Efficient 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. In this model, the only previously known protocol solving the self-stabilizing leader election problem 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. This 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 runs in asymptotically optimal $O(n)$ expected parallel time with an exponential number of states, as well as a protocol with a slightly worse expected time complexity of $O(n\log n)$ but with the asymptotically optimal $O(n)$ states. Without any silence or state space constraints, we show that it is possible to solve self-stabilizing leader election in asymptotically optimal expected parallel time of $O(\log n)$. All of our protocols (and also that of Cai et al.) work by solving the more difficult ranking problem: assigning agents the ranks $1,\ldots,n$.