Easy impossibility proofs for k-set agreement in message passing systems

Martin Biely, Peter Robinson, Ulrich Schmid

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

10 Scopus citations


Despite of being quite similar (agreement) problems, 1-set agreement (consensus) and general k-set agreement require surprisingly different techniques for proving the impossibility in asynchronous systems with crash failures: Rather than the relatively simple bivalence arguments as in the impossibility proof for consensus in the presence of a single crash failure, known proofs for the impossibility of k-set agreement in shared memory systems with f ≥ k > 1 crash failures use algebraic topology or a variant of Sperner's Lemma. In this paper, we present a generic theorem for proving the impossibility of k-set agreement in various message passing settings, which is based on a reduction to the consensus impossibility in a certain subsystem resulting from a partitioning argument. We demonstrate the broad applicability of our result by exploring the possibility/impossibility border of k-set agreement in several message-passing system models: (i) asynchronous systems with crash failures, (ii) partially synchronous processes with (initial) crash failures, and, most importantly, (iii) asynchronous systems augmented with failure detectors. In (i), (ii), and (iii), the impossibility part is an instantiation of our main theorem, whereas the possibility of achieving k-set agreement in (ii) follows by generalizing the consensus algorithm for initial crashes by Fisher, Lynch and Patterson. In (iii), applying our technique reveals the exact border for the parameter k where k-set agreement is solvable with the failure detector class (∑k, Ωk) 1≤k≤n-1 of Bonnet and Raynal. As ∑k was shown to be necessary for solving k-set agreement, this result yields new insights on the quest for the weakest failure detector.

Original languageEnglish (US)
Title of host publicationPrinciples of Distributed Systems - 15th International Conference, OPODIS 2011, Proceedings
Number of pages14
StatePublished - 2011
Externally publishedYes
Event15th International Conference on Principles of Distributed Systems, OPODIS 2011 - Toulouse, France
Duration: Dec 13 2011Dec 16 2011

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume7109 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference15th International Conference on Principles of Distributed Systems, OPODIS 2011


  • consensus
  • failure detectors
  • impossibility proofs
  • k-set agreement

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Computer Science


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