Toward a topological characterization of asynchronous complexity

Gunnar Hoest, Nir Shavit

Research output: Contribution to journalReview articlepeer-review

Abstract

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 US model: ⌈log d max-input-min-input/ε⌉ 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.

Original languageEnglish
Pages (from-to)457-497
Number of pages41
JournalSIAM Journal on Computing
Volume36
Issue number2
DOIs
StatePublished - 2006

Keywords

  • Approximate agreement
  • Asynchronous systems
  • Immediate snapshots
  • Shared memory
  • Simplicial complexes
  • Subdivisions
  • Topology

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