The match-maker: Constant-space distributed majority via random walks

We propose and analyze here a simple protocol for consensus on the majority color in networks whose nodes are initially one of two colors. Our protocol guarantees that, if a majority exists, then eventually each node learns of the majority color. Our protocol requires only 2 bits of memory per node and uses a simple token message, of also 2 bits size, that performs random walks. We show correctness of our protocol for any connected graph (even unknown to the nodes) and even for a natural class of dynamic graphs. We show upper and lower bounds on the convergence time of our protocol.We discuss termination and we also provide a variant of our protocol which the token uses a counter that can count only up to √n log n, where n is the number of network nodes. Our basic (memoryless) protocol takes only O(n log n) expected time on the clique which surprisingly does not deviate from the cover time of the random walk, and O(n²m) time on any connected undirected network of m edges and this bound is met from below by an argument on the line. Finally, we also consider random walks that can count the difference of colors and we show upper bounds on the counter value by using coupling arguments.