The Space Complexity of Consensus from Swap

Abstract

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 2 n − 1 readable swap objects with domain size 2, asymptotically matching our lower bound.