Consensus in Directed Dynamic Networks with Short-Lived Stability.

We consider the problem of deterministically solving consensus in a synchronous dynamic network with unreliable, unidirectional point-to-point links, which are under the control of a message adversary. As the nodes in practical dynamic networks typically start in a more or less uncoordinated way, and only eventually reach normal operating conditions, we focus on eventually stabilizing message adversaries here: Whereas the communication graphs of an unknown number of initial rounds may be quite arbitrary, a "stable period" of x consecutive rounds with a single common root component must eventually occur. Earlier work has established that such strongly connected components without incoming edges, which consist of the same set of nodes (with possibly varying interconnect topology) during at least x=2D+1 rounds allow to solve consensus. Herein, D is an upper bound on the number of rounds required by any root member to reach every other node in the system. In this paper, we complete the characterization of consensus solvability in this model by also considering short-lived stabilization and non-uniform algorithms: We prove that it is impossible to solve consensus for 0 < x < D+1 and D+1 < x < 2D+1 if at most a constant-factor upper bound on the number of nodes n is known. Surprisingly, though, consensus can be solved for x=D+1 by means of a novel non-uniform algorithm presented and proved correct in this paper.