Time efficient centralized gossiping in radio networks

In this paper we study the gossiping problem (all-to-all communication) in radio networks where all nodes are aware of the network topology. We start our presentation with a deterministic gossiping algorithm that works in at most n units of time in any radio network of size n. This algorithm is optimal in the worst case scenario since there exist radio network topologies, such as lines, stars and complete graphs in which radio gossiping cannot be completed in less than n communication rounds. Furthermore, we show that there does not exist any radio network topology in which the gossiping task can be solved in less than blog(n−1)c+2 rounds. We also show that this lower bound can be matched from above for a fraction of all possible integer values of n, and for all other values of n we propose a solution which accomplishes gossiping in dlog(n−1)e+2 rounds. Then we show an almost optimal radio gossiping algorithm in trees, which misses the optimal time complexity by a single round. Finally, we study asymptotically optimal O(D)-time gossiping (where D is the diameter of the network) in graphs with the maximum degree ∆ = O( D 1−1/(i+1) logi n ), for any integer constant i ≥ 0 and D large enough. c © 2007 Elsevier B.V. All rights reserved.