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 Gasieniec et al. [14], who established an upper bound of D + O(kn~(1/(k-2)) (log n)~2) and a lower bound of D + Ω((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/(2k)) (log n)~((2+1)/n)) on undirected graphs, and a lower bound of D + Ω(k·(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, log n})·n~(1/(k-1)) log n) rounds with high probability. Moreover, we present an Θ(log n)-shot broadcasting algorithm that completes broadcasting in at most O(D log n + (log n)~2) 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.
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