Abstract
Fault-tolerant distributed consensus is a fundamental problem in secure distributed computing. In this work, we consider the problem of distributed consensus in directed graphs tolerating crash failures. Tseng and Vaidya (PODC’15) presented necessary and sufficient condition for the existence of consensus protocols in directed graphs. We improve the round and communication complexity of their protocol. Moreover, we prove that our protocol requires the optimal number of communication rounds, required by any protocol belonging to a restricted class of crash-tolerant consensus protocols in directed graphs.
P. Sarkar and G. Garimella—The research was conducted while these two authors were at the Indian Institute of Science and International Institute of Information Technology Bangalore, respectively.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
In undirected graphs, \(f+1\) node connectivity is both and necessary and sufficient for the existence of crash-tolerant consensus.
- 2.
In the binary consensus problem, the inputs of each node is a binary value. On the other hand in the multi-valued case, the inputs belong to a publicly known domain.
- 3.
This argument also shows that the binary consensus protocol of [15] with \(2f+2\) alternate min-max phases will work for the multi-valued case as well, with no modifications; this is because there will be always two consecutive crash-free phases.
- 4.
Note that a directed path will exist from \(v_i\) to \(v_j\) and from \(v_j\) to \(v_i\) in \(\mathbf G _{\textsf {F}}\) as both \(v_i\) and \(v_j\) are source nodes.
- 5.
The number of rounds in each phase need to be finite so that \(\varPi _{\mathsf {MinMax}}\) should terminate for each node.
- 6.
Note that if \(p_{\ell } = p_{f+1}\) then there is no phase \(p_{\ell + 2}\) in \(\varPi \). If phase \(p_{\ell + 2}\) exists in \(\varPi \) then it will have either d or \(d+1\) rounds because in \(\varPi \) each phase has at least d rounds.
References
Alchieri, E.A.P., Bessani, A.N., da Silva Fraga, J., Greve, F.: Byzantine consensus with unknown participants. In: Baker, T.P., Bui, A., Tixeuil, S. (eds.) OPODIS 2008. LNCS, vol. 5401, pp. 22–40. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-92221-6_4
Attiya, H., Welch, J.: Distributed Computing: Fundamentals. Simulation and Advanced Topics, Wiley series on Parallel and Distributed Computing (2004)
Bansal, P., Gopal, P., Gupta, A., Srinathan, K., Vasishta, P.K.: Byzantine agreement using partial authentication. In: Peleg, D. (ed.) DISC 2011. LNCS, vol. 6950, pp. 389–403. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-24100-0_38
Bertsekas, D.P., Tsitsiklis, J.N.: Parallel and Distributed Computation: Numerical Methods. Athena Scientific, Optimization and Neural Computation Series (1997)
Charron-Bost, B., Függer, M., Nowak, T.: Approximate consensus in highly dynamic networks: the role of averaging algorithms. In: Halldórsson, M.M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) ICALP 2015. LNCS, vol. 9135, pp. 528–539. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-47666-6_42
Choudhury, A., Garimella, G., Patra, A., Ravi, D., Sarkar, P.: Crash-tolerant consensus in directed graph revisited. Cryptology ePrint Archive, Report 2018/436 (2018)
Dolev, D.: The Byzantine generals strike again. J. Algorithms 3(1), 14–30 (1982)
Fischer, M.J., Lynch, N.A., Merritt, M.: Easy impossibility proofs for distributed consensus problems. In: PODC, pp. 59–70. ACM (1985)
Fitzi, M.: Generalized communication and security models in byzantine agreement. Ph.D. thesis, ETH Zurich (2002)
Jadbabaie, A., Lin, J., Morse, A.S.: Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Trans. Automat. Contr. 48(6), 988–1001 (2003)
Lynch, N.A.: Distributed Algorithms. Morgan Kaufmann (1996)
Pease, M., Shostak, R.E., Lamport, L.: Reaching agreement in the presence of faults. JACM 27(2), 228–234 (1980)
Schmid, U., Weiss, B., Keidar, I.: Impossibility results and lower bounds for consensus under link failures. SIAM J. Comput. 38(5), 1912–1951 (2009)
Tseng, L.: Recent results on fault-tolerant consensus in message-passing networks. In: Suomela, J. (ed.) SIROCCO 2016. LNCS, vol. 9988, pp. 92–108. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-48314-6_7
Tseng, L., Vaidya, N.H.: Crash-tolerant consensus in directed graphs. CoRR, abs/1412.8532, 2014. Conference version appeared as [16]
Tseng, L., Vaidya, N.H.: Fault-tolerant consensus in directed graphs. In: PODC, pp. 451–460. ACM (2015)
Tseng, L., Vaidya, N.H.: A note on fault-tolerant consensus in directed networks. SIGACT News 47(3), 70–91 (2016)
Acknowledgments
We thank the anonymous referees of SIROCCO 2018 for their helpful comments. The work of the first two authors is financially supported by Infosys foundation. The third author would like to acknowledge the financial support by SERB Women Excellence Award from Science and Engineering Research Board of India and INSPIRE Faculty Fellowship from Department of Science & Technology, India
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this paper
Cite this paper
Choudhury, A., Garimella, G., Patra, A., Ravi, D., Sarkar, P. (2018). Crash-Tolerant Consensus in Directed Graph Revisited (Extended Abstract). In: Lotker, Z., Patt-Shamir, B. (eds) Structural Information and Communication Complexity. SIROCCO 2018. Lecture Notes in Computer Science(), vol 11085. Springer, Cham. https://doi.org/10.1007/978-3-030-01325-7_10
Download citation
DOI: https://doi.org/10.1007/978-3-030-01325-7_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-01324-0
Online ISBN: 978-3-030-01325-7
eBook Packages: Computer ScienceComputer Science (R0)