Energy Efficient Randomized Communication in Unknown AdHoc Networks
  This paper studies broadcasting and gossiping algorithms in random and general AdHoc networks. Our goal is not only to minimise the broadcasting and gossiping time, but also to minimise the energy consumption, which is measured in terms of the total number of messages (or transmissions) sent. We assume that the nodes of the network do not know the network, and that they can only send with a fixed power, meaning they can not adjust the areas sizes that their messages cover. We believe that under these circumstances the number of transmissions is a very good measure for the overall energy consumption.   For random networks, we present a broadcasting algorithm where every node transmits at most once. We show that our algorithm broadcasts in $O(\log n)$ steps, w.h.p, where $n$ is the number of nodes. We then present a $O(d \log n)$ ($d$ is the expected degree) gossiping algorithm using $O(\log n)$ messages per node.   For general networks with known diameter $D$, we present a randomised broadcasting algorithm with optimal broadcasting time $O(D \log (n/D) + \log^2 n)$ that uses an expected number of $O(\log^2 n / \log (n/D))$ transmissions per node. We also show a tradeoff result between the broadcasting time and the number of transmissions: we construct a network such that any oblivious algorithmusing a time-invariant distribution requires $\Omega(\log^2 n / \log (n/D))$ messages per node in order to finish broadcasting in optimal time. This demonstrates the tightness of our upper bound. We also show that no oblivious algorithm can complete broadcasting w.h.p. using $o(\log n)$ messages per node. 
Petra Berenbrink and Colin Cooper and Zengjian Hu