Optimal Proportion Computation with Population Protocols

The computational model of population protocols is a formalism that allows the analysis of properties emerging from simple and pairwise interactions among a very large number of anonymous finite-state agents. Significant work has been done so far to determine which problems are solvable in this model and at which cost in terms of states used by the agents and time needed to converge. The problem tackled in this paper is the population problem: each agent starts independently from each other in one of two states, say A or B, and the objective is for each agent to determine the proportion of agents that initially started in state A, assuming that each agent only uses a finite set of states, and does not know the number n of agents. We pro pose a solution which guarantees that in presence of a uniform probabilistic scheduler every agent outputs the population proportion with any precision epsilon is an element of (0,1) with any high probability after having interacted O( log n) times. The number of states maintained by every agent is optimal and is equal to 2[3/(4 epsilon)] + 1. Finally, we show that our solution is optimal in time and space to solve the counting problem, a generalization of the proportion problem. Finally, simulation results illustrate our theoretical analysis.