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 $\Theta(n^2)$ 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 $1,\ldots,n$.