Concurrent Computing and Shellable Complexes
Roughly speaking, a simplicial complex is shellable if it can be constructed by gluing a sequence of n-simplexes to one another along (n − 1)-faces only. Shellable complexes have been studied in the combinatorial topology literature because they have many nice properties.
It turns out that many standard models of concurrent computation can be captured either as shellable complexes, or as the simple union of shellable complexes. We consider general adversaries in the synchronous, asynchronous, and semi-synchronous message-passing models, as well as asynchronous shared memory augmented by consensus and set agreement objects.
We show how to exploit their common shellability structure to derive new and remarkably succinct tight (or nearly so) lower bounds on connectivity of protocol complexes and hence on solutions to the k-set agreement task in these models.
KeywordsSimplicial Complex Protocol Complex Shellability Condition General Adversary Canonical Order
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- 3.Björner, A.: Topological methods, pp. 1819–1872. MIT Press, Cambridge (1995)Google Scholar
- 8.Gafni, E., Rajsbaum, S.: Distributed programming with tasks. Technical Report 100001, UCLA Computer Science Department, Los Angeles, CA, USA, november (2009)Google Scholar
- 11.Herlihy, M., Rajsbaum, S.: The topology of shared-memory adversaries. In: PODC ’10: Proceedings of the Fourteenth Annual ACM Symposium on Principles of Distributed Computing (to appear, 2010)Google Scholar
- 15.Junqueira, F., Marzullo, K.: Designing algorithms for dependent process failures. In: Future Directions in Distributed Computing, pp. 24–28 (2003)Google Scholar
- 16.Kozlov, D.: Combinatorial Algebraic Topology. Springer, Heidelberg (2007)Google Scholar