# On the optimal space complexity of consensus for anonymous processes

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## Abstract

The optimal space complexity of consensus in asynchronous shared memory was an open problem for two decades. For a system of *n* processes, no algorithm using a sublinear number of registers is known. Up until very recently, the best known lower bound due to Fich, Herlihy, and Shavit was \({\varOmega }(\sqrt{n})\) registers. Fich, Herlihy, and Shavit first proved their lower bound for the special case of the problem where processes are anonymous (i.e. they run the same algorithm) and then extended it to the general case. In this paper we close the gap for the anonymous case of the problem. We show that any consensus algorithm from read–write registers for anonymous processes that satisfies nondeterministic solo termination has to use \({\varOmega }(n)\) registers in some execution. This implies an \({\varOmega }(n)\) lower bound on the space complexity of deterministic obstruction-free and randomized wait-free consensus, matching the upper bound. We introduce new techniques for marshalling anonymous processes and their executions, in particular, the concepts of *leader–follower pairs* and *reserving executions*, that play a critical role in the lower bound argument and will hopefully be more generally applicable.

## Keywords

Consensus Anonymous processes Space complexity Registers## Notes

### Acknowledgements

The author would like to thank Nir Shavit, Michael Coulombe, Dan Alistarh, Faith Ellen and Leqi Zhu for helpful conversations and feedback, and anonymous reviewers for their excellent comments.

## References

- 1.Abrahamson, K.: On achieving consensus using a shared memory. In: Proceedings of the 7th Annual ACM Symposium on Principles of Distributed Computing, pp. 291–302. ACM (1988)Google Scholar
- 2.Aspnes, J., Herlihy, M.: Fast randomized consensus using shared memory. J. Algorithms
**11**(3), 441–461 (1990)MathSciNetCrossRefzbMATHGoogle Scholar - 3.Attiya, H., Ellen, F.: Impossibility Results for Distributed Computing, vol. 5. Morgan & Claypool Publishers, San Rafael (2014)zbMATHGoogle Scholar
- 4.Ben-Or, M.: Another advantage of free choice (extended abstract): completely asynchronous agreement protocols. In: Proceedings of the 2nd Annual ACM Symposium on Principles of Distributed Computing, pp. 27–30. ACM (1983)Google Scholar
- 5.Bouzid, Z., Raynal, M., Sutra, P.: Brief announcement: Anonymous obstruction-free \((n, k)\)-set agreement with \(n- k+ 1\) atomic read/write registers. In: Proceedings of the 29th International Symposium on Distributed Computing, p. 669. Springer (2015)Google Scholar
- 6.Bowman, J.: Obstruction-free snapshot, obstruction-free consensus, and fetch-and-add modulo k. Tech. Rep. TR2011-681, Dartmouth College, Computer Science, Hanover, NH (2011)Google Scholar
- 7.Ellen, F., Gelashvili, R., Zhu, L.: Revisionist simulations: a new approach to proving space lower bounds. ArXiv preprint arXiv:1711.02455 (2018)
- 8.Fich, F., Herlihy, M., Shavit, N.: On the space complexity of randomized synchronization. J. ACM (JACM)
**45**(5), 843–862 (1998)MathSciNetCrossRefzbMATHGoogle Scholar - 9.Fischer, M.J., Lynch, N.A., Paterson, M.S.: Impossibility of distributed consensus with one faulty process. J. ACM (JACM)
**32**(2), 374–382 (1985)MathSciNetCrossRefzbMATHGoogle Scholar - 10.Giakkoupis, G., Helmi, M., Higham, L., Woelfel, P.: An \({\cal O\it }(\sqrt{n})\) space bound for obstruction-free leader election. In: Proceedings of the 27th International Symposium on Distributed Computing, pp. 46–60. Springer (2013)Google Scholar
- 11.Giakkoupis, G., Helmi, M., Higham, L., Woelfel, P.: Test-and-set in optimal space. In: Proceedings of the 47th Annual ACM on Symposium on Theory of Computing, pp. 615–623. ACM (2015)Google Scholar
- 12.Guerraoui, R., Ruppert, E.: What can be implemented anonymously? In: Proceedings of the 19th International Symposium on Distributed Computing, pp. 244–259. Springer (2005)Google Scholar
- 13.Saks, M., Shavit, N., Woll, H.: Optimal time randomized consensus—making resilient algorithms fast in practice. In: Proceedings of the 2nd Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 351–362. Society for Industrial and Applied Mathematics (1991)Google Scholar
- 14.Styer, E., Peterson, G.L.: Tight bounds for shared memory symmetric mutual exclusion problems. In: Proceedings of the 8th Annual ACM Symposium on Principles of Distributed Computing, pp. 177–191. ACM (1989)Google Scholar
- 15.Zhu, L.: Brief announcement: Tight space bounds for memoryless anonymous consensus. In: Proceedings of the 29th International Symposium on Distributed Computing, p. 665. Springer (2015)Google Scholar
- 16.Zhu, L.: A tight space bound for consensus. In: Proceedings of the 48th Annual ACM Symposium on Theory of Computing, pp. 345–350. ACM (2016)Google Scholar