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.
The question of broadcasting within the giant component of a RGG above the percolation threshold was recently analyzed by Bradonjić et al. REF , who also show that the graph distance between any two (sufficiently distant) nodes is at most a constant factor larger than their Euclidean distance.
Efficient broadcast on random geometric graphs
SODA '10
Mathematics Computer Science