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 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 (an external module providing supplementary information useful to solve a problem). Thus, Fischer and Jiang have proposed solutions to SSLE, for complete communication graphs and rings, using an oracle Omega?, called the eventual leader detector. In this paper, we extend their results. First, we show that Omega? is powerful enough to allow solving SSLE also over arbitrary communication graphs of bounded degree. Then, over rings, we show that Omega? has the minimum necessary power (or provides the minimum supplementary information) to solve SSLE. We prove this by implementing Omega? using SSLE. This result, together with the previous one, provides an equivalence between Omega? and SSLE. This equivalence also means that Omega? is the weakest (i.e., necessary and sucient) oracle to solve SSLE over rings. Moreover, we conjecture that the result over rings can be extended to regular graphs of bounded diameter. On the negative side, we prove that the equivalence between Omega? and SSLE does not hold on all graphs; in particular, it does not hold on complete graphs and on graphs of bounded degree.
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