Radio communication in random graphs

One of the most frequently studied problems in the context of information dissemination in communication networks is the broadcasting problem. We propose here several time efficient, centralized as well as fully distributed procedures for the broadcasting problem in random radio networks. In particular, we show how to perform a centralized broadcast in a random graph G_p = ( V, E ) of size n = | V | and expected average degree d = pn in time O ( ln n / ln d + ln d ). Later we present a randomized distributed broadcasting algorithm with the running time O ( ln n ). In both cases we show that the presented algorithms are asymptotically optimal by deriving lower bounds on the complexity of radio broadcasting in random graphs. In these proofs we determine some structural properties of random graphs and develop new techniques which might be useful for further research in this field. We should note here that the results of this paper hold with probability 1 - o ( 1 / n ).