The Complexity of Multi-Message Broadcast in Radio Networks with Known Topology

Broadcast is a fundamental communication primitive in wireless networks. In this paper, we give novel lower and matching upper bounds for the multi-message broadcast problem and its well studied special cases. These problems ask to broadcast k messages, initially residing in an arbitrary set of nodes, to all nodes in a radio network. This work is the first to show that the throughput of these tasks differs quadratically depending on whether nodes route information or use network coding. In particular, we show a direct sum type lower bound for routing: Alon et al. showed that in radius-2 networks with n nodes broadcasting one message needs Omega(log^2 n) time. We show that in the same family of networks k * \tilde{Omega}(log^2 n) rounds are needed to distribute k messages using routing, even in a centralized setting. This stands in contrast to network coding based solutions that complete in Theta(k log n + log^2 n) rounds. Besides identifying and fully characterizing the coding gap in radio networks, our routing lower bound also disproves a previously published O(n log n) routing algorithm for gossiping k=n messages. This work is also the first to give optimal (distributed) k-message broadcast algorithms for radio networks with known topology. Their running times are O(D + k log n + log^2 n) using network coding and O(D + k log^2 n) using routing, where D is the diameter of the network. Lastly, we show that the direct sum result for routing does not hold in radius-2 networks with small maximum receiver degree Delta < log n. We give a novel routing scheme based on Baranyai's celebrated theorem that achieves a complexity of k * O(Delta^2). We also present a k * \tilde{Omega}(Delta) lower bound. These two results show the routing throughput to lie asymptotically strictly between the throughput achievable for one message and the throughput of network coding in these networks.