  In this paper we study the inherent trade-off between time and communication
complexity for the distributed consensus problem. In our model, communication
complexity is measured as the maximum data throughput (in bits per second) sent
through the network at a given instant. Such a notion of communication
complexity, referred to as bandwidth complexity, is related to the frequency
bandwidth a designer should collectively allocate to the agents if they were to
communicate via a wireless channel, which represents an important constraint
for dense robotic networks. We prove a lower bound on the bandwidth complexity
of the consensus problem and provide a consensus algorithm that is
bandwidth-optimal for a wide class of consensus functions. We then propose a
distributed algorithm that can trade communication complexity versus time
complexity as a function of a tunable parameter, which can be adjusted by a
system designer as a function of the properties of the wireless communication
channel. We rigorously characterize the tunable algorithm's worst-case
bandwidth complexity and show that it compares favorably with the bandwidth
complexity of well-known consensus algorithm.
