Population protocols are a popular model of distributed computing , in which n agents with limited local state interact randomly, and cooperate to collectively compute global predicates. Inspired by recent developments in DNA programming, an extensive series of papers, across different communities, has examined the com-putability and complexity characteristics of this model. Majority, or consensus, is a central task in this model, in which agents need to collectively reach a decision as to which one of two states A or B had a higher initial count. Two metrics are important: the time that a protocol requires to stabilize to an output decision, and the state space size that each agent requires to do so. It is known that majority requires Ω(log log n) states per agent to allow for fast (poly-logarithmic time) stabilization, and that O(log 2 n) states are sufficient. Thus, there is an exponential gap between the space upper and lower bounds for this problem. This paper addresses this question. On the negative side, we provide a new lower bound of Ω(log n) states for any protocol which stabilizes in O(n 1−c) expected time, for any constant c > 0. This result is conditional on monotonicity and output assumptions, satisfied by all known protocols. Technically, it represents a departure from previous lower bounds, in that it does not rely on the existence of dense configurations. Instead, we introduce a new generalized surgery technique to prove the existence of incorrect executions for any algorithm which would contradict the lower bound. Subsequently, our lower bound also applies to general initial configurations, including ones with a leader. On the positive side, we give a new algorithm for majority which uses O(log n) states, and stabilizes in O(log 2 n) expected time. Central to the algorithm is a new leaderless phase clock technique , which allows agents to synchronize in phases of Θ(n log n) consecutive interactions using O(log n) states per agent, exploiting a new connection between population protocols and power-of-two-choices load balancing mechanisms. We also employ our phase * Dan

Abstract. This paper considers the fundamental problem of self-stabilizing leader election (SSLE) in the model of population protocols. In this model, an unknown number of asynchronous, anonymous and nite state mobile agents interact in pairs over a given communication graph. SSLE has been shown to be impossible in the original model. This impossibility can been circumvented by a modular technique augmenting the system with an oracle -an external module abstracting the added assumption about the system. Fischer and Jiang have proposed solutions to SSLE, for complete communication graphs and rings, using an oracle Ω?, called the eventual leader detector. In this work, we present a solution for arbitrary graphs, using a composition of two copies of Ω?. We also prove that the di culty comes from the requirement of self-stabilization, by giving a solution without oracle for arbitrary graphs, when an uniform initialization is allowed. Finally, we prove that there is no self-stabilizing implementation of Ω? using SSLE, in a sense we de ne precisely.