Advertisement

The Read/Write Protocol Complex Is Collapsible

  • Fernando Benavides
  • Sergio RajsbaumEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9644)

Abstract

The celebrated asynchronous computability theorem provides a characterization of the class of decision tasks that can be solved in a wait-free manner by asynchronous processes that communicate by writing and taking atomic snapshots of a shared memory. Several variations of the model have been proposed (immediate snapshots and iterated immediate snapshots), all equivalent for wait-free solution of decision tasks, in spite of the fact that the protocol complexes that arise from the different models are structurally distinct. The topological and combinatorial properties of these snapshot protocol complexes have been studied in detail, providing explanations for why the asynchronous computability theorem holds in all the models.

In reality concurrent systems do not provide processes with snapshot operations. Instead, snapshots are implemented (by a wait-free protocol) using operations that write and read individual shared memory locations. Thus, read/write protocols are also computationally equivalent to snapshot protocols. However, the structure of the read/write protocol complex has not been studied. In this paper we show that the read/write iterated protocol complex is collapsible (and hence contractible). Furthermore, we show that a distributed protocol that wait-free implements atomic snapshots in effect is performing the collapses.

References

  1. 1.
    Afek, Y., Attiya, H., Dolev, D., Gafni, E., Merritt, M., Shavit, N.: Atomic snapshots of shared memory. J. ACM 40(4), 873–890 (1993)CrossRefzbMATHGoogle Scholar
  2. 2.
    Attiya, H., Rajsbaum, S.: The combinatorial structure of wait-free solvable tasks. SIAM J. Comput. 31(4), 1286–1313 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Biran, O., Moran, S., Zaks, S.: A combinatorial characterization of the distributed 1-solvable tasks. J. Algorithms 11(3), 420–440 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Borowsky, E., Gafni, E.: Generalized FLP impossibility result for t-resilient asynchronous computations. In: Proceedings of the 25th Annual ACM Symposium on Theory of Computing, STOC, pp. 91–100. ACM, New York (1993)Google Scholar
  5. 5.
    Borowsky, E., Gafni, E.: Immediate atomic snapshots and fast renaming. In: Proceedings of the 12th ACM Symposium on Principles of Distributed Computing, PODC, pp. 41–51. ACM, New York (1993)Google Scholar
  6. 6.
    Borowsky, E., Gafni, E.: A simple algorithmically reasoned characterization of wait-free computation (extended abstract). In: Proceedings of the Sixteenth Annual ACM Symposium on Principles of Distributed Computing, PODC 1997, pp. 189–198. ACM, New York (1997)Google Scholar
  7. 7.
    Borowsky, E., Gafni, E., Lynch, N., Rajsbaum, S.: The BG distributed simulation algorithm. Distrib. Comput. 14(3), 127–146 (2001)CrossRefGoogle Scholar
  8. 8.
    Fischer, M., Lynch, N.A., Paterson, M.S.: Impossibility of distributed commit with one faulty process. J. ACM 32(2), 374–382 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gafni, E., Rajsbaum, S.: Recursion in distributed computing. In: Dolev, S., Cobb, J., Fischer, M., Yung, M. (eds.) SSS 2010. LNCS, vol. 6366, pp. 362–376. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  10. 10.
    Goubault, E., Mimram, S., Tasson, C.: Iterated chromatic subdivisions are coll apsible. Appl. Categorical Struct. 23(6), 777–818 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Havlicek, J.: Computable obstructions to wait-free computability. Distrib. Comput. 13(2), 59–83 (2000)CrossRefGoogle Scholar
  12. 12.
    Havlicek, J.: A note on the homotopy type of wait-free atomic snapshot protocol complexes. SIAM J. Comput. 33(5), 1215–1222 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Herlihy, M., Kozlov, D., Rajsbaum, S.: Distributed Computing Through Combinatorial Topology. Elsevier, Imprint Morgan Kaufmann, Boston (2013)zbMATHGoogle Scholar
  14. 14.
    Herlihy, M., Rajsbaum, S.: Simulations and reductions for colorless tasks. In: Proceedings of the ACM Symposium on Principles of Distributed Computing, PODC 2012, pp. 253–260. ACM, New York (2012)Google Scholar
  15. 15.
    Herlihy, M., Shavit, N.: The topological structure of asynchronous computability. J. ACM 46(6), 858–923 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Herlihy, M., Shavit, N.: The Art of Multiprocessor Programming. Morgan Kaufmann Publishers Inc., San Francisco (2008)Google Scholar
  17. 17.
    Hoest, G., Shavit, N.: Towards a topological characterization of asynchronous complexity. In: Proceedings of the 16th ACM Symposium Principles of Distributed Computing, PODC, pp. 199–208. ACM, New York (1997)Google Scholar
  18. 18.
    Jonsson, J.: Simplicial Complexes of Graphs. Lecture Notes in Mathematics. Springer, Heidelberg (2008). doi: 10.1007/978-3-540-75859-4 CrossRefzbMATHGoogle Scholar
  19. 19.
    Kozlov, D.N.: Chromatic subdivision of a simplicial complex. Homology Homotopy Appl. 14(2), 197–209 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kozlov, D.N.: Topology of the immediate snapshot complexes. Topology Appl. 178, 160–184 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kozlov, D.N.: Topology of the view complex. Homology Homotopy Appl. 17(1), 307–319 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Loui, M.C., Abu-Amara, H.H.: Memory requirements for agreement among unreliable asynchronous processes 4, 163–183 (1987). JAI PressMathSciNetGoogle Scholar
  23. 23.
    Rajsbaum, S., Raynal, M., Travers, C.: The iterated restricted immediate snapshot model. In: Hu, X., Wang, J. (eds.) COCOON 2008. LNCS, vol. 5092, pp. 487–497. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  24. 24.
    Saks, M., Zaharoglou, F.: Wait-free k-set agreement is impossible: the topology of public knowledge. SIAM J. Comput. 29(5), 1449–1483 (2000)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Instituto de MatemáticasUniversidad Nacional Autónoma de México, Ciudad UniversitariaMexico CityMexico
  2. 2.Departamento de Matemáticas y EstadísticaUniversidad de NariñoSan Juan de PastoColombia

Personalised recommendations