Byzantine Consensus in Directed Graphs∗

Consider a synchronous point-to-point network of n nodes connected by directed links, wherein each node has a binary input. This paper proves a tight necessary and sufficient condition for achieving Byzantine consensus among these nodes in the presence of up to f Byzantine faults. We derive two forms of the necessary condition. We provide a constructive proof of sufficiency by presenting a Byzantine consensus algorithm for directed graphs that satisfy the necessary condition. Prior work has developed analogous necessary and sufficient conditions for undirected graphs. It is known that, for undirected graphs, the following two conditions are together necessary and sufficient [4, 2]: (i) n ≥ 3f+1, and (ii) network connectivity greater than 2f . However, these conditions are not adequate to completely characterize Byzantine consensus in directed graphs. ∗This research is supported in part by Army Research Office grant W-911-NF-0710287. Any opinions, findings, and conclusions or recommendations expressed here are those of the authors and do not necessarily reflect the views of the funding agencies or the U.S. government. †This is a revised version of Exact Byzantine Consensus in Directed Graphs (Oct. 2012) of the previous report and version 3 of that report (on arxiv.org/abs/1208.5075).