Self-stabilizing Leader Election in Population Protocols over Arbitrary Communication Graphs

This paper considers the fundamental problem of \emph{self-stabilizing leader election} ($\mathcal{SSLE}$) in the model of \emph{population protocols}. In this model, an unknown number of asynchronous, anonymous and finite state mobile agents interact in pairs over a given communication graph. $\mathcal{SSLE}$ has been shown to be impossible in the original model. This impossibility can been circumvented by a modular technique augmenting the system with an \emph{oracle} - an external module abstracting the added assumption about the system. Fischer and Jiang have proposed solutions to $\mathcal{SSLE}$, for complete communication graphs and rings, using an oracle $\Omega?$, called the \emph{eventual leader detector}. In this work, we present a solution for arbitrary graphs, using a \emph{composition} of two copies of $\Omega?$. We also prove that the difficulty comes from the requirement of self-stabilization, by giving a solution without oracle for arbitrary graphs, when an uniform initialization is allowed. Finally, we prove that there is no self-stabilizing \emph{implementation} of $\Omega?$ using $\mathcal{SSLE}$, in a sense we define precisely