The Space Complexity of Consensus from Swap

Abstract

Nearly thirty years ago, it was shown that $Ω(\sqrt{n})$ 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 $Ω(\sqrt{n})$ 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. 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. 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, exactly matching 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. 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 $n-2$. 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.