local. Such algorithms are suitable for the relevant applications (e.g., wireless computing). Under the centralized model, we present an $O(n\lg n)$ time algorithm that computes a bounded-degree plane spanner of a complete Euclidean graph, 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 complete Euclidean graph. Under the distributed model, we give the first local algorithm for computing a spanner of a unit disk graph that is of bounded degree and plane. The upper bounds on the degree, stretch factor, and the locality of the algorithm dramatically improve the previous results, as shown in the paper. This algorithm can also be extended to compute a bounded-degree plane lightweight spanner of a unit disk graph. Our algorithms rely on structural and geometric results that we develop in this paper.