Finite time distributed averaging over 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 iterations. The algorithm is tailored to ring networks with bidirectional pairwise communications. If the number of agents $m$ is even, say $m=2n$, then, the number of iterations needed is equal to $n$, which in this case is the diameter of the network, whereas the number of iterations grows to $3n$ if the number of agents is odd and equal to $m=2n+1$. 
Alessandro Falsone, Kostas Margellos, Simone Garatti, Maria Prandini
math.OC