Reaching Approximate Byzantine Consensus with Multi-hop Communication
We address the problem of reaching approximate consensus in the presence of Byzantine faults in a synchronous system. We analyze iterative algorithms that maintain minimal state, and impose the constraint that in each iteration the nodes may only communicate with other nodes that are up to l hops away. For a given l, we prove a necessary and sufficient condition on the network structure for the existence of correct iterative algorithms that achieve approximate Byzantine consensus. We prove sufficiency of the condition by designing a correct algorithm, which uses a trim function based on a minimal messages cover property introduced in this paper. Our necessary and sufficient condition generalizes the tight condition identified in prior work for \(l=1\). For \(l\ge l^*\), where \(l^*\) is the length of a longest cycle-free path in the given network, our condition is equivalent to the necessary and sufficient conditions for exact consensus in undirected and directed networks both.
KeywordsApproximate byzantine consensus Iterative algorithm Synchronous system Incomplete network Bounded length communication paths
Unable to display preview. Download preview PDF.
- 2.Bnzit, F., Blondel, V., Thiran, P., Tsitsiklis, J., Vetterli, M.: Weighted gossip:Distributed averaging using non-doubly stochastic matrices. In: 2010 IEEE International Symposium on Information Theory Proceedings (ISIT), pp. 1753–1757, June 2010Google Scholar
- 3.Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C., et al.: Introduction toalgorithms, vol. 2. MIT Press Cambridge (2001)Google Scholar
- 6.Fischer, M.J., Lynch, N.A., Merritt, M.: Easy impossibility proofs for distributed consensus problems. In: Proceedings of the Fourth Annual ACM Symposium on Principles of Distributed Computing, PODC 1985, pp. 59–70. ACM, New York (1985)Google Scholar
- 8.LeBlanc, H.J., Zhang, H., Sundaram, S., Koutsoukos, X.: Consensus of multi-agent networks in the presence of adversaries using only local information. In: Proceedings of the 1st International Conference on High Confidence Networked Systems, HiCoNS 2012, pp. 1–10. ACM, New York (2012)Google Scholar
- 10.Su, L., Vaidya, N.: Reaching approximate byzantine consensus with multi-hop communication (2014). arXiv preprint arXiv:1411.5282
- 11.Tseng, L., Vaidya, N.: Iterative approximate consensus in the presence of byzantine link failures. In: Noubir, G., Raynal, M. (eds.) NETYS 2014. LNCS, vol. 8593, pp. 84–98. Springer, Heidelberg (2014) Google Scholar
- 12.Tseng, L., Vaidya, N.H.: Fault-tolerant consensus in directed graphs. In: Proceedings of the 2015 ACM Symposium on Principles of Distributed Computing. ACM (to appear, 2015)Google Scholar
- 13.Vaidya, N.H.: Matrix representation of iterative approximate byzantine consensus in directed graphs. CoRR, arXiv:1203.1888 (2012)
- 14.Vaidya, N.H., Tseng, L., Liang, G.: Iterative approximate byzantine consensus in arbitrary directed graphs. In: Proceedings of the 2012 ACM Symposium on Principles of Distributed Computing, pp. 365–374. ACM (2012)Google Scholar
- 15.West, D.B., et al.: Introduction to graph theory, vol. 2. Prentice Hall, Upper Saddle River (2001)Google Scholar
- 16.Zhang, H., Sundaram, S.: Robustness of information diffusion algorithms to locally bounded adversaries. In: American Control Conference (ACC 2012), pp. 5855–5861 (2012)Google Scholar