Time lower bounds for implementations of multi-writer snapshots
Journal of the ACM (JACM)
A non-topological proof for the impossibility of k-set agreement
SSS'11 Proceedings of the 13th international conference on Stabilization, safety, and security of distributed systems
Lower Bounds for Randomized Consensus under a Weak Adversary
SIAM Journal on Computing
Electronic Notes in Theoretical Computer Science (ENTCS)
A non-topological proof for the impossibility of k-set agreement
Theoretical Computer Science
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This paper introduces the use of topological models and methods, formerly used to analyze computability, as tools for the quantification and classification of asynchronous complexity. We present the first asynchronous complexity theorem, applied to decision tasks in the iterated immediate snapshot (IIS) model of Borowsky and Gafni. We do so by introducing a novel form of topological tool called the nonuniform chromatic subdivision. Building on the framework of Herlihy and Shavit's topological computability model, our theorem states that the time complexity of any asynchronous algorithm is directly proportional to the level of nonuniform chromatic subdivisions necessary to allow a simplicial map from a task's input complex to its output complex. To show the power of our theorem, we use it to derive a new tight bound on the time to achieve n process approximate agreement in the IIS model: $\bigl\lceil \log_d \frac{\max\_input - \min\_input}{\epsilon} \bigr\rceil$, where d = 3 for two processes and d = 2 for three or more. This closes an intriguing gap between the known upper and lower bounds implied by the work of Aspnes and Herlihy. More than the new bounds themselves, the importance of our asynchronous complexity theorem is that the algorithms and lower bounds it allows us to derive are intuitive and simple, with topological proofs that require no mention of concurrency at all.