Simple and fast approximate counting and leader election in populations

We study leader election and population size counting for population protocols: networks of finite-state anonymous agents that interact randomly. We provide simple protocols for approximate counting of the size of the population and for leader election. We show a protocol for leader election that terminates in O(log2⁡nlog⁡m) parallel time, where m is a parameter that belongs to Ω(log⁡n) and O(n), using O(max⁡{log⁡m,log⁡log⁡n}) bits. By adjusting m between log⁡n and n, we obtain a leader election protocol whose time and space can be smoothly traded off between O(log2⁡nlog⁡log⁡n) to O(log⁡n) time and O(log⁡log⁡n) to O(log⁡n) bits. We also give a protocol which provides a constant factor approximation of log⁡n of the population size n, or an upper bound ne which is at most na for some constant a>1. This protocol assumes the existence of a unique leader and stabilizes in Θ(log⁡n) parallel time.