On radio broadcasting in random geometric graphs

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 n nodes are placed uniformly at random in [0, √n]², and there is a (directed) edge from a node u to a node v in the corresponding graph iff the distance between u and v is 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 > r_min according to a probability density function ρ(r) ∼ r^-α (more precisely, ρ(r) = (α - 1)r_min^α-1 r^-α), where α ∈(1,3) and r_min > δ √log n with δ 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 > c according to the probability density function ρ(r) = (α - 1)c ^α-1 r^-α , where α is drawn from the same range as before and c is 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 (log log n)²) steps, with high probability, where D denotes 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 exponentially faster than in the (extensively studied) case where all nodes have the same transmission power.