Efficient kk-shot broadcasting in radio networks

The paper concerns time-efficient kk-shot broadcasting in undirected radio networks for nn-node graphs of diameter DD. In a kk-shot   broadcasting algorithm, each node in the network is allowed to transmit at most kk times. Both known and unknown topology models are considered. For the known topology model, the problem has been studied before by Ga̧sieniec et al. (2008).We improve both the upper and the lower bound of that paper providing a randomized algorithm for constructing a kk-shot broadcasting schedule of length D+O(kn1/2klog2+1/kn)D+O(kn1/2klog2+1/kn) on undirected graphs, and a lower bound of D+Ω(k⋅(n−D)1/2k)D+Ω(k⋅(n−D)1/2k), which almost closes the gap between these bounds. For the unknown topology model, we provide the first kk-shot broadcasting algorithm.Assuming that each node knows only the network size nn (or a linear upper bound on it), our randomized kk-shot broadcasting algorithm completes broadcasting in O((D+min{D⋅k,logn})⋅n1/(k−1)logn)O((D+min{D⋅k,logn})⋅n1/(k−1)logn) rounds with high probability for k≥2k≥2, and in O(D⋅n2logn)O(D⋅n2logn) rounds with high probability for k=1k=1. Moreover, we present an Θ(logn)Θ(logn)-shot broadcasting algorithm that completes broadcasting in at most O(Dlogn+log2n)O(Dlogn+log2n) rounds with high probability. This algorithm matches the broadcasting time of the algorithm of Bar-Yehuda et al. (1992), which assumes no limitation on the maximum number of transmissions per node.