The Adversarial Noise Threshold for Distributed Protocols

We consider the problem of implementing distributed protocols, despite adversarial channel errors, on synchronous-messaging networks with arbitrary topology. In our first result we show that any n-party T -round protocol on an undirected communication network G can be compiled into a robust simulation protocol on a sparse (O(n) edges) subnetwork so that the simulation tolerates an adversarial error rate of Ω ( 1 n ) ; the simulation has a round complexity of O ( m log n n T ) , where m is the number of edges in G. (So the simulation is work-preserving up to a log factor.) The adversary’s error rate is within a constant factor of optimal. Given the error rate, the round complexity blowup is within a factor of O(k log n) of optimal, where k is the edge connectivity of G. We also determine that the maximum tolerable error rate on directed communication networks is Θ(1/s) where s is the number of edges in a minimum equivalent digraph. Next we investigate adversarial per-edge error rates, where the adversary is given an error budget on each edge of the network. We determine the limit for tolerable per-edge error rates on an arbitrary directed graph to within a factor of 2. However, the construction that approaches this limit has exponential round complexity, so we give another compiler, which transforms T -round protocols into O(mT )-round simulations, and prove that for polynomial-query black box compilers, the per-edge error rate tolerated by this last compiler is within a constant factor of optimal.