Geometric spanner for routing in mobile networks
MobiHoc '01 Proceedings of the 2nd ACM international symposium on Mobile ad hoc networking & computing
Distributed Spanner with Bounded Degree for Wireless Ad Hoc Networks
IPDPS '02 Proceedings of the 16th International Parallel and Distributed Processing Symposium
The r-Neighborhood Graph: An Adjustable Structure for Topology Control in Wireless Ad Hoc Networks
IEEE Transactions on Parallel and Distributed Systems
Computational Geometry: Theory and Applications
Journal of Discrete Algorithms
IEEE/ACM Transactions on Networking (TON)
Planar hop spanners for unit disk graphs
ALGOSENSORS'10 Proceedings of the 6th international conference on Algorithms for sensor systems, wireless adhoc networks, and autonomous mobile entities
kthorder geometric spanners for wireless ad hoc networks
ICDCIT'11 Proceedings of the 7th international conference on Distributed computing and internet technology
On a family of strong geometric spanners that admit local routing strategies
Computational Geometry: Theory and Applications
Some properties of k-Delaunay and k-Gabriel graphs
Computational Geometry: Theory and Applications
On a family of strong geometric spanners that admit local routing strategies
WADS'07 Proceedings of the 10th international conference on Algorithms and Data Structures
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The spanning ratio of a graph defined on n points in the Euclidean plane is the maximum ratio over all pairs of data points (u,v) of the minimum graph distance between u and v divided by the Euclidean distance between u and v. A connected graph is said to be an S-spanner if the spanning ratio does not exceed S. For example, for any S there exists a point set whose minimum spanning tree is not an S-spanner. At the other end of the spectrum, a Delaunay triangulation is guaranteed to be a 2.42-spanner [J. M. Keil and C. A. Gutwin, Discrete Comput. Geom., 7 (1992), pp. 13-28]. For proximity graphs between these two extremes, such as Gabriel graphs [K. R. Gabriel and R. R. Sokal, Systematic Zoology, 18 (1969), pp. 259-278], relative neighborhood graphs [G. T. Toussaint, Pattern Recognition, 12 (1980), pp. 261-268], and $\beta$-skeletons [D. G. Kirkpatrick and J. D. Radke, Comput. Geom., G. T. Toussaint, ed., Elsevier, Amsterdam, 1985, pp. 217-248] with $\beta$ in [0,2] some interesting questions arise. We show that the spanning ratio for Gabriel graphs (which are $\beta$-skeletons with $\beta$ = 1) is $\Theta ( \sqrt{n})$ in the worst case. For all $\beta$-skeletons with $\beta$ in [0,1], we prove that the spanning ratio is at most $O(n^\gamma)$, where $\gamma = (1-\log_2(1+\sqrt{1-\beta^2}))/2$. For all $\beta$-skeletons with $\beta$ in [1,2], we prove that there exist point sets whose spanning ratio is at least $\left( \frac{1}{2} - o(1) \right) \sqrt{n} $. For relative neighborhood graphs [G. T. Toussaint, Pattern Recognition, 12 (1980), pp. 261-268] (skeletons with $\beta$ = 2), we show that there exist point sets where the spanning ratio is $\Omega(n)$. For points drawn independently from the uniform distribution on the unit square, we show that the spanning ratio of the (random) Gabriel graph and all $\beta$-skeletons with $\beta$ in [1,2] tends to $\infty$ in probability as $\sqrt{\log n / \log \log n}$.