Time efficient gossiping in known radio networks

We study here the gossiping problem (all-to-all communication) in known radio networks, i.e., when all nodes are aware of the network topology. We start our presentation with a deterministic algorithm for the gossiping problem that works in at most n units of time in any radio network of size n. This is an optimal algorithm in the sense that there exist radio network topologies, such as: a line, a star and a complete graph in which the radio gossiping cannot be completed in less then n units of time. Furthermore, we show that there isn't any radio network topology in which the gossiping task can be solved in time < ⌊log(n - 1)⌋ + 2. We show also 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 admitting gossiping in time ⌈log(n - 1)⌉ + 2. Finally we study asymptotically optimal O(D)-time gossiping (where D is a diameter of the network) in graphs with max-degree Δ = O(D^1-1/(i+1)/ logⁱ n), for any integer constant i ≥ 0 and D large enough.