Time-Optimal Self-Stabilizing Leader Election in Population Protocols

Abstract

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 $\Theta(n^2)$ 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 $\Omega(n)$ 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 $O(\log n)$, but using at least exponential states (a quasi-polynomial 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 $1,\ldots,n$.