Finite time distributed averaging over gossip-constrained ring networks

We consider a multi-agent system where each agent has its own estimate of a given quantity and the goal is to reach consensus on the average. To this purpose, we propose a distributed consensus algorithm that guarantees convergence to the average in a finite number of communication rounds. The algorithm is tailored to ring networks subject to a gossip constraint. If the number of agents m is even, say m = 2n, then, the number of communication rounds needed is equal to n, which in this case is the diameter of the network, whereas it grows to 3n if the number of agents is odd and equal to m = 2n + 1