Gracefully Degrading Consensus and $k$-Set Agreement in Directed Dynamic Networks (1501.02716v1)

Abstract: We study distributed agreement in synchronous directed dynamic networks, where an omniscient message adversary controls the availability of communication links. We prove that consensus is impossible under a message adversary that guarantees weak connectivity only, and introduce vertex-stable root components (VSRCs) as a means for circumventing this impossibility: A VSRC(k, d) message adversary guarantees that, eventually, there is an interval of $d$ consecutive rounds where every communication graph contains at most $k$ strongly (dynamic) connected components consisting of the same processes, which have at most outgoing links to the remaining processes. We present a consensus algorithm that works correctly under a VSRC(1, 4H + 2) message adversary, where $H$ is the dynamic causal network diameter. On the other hand, we show that consensus is impossible against a VSRC(1, H - 1) or a VSRC(2, $\infty$) message adversary, revealing that there is not much hope to deal with stronger message adversaries. However, we show that gracefully degrading consensus, which degrades to general $k$-set agreement in case of unfavourable network conditions, is feasible against stronger message adversaries: We provide a $k$-uniform $k$-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 VSRC(n, d) + MAJINF(k) message adversary, which combines VSRC(n, d) (for some small $d$, ensuring termination) with some information flow guarantee MAJINF(k) between certain VSRCs (ensuring $k$-agreement). Our results provide a significant step towards the exact solvability/impossibility border of general $k$-set agreement in directed dynamic networks.