Time-optimal self-stabilizing leader election in population protocols

We consider the standard population protocol model, where ( a priori ) indistinguishable and anonymous agents interact in pairs according to uniformly random scheduling. The self-stabilizing leader election problem requires the protocol to converge on a single leader agent from any possible initial configuration. We initiate the study of time complexity of population protocols solving this problem in its original setting: with probability 1, in a complete communication graph. The only previously known protocol by Cai, Izumi, and Wada [Theor. Comput. Syst. 50] runs in expected parallel time $$\Theta (n^2)$$ Θ ( n 2 ) and has the optimal number of n states in a population of n agents. The existing protocol has the additional property that it becomes silent, i.e., the agents’ states eventually stop changing. Observing that any silent protocol solving self-stabilizing leader election requires $$\Omega (n)$$ Ω ( n ) expected parallel time, we introduce a silent protocol that uses optimal O ( n ) parallel time and states. Without any silence constraints, we show that it is possible to solve self-stabilizing leader election in asymptotically optimal expected parallel time of $$O(\log n)$$ O ( log n ) , but using at least exponential states (a quasipolynomial number of bits). All of our protocols (and also that of Cai et al.) work by solving the more difficult ranking problem: assigning agents the ranks $$1,\ldots ,n$$ 1 , … , n .