The Space Complexity of Consensus from Swap

Abstract

Nearly thirty years ago, it was shown that Ω( √ n) registers are needed to solve obstruction-free consensus among n processes. This lower bound was improved to n registers in 2018, which exactly matches the best upper bound. The Ω( √ n) space complexity lower bound actually applies to a class of objects called historyless objects, which includes registers, testand-set objects, and readable swap objects. However, every known nprocess 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 kset agreement algorithm from swap objects uses at least ⌈ n k ⌉ − 1 objects. 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 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 n−2 3b+1 objects. For b = 2, we show a slightly better lower bound of n − 2. The best known obstruction-free binary consensus algorithm from readable swap objects with domain size 2 uses 2n − 1 objects, asymptotically matching our lower bound.