Collaborative Broadcast in O(log log n) Rounds

We consider the multihop broadcasting problem for $n$ nodes placed uniformly at random in a disk and investigate the number of hops required to transmit a signal from the central node to all other nodes under three communication models: Unit-Disk-Graph (UDG), Signal-to-Noise-Ratio (SNR), and the wave superposition model of multiple input/multiple output (MIMO). In the MIMO model, informed nodes cooperate to produce a stronger superposed signal. We do not consider the problem of transmitting a full message nor do we consider interference. In each round, the informed senders try to deliver to other nodes the required signal strength such that the received signal can be distinguished from the noise. We assume sufficiently high node density $\rho= \Omega(\log n)$. In the unit-disk graph model, broadcasting needs $O(\sqrt{n/\rho})$ rounds. In the other models, we use an Expanding Disk Broadcasting Algorithm, where in a round only triggered nodes within a certain distance from the initiator node contribute to the broadcasting operation. This algorithm achieves a broadcast in only $O(\frac{\log n}{\log \rho})$ rounds in the SNR-model. Adapted to the MIMO model, it broadcasts within $O(\log \log n - \log \log \rho)$ rounds. All bounds are asymptotically tight and hold with high probability, i.e. $1- n^{-O(1)}$.