Finite-Time Distributed Consensus in Graphs with Time-Invariant Topologies

We present a method for achieving consensus in distributed systems in a finite number of time-steps. Our scheme involves a linear iteration where, at each time-step, each node updates its value to be a weighted average of its own previous value and those of its neighbors. If D denotes the degree of the minimal polynomial of the weight matrix associated with the linear iteration, we show that each node can immediately calculate the consensus value as a linear combination of its own past values over at most D time-steps. We also show that each node can determine the coefficients for this linear combination in a decentralized manner. The proposed scheme has the potential to significantly reduce the time and communication required to reach consensus in distributed systems.