On the Optimal Space Complexity of Consensus for Anonymous Processes

TL;DR

This paper proves that in systems of anonymous processes, any consensus algorithm must use linear space, establishing a tight lower bound and solving a longstanding open problem in shared memory consensus.

Contribution

It establishes an (n) lower bound on space complexity for anonymous processes, matching known upper bounds and resolving the symmetric case of the open problem.

Findings

Any anonymous consensus algorithm with nondeterministic solo termination requires (n) registers.

The (n) lower bound applies to deterministic obstruction-free and randomized wait-free consensus.

This result closes the symmetric case of the open problem on space complexity in shared memory consensus.

Abstract

The optimal space complexity of consensus in shared memory is a decades-old open problem. For a system of $n$ processes, no algorithm is known that uses a sublinear number of registers. However, the best known lower bound due to Fich, Herlihy, and Shavit requires $\Omega(\sqrt{n})$ registers. The special symmetric case of the problem where processes are anonymous (run the same algorithm) has also attracted attention. Even in this case, the best lower and upper bounds are still $\Omega(\sqrt{n})$ and $O(n)$. Moreover, Fich, Herlihy, and Shavit first proved their lower bound for anonymous processes, and then extended it to the general case. As such, resolving the anonymous case might be a significant step towards understanding and solving the general problem. In this work, we show that in a system of anonymous processes, any consensus algorithm satisfying nondeterministic solo termination has to use $\Omega(n)$ read-write registers in some execution. This implies an $\Omega(n)$ lower bound on the space complexity of deterministic obstruction-free and randomized wait-free consensus, matching the upper bound and closing the symmetric case of the open problem.