Advertisement

Applied Categorical Structures

, Volume 23, Issue 6, pp 777–818 | Cite as

Iterated Chromatic Subdivisions are Collapsible

  • Éric Goubault
  • Samuel MimramEmail author
  • Christine Tasson
Article

Abstract

The standard chromatic subdivision of the standard simplex is a combinatorial algebraic construction, which was introduced in theoretical distributed computing, motivated by the study of the view complex of layered immediate snapshot protocols. A most important property of this construction is the fact that the iterated subdivision of the standard simplex is contractible, implying impossibility results in fault-tolerant distributed computing. Here, we prove this result in a purely combinatorial way, by showing that it is collapsible, studying along the way fundamental combinatorial structures present in the category of colored simplicial complexes.

Keywords

Abstract simplicial complex Standard chromatic subdivision Collapsibility Iterated subdivision 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bing, R.H.: Some aspects of the topology of 3-manifolds related to the poincaré conjecture. Lect. Mod. Math. II, 93–128 (1964)MathSciNetGoogle Scholar
  2. 2.
    Biran, O., Moran, S., Zaks, S.: A combinatorial characterization of the distributed tasks which are solvable in the presence of one faulty processor. In: Proceedings of the seventh annual ACM symposium on principles of distributed computing, pp 263–275. ACM (1988)Google Scholar
  3. 3.
    Borowsky, E., Gafni, E.: Generalized FLP impossibility result for t-resilient asynchronous computations. In: Proceedings of the 25th STOC. ACM Press (1993)Google Scholar
  4. 4.
    Castañeda, A., Rajsbaum, S.: New combinatorial topology bounds for renaming: The upper bound. J. ACM 59(1), 3 (2012)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cohen, M.M.: A course in simple-homotopy theory, vol.10 of Graduate Texts in Mathematics. Springer, New York (1973)Google Scholar
  6. 6.
    Ehlers, P., Porter, T.: Joins for (augmented) simplicial sets. J. Pure and Appl. Algebra 145(1), 37–44 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Fajstrup, L., Goubault, É., Haucourt, E., Mimram, S., Raußen, M.: Trace spaces: An efficient new technique for state-space reduction. In: ESOP, pp 274–294 (2012)Google Scholar
  8. 8.
    Fischer, M.J, Lynch, N.A., Paterson, M.S.: Impossibility of distributed consensus with one faulty process. J.ACM 32(2), 374–382 (1985)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory, vol. 174. Springer (2009)Google Scholar
  10. 10.
    Goubault, É., Mimram, S.: Trace spaces: algorithmics and applications. Presentation in Applied Algebraic Topology conference (2011)Google Scholar
  11. 11.
    Grandis, M.: Directed Algebraic Topology: Models of Non-Reversible Worlds. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar
  12. 12.
    Herlihy, M., Kozlov, D., Rajsbaum, S.: Distributed Computing Through Combinatorial Topology. Elsevier, New York (2014)zbMATHGoogle Scholar
  13. 13.
    Herlihy, M., Shavit, N.: The asynchronous computability theorem for t-resilient tasks. In: Proceedings of the twenty-fifth annual ACM symposium on theory of computing, pp 111–120. ACM (1993)Google Scholar
  14. 14.
    Herlihy, M., Shavit, N.: The topological structure of asynchronous computability. J. ACM 46(6), 858–923 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Kozlov, D.: Combinatorial Algebraic Topology, vol. 21. Springer, Berlin Heidelberg (2008)CrossRefGoogle Scholar
  16. 16.
    Kozlov, D.: Chromatic subdivision of a simplicial complex. Homology, Homotopy and Appl. 14(2), 197–209 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Kozlov, D.: Topology of the view complex. arXiv preprint arXiv:1311.7283. (2013)
  18. 18.
    Lynch, N.A.: Distributed Algorithms. Morgan Kaufmann, San Mateo (1996)zbMATHGoogle Scholar
  19. 19.
    MacLane, S.: Categories for the Working Mathematician, vol. 5. Springer, Berlin Heidelberg (1998)Google Scholar
  20. 20.
    MacLane, S., Moerdijk, I.: Sheaves in Geometry and Logic: A First Introduction to Topos Theory. Springer, Berlin Heidelberg (1992)CrossRefGoogle Scholar
  21. 21.
    Saks, M.E., Zaharoglou, F.: Wait-free k-set agreement is impossible: the topology of public knowledge. In: STOC, pp 101–110 (1993)Google Scholar
  22. 22.
    Whitehead, J.H.C.: Simplicial spaces, nuclei and m-groups. Proc. Lond. Math. Soc. 2(1), 243–327 (1939)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Éric Goubault
    • 1
  • Samuel Mimram
    • 1
    Email author
  • Christine Tasson
    • 2
  1. 1.CEALIST / École PolytechniqueGif-sur-YvetteFrance
  2. 2.PPSCNRS / Université Paris DiderotParis Cedex 13France

Personalised recommendations