The Space Complexity of Consensus from Swap.

Nearly thirty years ago, it was shown that \(\Omega (\sqrt {n})\) read/write registers are needed to solve randomized wait-free consensus among n processes. This lower bound was improved to n registers in 2018, which exactly matches known algorithms. The \(\Omega (\sqrt {n})\) space complexity lower bound actually applies to a class of objects called historyless objects, which includes registers, test-and-set objects, and readable swap objects. However, every known n-process obstruction-free consensus algorithm from historyless objects uses Ω (n) objects. In this paper, we give the first Ω (n) space complexity lower bounds on consensus algorithms for two kinds of historyless objects. First, we show that any obstruction-free consensus algorithm from swap objects uses at least n-1 objects. More generally, we prove that any obstruction-free k-set agreement algorithm from swap objects uses at least \(\lceil \frac{n}{k}\rceil - 1\) objects. The k-set agreement problem is a generalization of consensus in which processes agree on no more than k different output values. This is the first non-constant lower bound on the space complexity of solving k-set agreement with swap objects when k > 1. We also present an obstruction-free k-set agreement algorithm from n-k swap objects, which exactly matches our lower bound when k=1. Second, we show that any obstruction-free binary consensus algorithm from readable swap objects with domain size b uses at least \(\frac{n-2}{3b+1}\) objects. When b is a constant, this asymptotically matches the best known obstruction-free consensus algorithms from readable swap objects with unbounded domains. Since any historyless object can be simulated by a readable swap object with the same domain, our results imply that any obstruction-free consensus algorithm from historyless objects with domain size b uses at least \(\frac{n-2}{3b+1}\) objects. For b = 2, we show a slightly better lower bound of n-2. There is an obstruction-free binary consensus algorithm using 2n-1 readable swap objects with domain size 2, asymptotically matching our lower bound. [ABSTRACT FROM AUTHOR]