Skip to main content
SpringerLink
Log in

The topology of distributed adversaries

  • Published:
Distributed Computing Aims and scope Submit manuscript

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 widely studied because they have nice combinatorial properties. It turns out that several standard models of concurrent computation can be constructed from shellable complexes. We consider adversarial schedulers in the synchronous, asynchronous, and semi-synchronous message-passing models, as well as asynchronous shared memory. 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. Earlier versions of material in this article appeared in the 2010 ACM Symposium on Principles of Distributed Computing (Herlihy and Rajsbaum 2010), and the International Conference on Distributed Computing (Herlihy and Rajsbaum 2010, doi:10.1145/1835698.1835724).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

Notes

  1. Choosing \(n+1\) processes rather than \(n\) simplifies the topological notation but slightly complicates the computing notation. Choosing \(n\) processes makes the opposite trade-off. We choose \(n+1\) for compatibility with prior work.

References

  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)

    Article  MATH  Google Scholar 

  2. Attiya, H., Bar-Noy, A., Dolev, D., Peleg, D., Reischuk, R.: Renaming in an asynchronous environment. J. ACM 37(3), 524–548 (1990)

    Google Scholar 

  3. Attiya, H., Borran, F., Hutle, M., Milosevic, Z., Schiper, A.: Structured Derivation of Semi-Synchronous Algorithms. In: Peleg, D. (ed.) Distributed Computing, Lecture Notes in Computer Science, vol. 6950, pp. 374–388. Springer, Berlin (2011). doi:10.1007/978-3-642-24100-0_37

  4. 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). doi:10.1145/174644.174649

    Article  MathSciNet  MATH  Google Scholar 

  5. Attiya, H., Welch, J.: Distributed Computing Fundamentals, Simulations, and Advanced Topics, 2nd Edn. Wiley, New York (2004). URL: http://www.cs.technion.ac.il/~hagit/DC/

  6. Biran, O., Moran, S., Zaks, S.: A combinatorial characterization of the distributed 1-solvable tasks. J. Algorithms 11(3), 420–440 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  7. Björner, A.: Shellable and Cohen-Macaulay partially ordered sets. Trans. Am. Math. Soc. 260(1), 159–183 (1980)

    Article  MATH  Google Scholar 

  8. Borowsky, E., Gafni, E.: Immediate atomic snapshots and fast renaming. In: Proceedings of the Twelfth Annual ACM Symposium on Principles of Distributed Computing, PODC ’93, pp. 41–51. ACM, New York, NY, USA (1993). doi:10.1145/164051.164056

  9. 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, NY, USA (1997)

  10. Castañeda, A., Herlihy, M., Rajsbaum, S.: An equivariance theorem with applications to renaming. In: Proceedings of the 10th Latin American International Conference on Theoretical Informatics, LATIN’12, pp. 133–144. Springer, Berlin (2012). doi:10.1007/978-3-642-29344-3_12

  11. Chaudhuri, S.: More choices allow more faults: set consensus problems in totally asynchronous systems. Inform. Comput. 105(1), 132–158 (1993)

    Article  MATH  Google Scholar 

  12. Chaudhuri, S., Herlihy, M., Lynch, N.A., Tuttle, M.R.: Tight bounds for k-set agreement. J. ACM 47(5), 912–943 (2000). doi:10.1145/355483.355489

    Article  MathSciNet  Google Scholar 

  13. Delporte-Gallet, C., Fauconnier, H., Guerraoui, R., Tielmann, A.: The disagreement power of an adversary. Distrib. Comput. 24(3), 137–147 (2011). doi:10.1007/s00446-010-0122-4

    Article  MATH  Google Scholar 

  14. Dolev, D., Strong, H.R.: Authenticated algorithms for byzantine agreement. SIAM J. Comput. 12(4), 656–666 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fischer, M., Lynch, N.A., Paterson, M.S.: Impossibility of distributed commit with one faulty process. J. ACM 32(2), 374–382 (1985)

    Google Scholar 

  16. Fischer, M.J., Lynch, N.A.: A lower bound for the time to assure interactive consistency. Inf. Process. Lett. 14(4), 183–186 (1982)

    Google Scholar 

  17. Gafni, E.: Round-by-round fault detectors (extended abstract): unifying synchrony and asynchrony. In: Proceedings of the Seventeenth Annual ACM Symposium on Principles of Distributed Computing, PODC ’98, pp. 143–152. ACM, New York, NY, USA (1998). doi:10.1145/277697.277724

  18. Gafni, E., Koutsoupias, E.: Three-processor tasks are undecidable. SIAM J. Comput. 28(3), 970–983 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gafni, E., Kuznetsov, P.: Relating \({\cal L}\)”-resilience and wait-freedom via hitting sets. In: Aguilera, M., Yu, H., Vaidya, N., Srinivasan, V., Choudhury, R. (eds.) Distributed Computing and Networking, Lecture Notes in Computer Science, vol. 6522, pp. 191–202. Springer, Berlin (2011)

  20. Gafni, E., Rajsbaum, S.: Distributed programming with tasks. In: Proceedings of the 14th International Conference on Principles of Distributed Systems, OPODIS’10, pp. 205–218. Springer, Berlin (2010). URL:http://portal.acm.org/citation.cfm?id=1940258

  21. Gafni, E., Rajsbaum, S., Herlihy, M.: Subconsensus tasks: Renaming is weaker than set agreement. In: Distributed Computing, 20th International Symposium, Stockholm, Sweden, September 18–20, 2006, Proceedings(DISC), Lecture Notes in Computer Science, vol. 4167, pp. 329–338. Springer, Berlin (2006)

  22. Halpern, J.Y., Moses, Y.: Knowledge and common knowledge in a distributed environment. J. ACM 37(3), 549–587 (1990). doi:10.1145/79147.79161

    Article  MathSciNet  MATH  Google Scholar 

  23. Herlihy, M.: Wait-free synchronization. ACM Trans. Program. Lang. Syst. 13(1), 124–149 (1991). doi:10.1145/114005.102808

    Article  Google Scholar 

  24. 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, NY, USA (1994)

  25. Herlihy, M., Rajsbaum, S.: The decidability of distributed decision tasks (extended abstract). In: STOC ’97: Proceedings of the Twenty-ninth Annual ACM Symposium on Theory of Computing, pp. 589–598. ACM, New York, NY, USA (1997). doi:10.1145/258533.258652

  26. Herlihy, M., Rajsbaum, S.: Algebraic spans. Math. Struct. Comput. Sci. 10(4), 549–573 (2000). http://journals.cambridge.org/action/displayAbstract?aid=54601

  27. Herlihy, M., Rajsbaum, S.: Concurrent computing and shellable complexes. In: Proceedings of the 24th International Conference on Distributed Computing, DISC’10, pp. 109–123. Springer, Berlin (2010). URL:http://dl.acm.org/citation.cfm?id=1888781.1888795

  28. Herlihy, M., Rajsbaum, S.: The topology of shared-memory adversaries. In: Proceeding of the 29th ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing, PODC ’10, pp. 105–113. ACM, New York, NY, USA (2010). doi:10.1145/1835698.1835724

  29. Herlihy, M., Rajsbaum, S.: Simulations and reductions for colorless tasks. In: Proceedings of the 2012 ACM Symposium on Principles of Distributed Computing, PODC ’12, pp. 253–260. ACM, New York, NY, USA (2012). doi:10.1145/2332432.2332483

  30. 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). doi:10.1016/j.entcs.2009.02.018

  31. 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, NY, USA (1998). doi:10.1145/277697.277722

  32. Herlihy, M., Shavit, N.: The topological structure of asynchronous computability. J. ACM 46(6), 858–923 (1999). doi:10.1145/331524.331529

    Article  MathSciNet  MATH  Google Scholar 

  33. Junqueira, F., Marzullo, K.: A framework for the design of dependent-failure algorithms: research articles. Concurr. Comput.: Pract. Exper. 19(17), 2255–2269 (2007)

    Article  Google Scholar 

  34. Kozlov, D.: Combinatorial Algebraic Topology, Algorithms and Computation in Mathematics, vol. 21, 1st edn. Springer, Berlin (2007). doi:10.1007/978-3-540-71962-5

  35. Kuznetsov, P.: Understanding non-uniform failure models. In: Bulletin of the EATCS, 106, p. 53.77. European Association for, Theoretical Computer Science (2012)

  36. Lamport, L.: On interprocess communication (part I): basic formalism. Distrib. Comput. 1, 77–101 (1986). URL:http://citeseer.nj.nec.com/227917.html

  37. Lamport, L.: On interprocess communication (part II): algorithms. Distrib. Comput. 1(1), 203–213 (1986)

    MathSciNet  Google Scholar 

  38. Loui, M.C., Abu-Amara, H.H.: Memory requirements for agreement among unreliable asynchronous processes, vol. 4. JAI press, Greenwick (1987)

    Google Scholar 

  39. Michailidis, D.: Fast set agreement in the presence of timing uncertainty. In: Proceedings of the Eighteenth Annual ACM Symposium on Principles of Distributed Computing, PODC ’99, pp. 249–256. ACM, New York, NY, USA (1999). doi:10.1145/301308.301365

  40. Moses, Y., Rajsbaum, S.: A layered analysis of consensus. SIAM J. Comput. 31(4), 989–1021 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  41. Munkres, J.: Elements of Algebraic Topology, 2nd edn. Prentice Hall, New York (1984). URL:http://www.worldcat.org/isbn/0131816292

  42. Saks, M., Zaharoglou, F.: Wait-free k-set agreement is impossible: the topology of public knowledge. SIAM J. Comput. 29(5), 1449–1483 (2000). URL:http://epubs.siam.org/doi/abs/10.1137/S0097539796307698?journalCode=smjcat

  43. Saks, M., Zaharoglou, F.: Wait-free k-set agreement is impossible: the topology of public knowledge. SIAM J. Comput. 29(5), 1449–1483 (2000). doi:10.1145/167088.167122

    Article  MathSciNet  MATH  Google Scholar 

  44. Spanier, E.H.: Algebraic Topology. Springer, New York (1981). URL:http://www.ams.org/mathscinet-getitem?mr=666554

  45. Wang, J., Song, M.: A new algorithm to solve synchronous consensus for dependent failures. In: Proceedings of the Sixth International Conference on Parallel and Distributed Computing Applications and Technologies, PDCAT ’05, pp. 371–375. IEEE Computer Society, Washington, DC, USA (2005). doi:10.1109/PDCAT.2005.23

Download references

Acknowledgments

The authors thank the anonymous referees for their very careful readings and helpful comments.

Author information

Authors and Affiliations

  1. Computer Science Department, Brown University, Providence, RI, 02912, USA

    Maurice Herlihy

  2. Instituto de Matemáticas, Universidad Nacional Autónoma de México, Ciudad Universitaria, Mexico, D.F., 04510, Mexico

    Sergio Rajsbaum

Authors
  1. Maurice Herlihy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sergio Rajsbaum
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maurice Herlihy.

Additional information

Supported by NSF 0830491 and partially supported by UNAM-PAPIIT.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Herlihy, M., Rajsbaum, S. The topology of distributed adversaries. Distrib. Comput. 26, 173–192 (2013). https://doi.org/10.1007/s00446-013-0189-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00446-013-0189-9

Keywords

Search

Navigation