Selecting a Leader in a Network of Finite State Machines

This paper studies a variant of the \emph{leader election} problem under the \emph{stone age} model (Emek and Wattenhofer, PODC 2013) that considers a network of $n$ randomized finite automata with very weak communication capabilities (a multi-frequency asynchronous generalization of the \emph{beeping} model's communication scheme). Since solving the classic leader election problem is impossible even in more powerful models, we consider a relaxed variant, referred to as \emph{$k$-leader selection}, in which a leader should be selected out of at most $k$ initial candidates. Our main contribution is an algorithm that solves $k$-leader selection for bounded $k$ in the aforementioned stone age model. On (general topology) graphs of diameter $D$, this algorithm runs in $\tilde{O}(D)$ time and succeeds with high probability. The assumption that $k$ is bounded turns out to be unavoidable: we prove that if $k = \omega (1)$, then no algorithm in this model can solve $k$-leader selection with a (positive) constant probability.