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M. Biely, P. Robinson, U. Schmid, M. Schwarz, K. Winkler:
"Gracefully Degrading Consensus and k-Set Agreement in Directed Dynamic Networks";
Report No. TUW-258404, 2016.

We study distributed agreement in synchronous directed dynamic networks, where an omniscient message adversary controls the presence/absence of communication links. We prove that consensus is impossible under a message adversary that guarantees weak connectivity only, and introduce eventually vertex-stable source components (VSSCs) as a means for circumventing this impossibility: A VSSC(k,d) message adversary guarantees that, eventually, there is an interval of d consecutive rounds where every communication graph contains at most k strongly connected components consisting of the same processes (with possibly varying interconnect topology), which have no incoming links from outside processes. We present a consensus algorithm that works correctly under a VSSC(1,4E+2) message adversary, where E is the dynamic causal network diameter. Our algorithm maintains local estimates of the communication graphs, and applies techniques for detecting network stability and univalent system configurations. Several related impossibility results and lower bounds, in particular, that neither a VSSC(1,E-1) message adversary nor a VSSC(2,infty) one allow to solve consensus, reveal that there is not much hope to deal with (much) stronger message adversaries here.
However, we show that gracefully degrading consensus, which degrades to general k-set agreement in case of unfavorable network conditions, allows to cope with stronger message adversaries: We provide a k-universal k-set agreement algorithm, where the number of system-wide decision values k is not encoded in the algorithm, but rather determined by the actual power of the message adversary in a run: Our algorithm guarantees at most k decision values under a VSSC(n,d)+MAJINF(k) message adversary, which combines VSSC(n,d) (with some small value of d, ensuring termination) with some information flow guarantee MAJINF(k) between certain VSSCs (ensuring k-agreement). Since related impossibility results reveal that a VSSC(k,d) message adversary is too strong for solving k-set agreement and that some information flow between VSSCs is mandatory for this purpose as well, our results provide a significant step towards the exact solvability/impossibility border of general k-set agreement in directed dynamic networks.
Finally, we relate (the eventually-forever-variants of) our message adversaries to failure detectors. It turns out that even though VSSC(1,infty) allows to solve consensus and to implement the Omega failure detector, it does not allow to implement Sigma. This contrasts the fact that, in asynchronous message-passing systems with a majority of process crashes, (Sigma,Omega) is a weakest failure detector for solving consensus. Similarly, although the message adversary VSSC(n,d)+MAJINF(k) allows to solve k-set agreement, it does not allow to implement the failure detector Sigma_k, which is known to be necessary for k-set agreement in asynchronous message-passing systems with a majority of process crashes. Consequently, it is not possible to adapt failure-detector-based algorithms to work in conjunction with our message adversaries.

Directed dynamic networks, consensus, k-set agreement, message adversaries, impossibility results, lower bounds, failure detectors

https://publik.tuwien.ac.at/files/publik_258404.pdf