# 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.

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