Exact 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 on the underlying communication topology for achieving Byzantine consensus among these nodes in the presence of up to f Byzantine faults. We derive a necessary condition, and then 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 [8, 2, 6]: (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. 
Lewis Tseng and Nitin Vaidya
cs.DC