There is a planar graph almost as good as the complete graph
SCG '86 Proceedings of the second annual symposium on Computational geometry
Functional approach to data structures and its use in multidimensional searching
SIAM Journal on Computing
New sparseness results on graph spanners
SCG '92 Proceedings of the eighth annual symposium on 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
Euclidean spanners: short, thin, and lanky
STOC '95 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing
Computational geometry: algorithms and applications
Computational geometry: algorithms and applications
Efficient algorithms for constructing fault-tolerant geometric spanners
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Approximating geometrical graphs via “spanners” and “banyans”
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
A new way to weigh Malnourished Euclidean graphs
Proceedings of the sixth annual ACM-SIAM symposium on Discrete algorithms
Dynamic algorithms for geometric spanners of small diameter: randomized solutions
Computational Geometry: Theory and Applications
Distributed computing: a locality-sensitive approach
Distributed computing: a locality-sensitive approach
Fast Greedy Algorithms for Constructing Sparse Geometric Spanners
SIAM Journal on Computing
Region-fault tolerant geometric spanners
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
A simple and efficient kinetic spanner
Proceedings of the twenty-fourth annual symposium on Computational geometry
Fault-tolerant spanners: better and simpler
Proceedings of the 30th annual ACM SIGACT-SIGOPS symposium on Principles of distributed computing
Fault tolerant interference-aware topology control for ad hoc wireless networks
ADHOC-NOW'11 Proceedings of the 10th international conference on Ad-hoc, mobile, and wireless networks
Fault-tolerant spanners for ad hoc networks
International Journal of Network Management
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We present two new results about vertex and edge fault-tolerant spanners in Euclidean spaces.We describe the first construction of vertex and edge fault-tolerant spanners having optimal bounds for maximum degree and total cost. We present a greedy algorithm that for any t 1 and any non-negative integer k, constructs a k-fault-tolerant t-spanner in which every vertex is of degree O(k) and whose total cost is O(k2) times the cost of minimum spanning tree; these bounds are asymptotically optimal.Our next contribution is an efficient algorithm for constructing good fault-tolerant spanners. We present a new, sufficient condition for a graph to be a k-fault-tolerant spanner. Using this condition, we design an efficient algorithm that finds fault-tolerant spanners with asymptotically optimal bound for the maximum degree and almost optimal bounds for the total cost.