Breaking the Ω (n) barrier: Fast consensus under a late adversary

Peter Robinson, Christian Scheideler, Alexander Setzer

Research output: Chapter in Book/Report/Conference proceedingConference contribution

4 Scopus citations


We study the consensus problem in a synchronous distributed system of n nodes under an adaptive adversary that has a slightly outdated view of the system and can block all incoming and outgoing communication of a constant fraction of the nodes in each round. Motivated by a result of Ben-Or and Bar-Joseph (1998), showing that any consensus algorithm that is resilient against a linear number of crash faults requires Ω (n) rounds in an n-node network against an adaptive adversary, we consider a late adaptive adversary, who has full knowledge of the network state at the beginning of the previous round and unlimited computational power, but is oblivious to the current state of the nodes. Our main contributions are randomized distributed algorithms that achieve consensus with high probability among all except a small constant fraction of the nodes (i.e., “almost-everywhere”) against a late adaptive adversary who can block up to ϵn nodes in each round, for a small constant ϵ > 0. Our first protocol achieves binary almost-everywhere consensus and also guarantees a decision on the majority input value, thus ensuring plurality consensus. We also present an algorithm that achieves the same time complexity for multi-value consensus. Both of our algorithms succeed in O(log n) rounds with high probability, thus showing an exponential gap to the Ω (n) lower bound of Ben-Or and Bar-Joseph for strongly adaptive crash-failure adversaries, which can be strengthened to Ω(n) when allowing the adversary to block nodes instead of permanently crashing them. Our algorithms are scalable to large systems as each node contacts only an (amortized) constant number of peers in each communication round. We show that our algorithms are optimal up to constant (resp. sub-logarithmic) factors by proving that every almost-everywhere consensus protocol takes Ω(logd n) rounds in the worst case, where d is an upper bound on the number of communication requests initiated per node in each round. We complement our theoretical results with an experimental evaluation of the binary almost-everywhere consensus protocol revealing a short convergence time even against an adversary blocking a large fraction of nodes.

Original languageEnglish (US)
Title of host publicationSPAA 2018 - Proceedings of the 30th ACM Symposium on Parallelism in Algorithms and Architectures
PublisherAssociation for Computing Machinery
Number of pages10
ISBN (Electronic)9781450357999
StatePublished - Jul 11 2018
Externally publishedYes
Event30th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA 2018 - Vienna, Austria
Duration: Jul 16 2018Jul 18 2018

Publication series

NameAnnual ACM Symposium on Parallelism in Algorithms and Architectures


Conference30th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA 2018


  • Adaptive adversary
  • Almost-everywhere consensus
  • Probabilistic analysis
  • Randomized Algorithm

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science
  • Hardware and Architecture


Dive into the research topics of 'Breaking the Ω (n) barrier: Fast consensus under a late adversary'. Together they form a unique fingerprint.

Cite this