On Radio Broadcasting in Random Geometric Graphs

• Authors: Robert Elsässer;Leszek Gąsieniec;Thomas Sauerwald
• Venue: DISC '08 Proceedings of the 22nd international symposium on Distributed Computing
• Year: 2008

One of the most frequently studied problems in the context of information dissemination in communication networks is the broadcasting problem. In this paper we consider radio broadcasting in random geometric graphs, in which nnodes are placed uniformly at random in $[0, \sqrt{n}]^2$, and there is a (directed) edge from a node uto a node vin the corresponding graph iff the distance between uand vis smaller than the transmission radius assigned to u. Throughout this paper we consider the distributed case, i.e., each node is only aware (apart from n) of its own coordinates and its own transmission radius, and we assume that the transmission radii of the nodes vary according to a power law distribution. First, we consider the model in which any node is assigned a transmission radius r rminaccording to a probability density function ρ(r) ~r茂戮驴 茂戮驴(more precisely, $\rho(r) = (\alpha-1)r_{\min}^{\alpha-1} r^{-\alpha}$), where 茂戮驴茂戮驴 (1,3) and $r_{\min}\delta \sqrt{\log n}$ with 茂戮驴being a large constant. For this case, we develop a simple radio broadcasting algorithm which has the running time O(loglogn), with high probability, and show that this result is asymptotically optimal. Then, we consider the model in which any node is assigned a transmission radius r caccording to the probability density function ρ(r) = (茂戮驴茂戮驴 1)c茂戮驴茂戮驴 1r茂戮驴 茂戮驴, where 茂戮驴is drawn from the same range as before and cis a constant. Since this graph is usually not strongly connected, we assume that the message which has to be spread to all nodes of the graph is placed initially in one of the nodes of the giant component. We show that there exists a fully distributed randomized algorithm which disseminates the message in O(D(loglogn)2) steps, with high probability, where Ddenotes the diameter of the giant component of the graph.Our results imply that by setting the transmission radii of the nodes according to a power law distribution, one can design energy efficient radio networks with low average transmission radius, in which broadcasting can be performed exponentiallyfaster than in the (extensively studied) case where all nodes have the same transmission power.