SIAM Journal on Applied Mathematics
Wireless integrated network sensors
Communications of the ACM
Random Geometric Graph Diameter in the Unit Ball
Algorithmica
On the cover time and mixing time of random geometric graphs
Theoretical Computer Science
The cover time of the giant component of a random graph
Random Structures & Algorithms
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
Broadcasting vs. mixing and information dissemination on Cayley graphs
STACS'07 Proceedings of the 24th annual conference on Theoretical aspects of computer science
Fast message dissemination in random geometric ad-hoc radio networks
ISAAC'07 Proceedings of the 18th international conference on Algorithms and computation
Managing random sensor networks by means of grid emulation
NETWORKING'06 Proceedings of the 5th international IFIP-TC6 conference on Networking Technologies, Services, and Protocols; Performance of Computer and Communication Networks; Mobile and Wireless Communications Systems
Proceedings of the 5th ACM workshop on Performance monitoring and measurement of heterogeneous wireless and wired networks
Rumor spreading on random regular graphs and expanders
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Asymptotically optimal randomized rumor spreading
ICALP'11 Proceedings of the 38th international conference on Automata, languages and programming - Volume Part II
Modeling and analysis of composite network embeddings
Proceedings of the 14th ACM international conference on Modeling, analysis and simulation of wireless and mobile systems
Mobile geometric graphs: detection, coverage and percolation
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Fast information spreading in graphs with large weak conductance
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Diameter and broadcast time of random geometric graphs in arbitrary dimensions
ISAAC'11 Proceedings of the 22nd international conference on Algorithms and Computation
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A Random Geometric Graph (RGG) in two dimensions is constructed by distributing n nodes independently and uniformly at random in [0, √n]2 and creating edges between every pair of nodes having Euclidean distance at most r, for some prescribed r. We 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 with probability 1 -- O(n-1) this algorithm informs every node in the largest connected component of an RGG within O(√n/r + log n) rounds. This holds for any value of r larger than the critical value for the emergence of a connected component with Ω(n) nodes. In order to prove this result, we show that for any two nodes sufficiently distant from each other in [0, √n]2, the length of the shortest path between them in the RGG, when such a path exists, is only a constant factor larger than the optimum. This result has independent interest and, in particular, gives that the diameter of the largest connected component of an RGG is Θ(√n/r), which surprisingly has been an open problem so far.