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Concurrent Computing and Shellable Complexes

  • Maurice Herlihy
  • Sergio Rajsbaum
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6343)

Abstract

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.

Keywords

Simplicial Complex Protocol Complex Shellability Condition General Adversary Canonical Order 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Attiya, H., Dwork, C., Lynch, N., Stockmeyer, L.: Bounds on the time to reach agreement in the presence of timing uncertainty. J. ACM 41(1), 122–152 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Björner, A.: Shellable and Cohen-Macaulay partially ordered sets. Transactions of the American Mathematical Society 260(1), 159–183 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Björner, A.: Topological methods, pp. 1819–1872. MIT Press, Cambridge (1995)Google Scholar
  4. 4.
    Borowsky, E., Gafni, E.: Generalized FLP impossibility result for t-resilient asynchronous computations. In: STOC ’93: Proceedings of the Twenty-Fifth Annual ACM Symposium on Theory of Computing, pp. 91–100. ACM, New York (1993)CrossRefGoogle Scholar
  5. 5.
    Borowsky, E., Gafni, E.: A simple algorithmically reasoned characterization of wait-free computations (extended abstract). In: PODC ’97: Proceedings of the Sixteenth Annual ACM Symposium on Principles of Distributed Computing, pp. 189–198. ACM, New York (1997)CrossRefGoogle Scholar
  6. 6.
    Borowsky, E., Gafni, E., Lynch, N., Rajsbaum, S.: The BG distributed simulation algorithm. Distributed Computing 14(3), 127–146 (2001)CrossRefGoogle Scholar
  7. 7.
    Delporte-Gallet, C., Fauconnier, H., Guerraoui, R., Tielmann, A.: The disagreement power of an adversary: extended abstract. In: Proceedings of the 28th ACM Symposium on Principles of Distributed Computing, pp. 288–289. ACM, New York (2009)CrossRefGoogle Scholar
  8. 8.
    Gafni, E., Rajsbaum, S.: Distributed programming with tasks. Technical Report 100001, UCLA Computer Science Department, Los Angeles, CA, USA, november (2009)Google Scholar
  9. 9.
    Herlihy, M., Rajsbaum, S.: Set consensus using arbitrary objects (preliminary version). In: PODC ’94: Proceedings of the Thirteenth Annual ACM Symposium on Principles of Distributed Computing, pp. 324–333. ACM, New York (1994)CrossRefGoogle Scholar
  10. 10.
    Herlihy, M., Rajsbaum, S.: Algebraic spans. Mathematical Structures in Computer Science 10(4), 549–573 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 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
  12. 12.
    Herlihy, M., Rajsbaum, S., Tuttle, M.: An axiomatic approach to computing the connectivity of synchronous and asynchronous systems. Electron. Notes Theor. Comput. Sci. 230, 79–102 (2009)CrossRefGoogle Scholar
  13. 13.
    Herlihy, M., Rajsbaum, S., Tuttle, M.R.: Unifying synchronous and asynchronous message-passing models. In: PODC ’98: Proceedings of the Seventeenth Annual ACM Symposium on Principles of Distributed Computing, pp. 133–142. ACM, New York (1998)CrossRefGoogle Scholar
  14. 14.
    Herlihy, M., Shavit, N.: The topological structure of asynchronous computability. J. ACM 46(6), 858–923 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Junqueira, F., Marzullo, K.: Designing algorithms for dependent process failures. In: Future Directions in Distributed Computing, pp. 24–28 (2003)Google Scholar
  16. 16.
    Kozlov, D.: Combinatorial Algebraic Topology. Springer, Heidelberg (2007)Google Scholar
  17. 17.
    Moses, Y., Rajsbaum, S.: A layered analysis of consensus. SIAM J. Computing 31(4), 989–1021 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Saks, M., Zaharoglou, F.: Wait-free k-set agreement is impossible: The topology of public knowledge. SIAM J. Comput. 29(5), 1449–1483 (2000)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Maurice Herlihy
    • 1
  • Sergio Rajsbaum
    • 2
  1. 1.Computer Science DepartmentBrown UniversityProvidence
  2. 2.Instituto de MatemáticasUniversidad Nacional Autónoma de México, Ciudad UniversitariaMexico

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