We consider the model of population protocols introduced by Angluin et al. (Computation in networks of passively mobile finite-state sensors, pp. 290-299. ACM, New York, 2004), in which anonymous finite-state agents stably compute a predicate of the multiset of their inputs via two-way interactions in the family of all-pairs communication networks. We prove that all predicates stably computable in this model (and certain generalizations of it) are semilinear, answering a central open question about the power of the model. Removing the assumption of two-way interaction, we also consider several variants of the model in which agents communicate by anonymous message-passing where the recipient of each message is chosen by an adversary and the sender is not identified to the recipient. These one-way models are distinguished by whether messages are delivered immediately or after a delay, whether a sender can record that it has sent a message, and whether a recipient can queue James Aspnes was supported in part by NSF grants CNS-0305258 and incoming messages, refusing to accept new messages until it has had a chance to send out messages of its own. We characterize the classes of predicates stably computable in each of these one-way models using natural subclasses of the semilinear predicates.

Fast algorithms are presented for performing computations in a probabilistic population model. This is a variant of the standard population protocol model-in which finite-state agents interact in pairs under the control of an adversary scheduler-where all pairs are equally likely to be chosen for each interaction. It is shown that when a unique leader agent is provided in the initial population, the population can simulate a virtual register machine in which standard arithmetic operations like comparison, addition, subtraction, multiplication, and division can be simulated in O(n log 4 n) interactions with high probability. Applications include a reduction of the cost of computing a semilinear predicate to O(n log 4 n) interactions from the previously best-known bound of O(n 2 log n) interactions and simulation of a LOGSPACE Turing machine using the same O(n log 4 n) interactions per step. These bounds on interactions translate into O(log 4 n) time per step in a natural model in which each agent participates in an expected Θ(1) interactions per time unit. The central method is the extensive use of epidemics to propagate information from and to the leader, combined with an epidemic-based phase clock used to detect when these epidemics are likely to be complete.

Population protocols are a popular model of distributed computing, in which randomly-interacting agents with little computational power cooperate to jointly perform computational tasks. Inspired by developments in molecular computation, and in particular DNA computing, recent algorithmic work has focused on the complexity of solving simple yet fundamental tasks in the population model, such as leader election (which requires convergence to a single agent in a special "leader" state), and majority (in which agents must converge to a decision as to which of two possible initial states had higher initial count). Known results point towards an inherent trade-off between the time complexity of such algorithms, and the space complexity, i.e. size of the memory available to each agent.In this paper, we explore this trade-off and provide new upper and lower bounds for majority and leader election. First, we prove a unified lower bound, which relates the space available per node with the time complexity achievable by a protocol: for instance, our result implies that any protocol solving either of these tasks for n agents using O(log log n) states must take Ω(n/polylogn) expected time. This is the first result to characterize time complexity for protocols which employ super-constant number of states per node, and proves that fast, poly-logarithmic running times require protocols to have relatively large space costs.On the positive side, we give algorithms showing that fast, poly-logarithmic convergence time can be achieved using O(log 2 n) space per node, in the case of both tasks. Overall, our results highlight a time complexity separation between O(log log n) and Θ(log 2 n) state space size for both majority and leader election in population protocols, and introduce new techniques, which should be applicable more broadly.

We describe and analyze a 3-state one-way population protocol to compute approximate majority in the model in which pairs of agents are drawn uniformly at random to interact. Given an initial configuration of x's, y's and blanks that contains at least one non-blank, the goal is for the agents to reach consensus on one of the values x or y. Additionally, the value chosen should be the majority non-blank initial value, provided it exceeds the minority by a sufficient margin. We prove that with high probability n agents reach consensus in O(n log n) interactions and the value chosen is the majority provided that its initial margin is at least ω( √ n log n). This protocol has the additional property of tolerating Byzantine behavior in o( √ n) of the agents, making it the first known population protocol that tolerates Byzantine agents.

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