Convergence Speed in Distributed Consensus and Averaging

We study the convergence speed of distributed iterative algorithms for the consensus and averaging problems, with emphasis on the latter. We first consider the case of a fixed communication topology. We show that a simple adaptation of a consensus algorithm leads to an averaging algorithm. We prove lower bounds on the worst-case convergence time for various classes of linear, time-invariant, distributed consensus methods, and provide an algorithm that essentially matches those lower bounds. We then consider the case of a time-varying topology, and provide a polynomial-time averaging algorithm.