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.
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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.
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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
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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
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