Efficient k-Shot Broadcasting in Radio Networks

Abstract

The paper concerns time-efficient k-shot broadcasting in undirected radio networks. In a k-shot broadcasting algorithm, each node in the network is allowed to transmit at most k times. Both known and unknown topology models are considered. For the known topology model, the problem has been studied before by Ga̧sieniec et al. [14], who established an upper bound of D + O(kn ^{1/(k − 2)}log² n) and a lower bound of $D+{\it \Omega}((n-D)^{1/2k})$ on the length of k-shot broadcasting schedules for n-node graphs of diameter D. We improve both the upper and the lower bound, providing a randomized algorithm for constructing a k-shot broadcasting schedule of length D + O(kn ^1/2klog^2 + 1/k n) on undirected graphs, and a lower bound of $D+{\it \Omega}(k\cdot(n-D)^{1/2k})$, which almost closes the gap between these bounds. For the unknown topology model, we provide the first k-shot broadcasting algorithm. Assuming that each node knows only the network size n (or a linear upper bound on it), our randomized k-shot broadcasting algorithm completes broadcasting in O((D + min {D ·k,logn}) ·n ^{1/(k − 1)}logn) rounds with high probability. Moreover, we present an ${\it \Theta}(\log n )$-shot broadcasting algorithm that completes broadcasting in at most O(Dlogn + log² n) rounds with high probability. This algorithm matches the broadcasting time of the algorithm of Bar-Yehuda et al. [3], which assumes no limitation on the maximum number of transmissions per node.

Supported in part by grants from the Minerva Foundation and the Israel Ministry of Science.