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Discrete Applied Mathematics
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Extremal Properties of Three-Dimensional Sensor Networks with Applications
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On the cover time and mixing time of random geometric graphs
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Sensor Position Determination with Flying Anchors in Three-Dimensional Wireless Sensor Networks
IEEE Transactions on Mobile Computing
The cover time of random geometric graphs
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
On the runtime and robustness of randomized broadcasting
Theoretical Computer Science
On Mixing and Edge Expansion Properties in Randomized Broadcasting
Algorithmica - Special Issue: Algorithms and Computation; Guest Editor: Takeshi Tokuyama
Broadcasting vs. mixing and information dissemination on Cayley graphs
STACS'07 Proceedings of the 24th annual conference on Theoretical aspects of computer science
Efficient broadcast on random geometric graphs
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Rumor spreading on random regular graphs and expanders
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A random geometric graph (RGG) is defined by placing n points uniformly at random in [0,n1/d]d, and joining two points by an edge whenever their Euclidean distance is at most some fixed r. We assume that r is larger than the critical value for the emergence of a connected component with Ω(n) nodes. We show that, with high probability (w.h.p.), for any two connected nodes with a minimum Euclidean distance of ω(logn), their graph distance is only a constant factor larger than their Euclidean distance. This implies that the diameter of the largest connected component is Θ(n1/d/r) w.h.p. We also analyze the following randomized broadcast algorithm on RGGs. At the beginning, only one node from the largest connected component of the RGG is informed. Then, in each round, each informed node chooses a neighbor independently and uniformly at random and informs it. We prove that w.h.p. this algorithm informs every node in the largest connected component of an RGG within Θ(n1/d/r+logn) rounds.