A sparse graph almost as good as the complete graph on points in K dimensions
Discrete & Computational Geometry
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
Sparse communication networks and efficient routing in the plane
Distributed Computing
Navigating nets: simple algorithms for proximity search
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Deformable spanners and applications
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
Computational Geometry: Theory and Applications - Special issue on the 14th Canadian conference on computational geometry — CCCG02
On hierarchical routing in doubling metrics
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Fast Construction of Nets in Low-Dimensional Metrics and Their Applications
SIAM Journal on Computing
Searching dynamic point sets in spaces with bounded doubling dimension
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Routing in Networks with Low Doubling Dimension
ICDCS '06 Proceedings of the 26th IEEE International Conference on Distributed Computing Systems
Fully dynamic geometric spanners
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Optimal scale-free compact routing schemes in networks of low doubling dimension
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
An Optimal Dynamic Spanner for Doubling Metric Spaces
ESA '08 Proceedings of the 16th annual European symposium on Algorithms
Fault-tolerant spanners for general graphs
Proceedings of the forty-first annual ACM symposium on Theory of computing
Proceedings of the twenty-fifth annual symposium on Computational geometry
Proximity algorithms for nearly-doubling spaces
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Fault Tolerant Spanners for General Graphs
SIAM Journal on Computing
Resilient and low stretch routing through embedding into tree metrics
WADS'11 Proceedings of the 12th international conference on Algorithms and data structures
Fast, precise and dynamic distance queries
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
The emergence of sparse spanners and greedy well-separated pair decomposition
SWAT'10 Proceedings of the 12th Scandinavian conference on Algorithm Theory
Optimal euclidean spanners: really short, thin and lanky
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
New doubling spanners: better and simpler
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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For a set S of points in ℝd, a t-spanner is a sparse graph on the points of S such that between any pair of points there is a path in the spanner whose total length is at most t times the Euclidean distance between the points. In this paper, we show how to construct a (1 + ε)-spanner with O(n/εd) edges and maximum degree O(1/εd) in time O(n log n). A spanner with similar properties was previously presented in [6, 8]. However, using our new construction (coupled with several other innovations) we obtain new results for two fundamental problems for constant doubling dimension metrics: The first result is an essentially optimal compact routing scheme. In particular, we show how to perform routing with a stretch of 1 + ∈, where the label size is [log n] and the size of the table stored at each point is only O(log n/εd). This routing problem was first considered by Peleg and Hassin [11], who presented a routing scheme in the plane. Later, Chan et al. [6] and Abraham et al. [1] considered this problem for doubling dimension metric spaces. Abraham et al. [1] were the first to present a (1 + ∈) routing scheme where the label size depends solely on the number of points. In their scheme labels are of size of [log n], and each point stores a table of size O(log2 n/εd). In our routing scheme, we achieve routing tables of size O(log n/εd), which is essentially the same size as a label (up to the factor of 1/εd). The second and main result of this paper is the first fully dynamic geometric spanner with poly-logarithmic update time for both insertions and deletions. We present an algorithm that allows points to be inserted into and deleted from S with an amortized update time of O(log3 n).