Tight Bounds for Asymptotic and Approximate Consensus

Agreeing on a common value among set of agents is fundamental problem in distributed computing, which occurs several variants: In contrast to exact consensus, approximate variants are studied systems where agreement not possible or required, e.g., human-made control and the analysis natural systems, such as bird flocking opinion dynamics. We study time complexity two classical problems: non-terminating asymptotic consensus terminating consensus. Asymptotic requires repeatedly their outputs that converge within convex hull initial values; eventually stop setting outputs, must then lie predefined distance each other. prove tight lower bounds contraction ratios algorithms subject oblivious message adversaries, from we deduce algorithms. particular, obtained show optimality presented by Charron-Bost et al. (ICALP’16) for certain including strongest adversary solvable. As corollary also obtain asymptotically asynchronous model with crashes. Central lower-bound proofs an extended notion valency, reachable limits algorithm starting given configuration. further relate topological properties valencies solvability shedding some light relation these three problems dynamic networks.