Efficient broadcasting in known geometric radio networks with non-uniform ranges

We study here deterministic broadcasting in geometric radio networks (GRN) whose nodes have complete knowledge of the network. Nodes of a GRN are deployed in the Euclidean plane (R²) and each of them can transmit within some range r assigned to it. We adopt model in which ranges of nodes are non-uniform and they are drawn from the predefined interval 0 ≤ r_min ≤r_max. All our results are in the conflict-embodied model where a receiving node must be in the range of exactly one transmitting node in order to receive the message. We derive several lower and upper bounds on the time of deterministic broadcasting in GRNs in terms of the number of nodes n, a distribution of nodes ranges, and the eccentricity D of the source node (i.e., the maximum length of a shortest directed path from the source node to another node in the network). In particular: (1) We show that D + Ω(log(n - D)) rounds are required to accomplish broadcasting in some GRN where each node has the transmission range set either to 1 or to 0. We also prove that the bound D + Ω(log(n - D)) is almost tight providing a broadcasting procedure that works in this type of GRN in time D + O(logn). (2) In GRNs with a wider choice of positive node ranges from r_min , ..., r_max, we show that broadcasting requires D + Ω(min{log r_max/r_min, log (n -D)}) rounds and that it can be accomplished in O(D log² r_max/r_min rounds subsuming the best currently known upper bound O(D(r_max/r_min)⁴) provided in [15]. (3) We also study the problem of simulation of minimum energy broadcasting in arbitrary GRNs. We show that energy optimal broadcasting that can be completed in h rounds in a conflict-free model may require up to h/2 additional rounds in the conflict-embodied model. This lower bound should be seen as a separation result between conflict-free and conflict-embodied geometric radio networks. Finally, we also prove that any h-hop broadcasting algorithm with the energy consumption ε in a GRN can be simulated within O(h log ψ) rounds in the conflict-embodied model using energy O(ε), where ψ is the ratio between the largest and the shortest Euclidean distance between a pair of nodes in the network.