On the Power of Oracle Omega? for Self-Stabilizing Leader Election in Population Protocols
Résumé
This paper considers the fundamental problem of
self-stabilizing leader election (SSLE) in the model of
population protocols. In this model an unknown number of
asynchronous, anonymous and finite state mobile agents interact in
pairs. SSLE has been shown to be impossible in this model without
additional assumptions. This impossibility can be circumvented for
instance by augmenting the system with an oracle (an external
module providing supplementary information useful to solve a
problem). Fischer and Jiang have proposed solutions to SSLE, for
complete communication graphs and rings, using the oracle $\Omega?$,
called the eventual leader detector. In this paper, we
investigate the power of $\Omega?$ on larger families of graphs. We
present two important results.
Our first result states that $\Omega?$ is powerful enough to allow
solving SSLE over arbitrary communication graphs of bounded
degree. Our second result states that, $\Omega?$ is the weakest (in
the sense of Chandra, Hadzilacos and Toueg) for solving SSLE over rings. We
also prove that this result does not extend to all graphs; in
particular not to the family of arbitrary graphs of bounded degree.
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