The Step Complexity of Multidimensional Approximate Agreement

Authors Hagit Attiya , Faith Ellen

Thumbnail PDF


  • Filesize: 0.62 MB
  • 12 pages

Document Identifiers

Author Details

Hagit Attiya
  • Department of Computer Science, Technion, Israel
Faith Ellen
  • Department of Computer Science, University of Toronto, Canada


We thank Sasho Nikolov for useful discussion. We also appreciate the helpful comments of the anonymous reviewers.

Cite AsGet BibTex

Hagit Attiya and Faith Ellen. The Step Complexity of Multidimensional Approximate Agreement. In 26th International Conference on Principles of Distributed Systems (OPODIS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 253, pp. 6:1-6:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


Approximate agreement allows a set of n processes to obtain outputs that are within a specified distance ε > 0 of one another and within the convex hull of the inputs. When the inputs are real numbers, there is a wait-free shared-memory approximate agreement algorithm [Moran, 1995] whose step complexity is in O(n log(S/ε)), where S, the spread of the inputs, is the maximal distance between inputs. There is another wait-free algorithm [Schenk, 1995] that avoids the dependence on n and achieves O(log(M/ε)) step complexity where M, the magnitude of the inputs, is the absolute value of the maximal input. This paper considers whether it is possible to obtain an approximate agreement algorithm whose step complexity depends on neither n nor the magnitude of the inputs, which can be much larger than their spread. On the negative side, we prove that Ω(min{(log M)/(log log M), (√log n)/(log log n)}) is a lower bound on the step complexity of approximate agreement, even when the inputs are real numbers. On the positive side, we prove that a polylogarithmic dependence on n and S/ε can be achieved, by presenting an approximate agreement algorithm with O(log n (log n + log(S/ε))) step complexity. Our algorithm works for multidimensional domains. The step complexity can be further restricted to be in O(min{log n (log n + log (S/ε)), log(M/ε)}) when the inputs are real numbers.

Subject Classification

ACM Subject Classification
  • Theory of computation → Shared memory algorithms
  • Theory of computation → Distributed algorithms
  • approximate agreement
  • conflict detection
  • shared memory
  • wait-freedom
  • step complexity


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Ittai Abraham, Yonatan Amit, and Danny Dolev. Optimal resilience asynchronous approximate agreement. In Proceedings of the 8th International Conference On Principles Of Distributed Systems (OPODIS), pages 229-239, 2004. Google Scholar
  2. Yehuda Afek, Hagit Attiya, Danny Dolev, Eli Gafni, Michael Merritt, and Nir Shavit. Atomic snapshots of shared memory. Journal of the ACM, 40(4):873-890, 1993. Google Scholar
  3. James Aspnes and Faith Ellen. Tight bounds for adopt-commit objects. Theory of Computing Systems, 55(3):451-474, 2014. Google Scholar
  4. Hagit Attiya and Faith Ellen. Impossibility Results for Distributed Computing. Synthesis Lectures on Distributed Computing Theory. Morgan & Claypool Publishers, 2014. Google Scholar
  5. Hagit Attiya, Nancy Lynch, and Nir Shavit. Are wait-free algorithms fast? Journal of the ACM, 41(4):725-763, 1994. Google Scholar
  6. Hagit Attiya and Ophir Rachman. Atomic snapshots in o (n log n) operations. SIAM Journal on Computing, 27(2):319-340, 1998. Google Scholar
  7. Danny Dolev, Nancy Lynch, Shlomit Pinter, Eugene Stark, and William Weihl. Reaching approximate agreement in the presence of faults. Journal of the ACM, 33(3):499-516, 1986. Google Scholar
  8. Faith Ellen, Rati Gelashvili, and Leqi Zhu. Revisionist simulations: A new approach to proving space lower bounds. In Proceedings of the 37th ACM Symposium on Principles of Distributed Computing (PODC), pages 61-70, 2018. Google Scholar
  9. Matthias Függer and Thomas Nowak. Fast multidimensional asymptotic and approximate consensus. In Proceedings of the 32nd International Symposium on DIStributed Computing (DISC), pages 27:1-27:16, 2018. Google Scholar
  10. Matthias Függer, Thomas Nowak, and Manfred Schwarz. Tight bounds for asymptotic and approximate consensus. Journal of the ACM, 68(6):46:1-46:35, 2021. Google Scholar
  11. Maurice Herlihy. Impossibility results for asynchronous PRAM. In Proceedings of the 3rd Annual ACM Symposium on Parallel Algorithms and Architectures (SPAA), pages 327-336, 1991. Google Scholar
  12. Gunnar Hoest and Nir Shavit. Toward a topological characterization of asynchronous complexity. SIAM Journal on Computing, 36(2):457-497, 2006. Google Scholar
  13. Michiko Inoue, Toshimitsu Masuzawa, Wei Chen, and Nobuki Tokura. Linear-time snapshot using multi-writer multi-reader registers. In Proceedings of the 8th International Workshop on Distributed Algorithms (WDAG), pages 130-140, 1994. Google Scholar
  14. Hammurabi Mendes and Maurice Herlihy. Multidimensional approximate agreement in Byzantine asynchronous systems. In Proceedings of the 45th Annual ACM Symposium on Theory of Computing (STOC), pages 391-400, 2013. Google Scholar
  15. Hammurabi Mendes, Maurice Herlihy, Nitin Vaidya, and Vijay K. Garg. Multidimensional agreement in Byzantine systems. Distributed Computing, 28(6):423-441, 2015. Google Scholar
  16. Shlomo Moran. Using approximate agreement to obtain complete disagreement: The output structure of input-free asynchronous computations. In Proceedings of the 3rd Israel Symposium on the Theory of Computing and Systems (ISTCS), pages 251-257, 1995. Google Scholar
  17. Eric Schenk. Faster approximate agreement with multi-writer registers. In Proceedings of the 36th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 714-723, 1995. Google Scholar
  18. Nitin H. Vaidya and Vijay K. Garg. Byzantine vector consensus in complete graphs. In Proceedings of the 32nd Annual ACM Symposium on Principles of Distributed Computing (PODC), pages 65-73, 2013. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail