Space-Optimal Proportion Consensus with Population Protocols

Abstract

Population protocols provide a distributed computing model in which a set of finite-state identical agents cooperate through random interactions, between neighbors in the interaction graph, to collectively carry out a computation in a distributed setting. Population protocols have become very popular in various research areas, such as distributed computing, sensor or social networks, as well as chemistry and biology. A central task in this model is majority computation, in which agents need to reach an agreement on the leading one of two possible initial opinions. In this paper we consider a generalization of the majority problem, named proportion consensus, which asks for an agreement on the proportion of one opinion, between two possible views (say \(\mathcal A\) or \(\mathcal B\)). The objective is to reach a configuration where all the agents agree on a range \(\gamma _A \subseteq [0,1]\) which contains the value of the fraction \(\rho _A\) of agents that started with view \(\mathcal A\); the goal is to get the size of \(\gamma _A\) as small as possible while also minimizing the number of states adopted by agents. We provide a lower bound on the trade-off between precision \(\epsilon \) (the size of \(\gamma _A\)) and the number of states required by any population protocol that solves the proportion consensus problem. In particular, we show that in any population protocol that solves the proportion consensus problem with precision \(\epsilon \), any agent must have at least \(\lceil 2/\epsilon \rceil \) states. We also provide a population protocol that exactly solves the proportion consensus problem with precision \(\epsilon \) and \(6\lceil 1/(2\epsilon ) \rceil -1\) states. We show that in case of an arbitrary interaction graph our protocol requires \(O(n^6/\epsilon )\) interactions (which corresponds to the number of rounds in the sequential communication model) on any network with n agents. On complete interaction networks, the expected number of required interactions is \(O(n^2 \log n)\). Using the random matching communication model, the expected number of rounds, required to reach a consensus, decreases to \(O(\varDelta n^4/\epsilon )\) in case of arbitrary interaction networks (where \(\varDelta \) denotes the maximum degree among the agents in the network) and \(O(n \log n)\) for complete networks.