Fast Convergence of k-Opinion Undecided State Dynamics in the Population Protocol Model

We analyze the convergence of the k-opinion Undecided State Dynamics (USD) in the population protocol model. For k =2 opinions it is well known that the USD reaches consensus with high probability within O(n logn) interactions. Proving that the process also quickly solves the consensus problem for k > 2 opinions has remained open, despite analogous results for larger k in the related parallel gossip model. In this paper we prove such convergence: under mild assumptions on k and on the initial number of undecided agents we prove that the USD achieves plurality consensus within O(kn logn) interactions with high probability, regardless of the initial bias. Moreover, if there is an initial additive bias of at least Omega(root n log n) we prove that the initial plurality opinion wins with high probability, and if there is a multiplicative bias the convergence time is further improved. Note that this is the first result for k > 2 for the USD in the population protocol model. Furthermore, it is the first result for the unsynchronized variant of the USD with k > 2 which does not need any initial bias.