The Space Complexity of Scannable Binary Objects

Ovens, Sean

doi:10.1145/3465084.3467916

Cited by 2 publications

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“…In [19], we introduced a new technique to prove that at least n + k − 2 binary objects are necessary to implement an n-process, k-component scannable binary object when n ≤ 2 k−1 + 2 [19]. A scannable binary object is a generalization of a binary snapshot object [20,21].…”

Section: Introductionmentioning

confidence: 99%

The Space Complexity of Consensus from Swap

Ovens

2022

Proceedings of the 2022 ACM Symposium on Principles of Distributed Computing

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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.

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Section: Introductionmentioning

confidence: 99%

The Space Complexity of Consensus from Swap

Ovens

2022

Proceedings of the 2022 ACM Symposium on Principles of Distributed Computing

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The Space Complexity of Consensus from Swap

Ovens

2024

J. ACM

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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.

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The Space Complexity of Scannable Binary Objects

Cited by 2 publications

References 36 publications

The Space Complexity of Consensus from Swap

The Space Complexity of Consensus from Swap

The Space Complexity of Consensus from Swap

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