Oracles for Self-Stabilizing Leader Election in Population Protocols
Résumé
This paper considers the fundamental problem of \emph{self-stabilizing leader election} ($SSLE$) in the model of \emph{population protocols}. In this model an unknown number of asynchronous, anonymous and finite state mobile agents interact in pairs. $SSLE$ was shown to be impossible in this model without additional assumptions. This impossibility can be circumvented for instance by augmenting the system with an oracle, like the eventual \emph{leader detector} $\Omega?$ of Fischer and Jiang, who presented a uniform protocol solving $SSLE$ with the help of $\Omega?$ on complete communication graphs and rings. In this paper, we extend their results. Our first contribution is a precise framework for dealing with oracles. This framework is independent of the notion of real time. Such a design choice avoids some known problems of traditional real time based frameworks. We then formally define $\Omega?$ as well as a \emph{stronger} oracle $\Omega\$$ and a \emph{weaker} one $W\Omega?$. The comparison between the oracles is based on the notion of \emph{implementation}. We prove that $SSLE$ can be implemented with $\Omega\$$ over weakly connected communication graphs, with $W\Omega?$ over oriented trees and with $\Omega?$ over weakly connected communication graphs of bounded degree. Finally, we show that $\Omega?$ can be implemented using $SSLE$ over rings, proving their equivalence. All these results allow to establish relations between the different oracles and $SSLE$, which we summarize in a figure.
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