Name
Abstract | ||
|---|---|---|
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. |
Year | DOI | Venue |
|---|---|---|
2010 | 10.5555/1873601.1873715 | SODA |
Keywords | Field | DocType |
random geometric graph,euclidean distance,n node,prescribed r,largest connected component,shortest path,critical value,connected component,efficient broadcast,informed node,log n,following randomized broadcast algorithm,giant component,two dimensions,geometric graph | Broadcasting,Binary logarithm,Discrete mathematics,Graph,Combinatorics,Open problem,Shortest path problem,Euclidean distance,Connected component,Random geometric graph,Mathematics | Conference |
Volume | ISBN | Citations |
135 | 978-0-89871-698-6 | 14 |
PageRank | References | Authors |
0.80 | 11 | 5 |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Milan Bradonjic | 1 | 65 | 9.36 |
| Robert Elsässer | 2 | 14 | 0.80 |
| Tobias Friedrich | 3 | 211 | 13.48 |
| Thomas Sauerwald | 4 | 575 | 39.99 |
| Alexandre O. Stauffer | 5 | 130 | 11.34 |