Majority and Unanimity in Synchronous Networks with Ubiquitous Dynamic Faults

  • Nicola Santoro
  • Peter Widmayer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3499)


In this paper we are interested in synchronous distributed systems subject to transient and ubiquitous failures. This includes systems where failures will occur on any communication link, systems where every processor will fail at one time or another, etc., and, following a failure, normal functioning can resume after a finite (although unpredictable) amount of time. Notice that these cases cannot be handled by the traditional component failure models.

The model we use is the transmission failure model, known also as the dynamic faults model. Using this model, we study the fundamental problem of agreement in synchronous systems of arbitrary topology.We establish bounds on the number of dynamic faults that make any non-trivial form of agreement (even strong majority) impossible; in turn, these bounds express connectivity requirements which must be met to achieve any meaningful form of agreement. We also provide, constructively, bounds on the number of dynamic faults in spite of which any non-trivial form of agreement (even unanimity) is possible.

These bounds are shown to be tight for a large class of networks, that includes hypercubes, toruses, rings, and complete graphs; incidentally, we close the existing gap between possibility and impossibility of non-trivial agreement in complete graphs in presence of dynamic Byzantine faults.

None of these results is derivable in the component failure models; in particular, all our possibility results hold in situations for which those models indicate impossibility.


Complete Graph Clock Cycle Failure Model Dynamic Fault Information Processing Letter 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aguilera, M.K., Chen, W., Toueg, S.: Failure detection and consensus in the crash-recovery model. Distributed Computing 13(2), 99–125 (2000)CrossRefGoogle Scholar
  2. 2.
    Aguilera, M.K., Toueg, S.: A simple bivalency proof that t-resilient consensus requires t+1 rounds. Information Processing Letters 71, 155–158 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bar-Joseph, Z., Ben-Or, M.: A tight lower bound for randomized synchronous consensus. In: Proc. ACM Symp. on Principles of Distributed Computing (PODC 1998), Puerto Vallarta, pp. 193–199 (1998)Google Scholar
  4. 4.
    Ben-Or, M., Ron, D.: Agreement in presence of faults on networks of bounded degree. Information Processing Letters 57(6), 329–334 (1996)zbMATHCrossRefGoogle Scholar
  5. 5.
    Chlebus, B.S., Diks, K., Pelc, A.: Broadcasting in synchronous networks with dynamic faults. Networks 27, 309–318 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Cristian, F., Aghili, H., Strong, R., Dolev, D.: Atomic broadcast: From simple message diffusion to Byzantine agreement. Information and Computation 118(1), 158–179 (1995)CrossRefMathSciNetGoogle Scholar
  7. 7.
    De Marco, G., Rescigno, A.: Tighter bounds on broadcasting in torus networks in presence of dynamic faults. Parallel Processing Letters 10, 39–49 (2000)zbMATHCrossRefGoogle Scholar
  8. 8.
    De Marco, G., Vaccaro, U.: Broadcasting in hypercubes and star graphs with dynamic faults. Information Processing Letters 66, 309–318 (1998)CrossRefGoogle Scholar
  9. 9.
    Dobrev, S.: Computing input multiplicity in anonymous synchronous networks with dynamic faults. In: Brandes, U., Wagner, D. (eds.) WG 2000. LNCS, vol. 1928, pp. 137–148. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  10. 10.
    Dobrev, S.: Communication-efficient broadcasting in complete networks with dynamic faults. In: Proc. 9th Coll. on Structural Information and Communication complexity (SIROCCO 2002), pp. 101–113 (2002)Google Scholar
  11. 11.
    Dobrev, S., Vrt’o, I.: Optimal broadcasting in hypercubes with dynamic faults. Information Processing Letters 71, 81–85 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Dobrev, S., Vrt’o, I.: Optimal broadcasting in even tori with dynamic faults. Parallel Processing Letters 12, 17–22 (2002)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Dolev, D.: The Byzantine generals strike again. J. Algorithms 3(1), 14–30 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Dolev, D., Strong, H.R.: Polynomial algorithms for multiple processor agreement. In: Proc. 14th ACM Symp. on Theory of Computing (STOC 1982), pp. 401–407 (1982)Google Scholar
  15. 15.
    Dwork, C., Peleg, D., Pippenger, N., Upfal, E.: Fault tolerance in networks of bounded degree. SIAM J. Computing 17(5), 975–988 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Fischer, M.J., Lynch, N.A.: A lower bound for the time to assure interactive consistency. Information Processing Letters 14(4), 183–186 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Fischer, M.J., Lynch, N.A., Merritt, M.: Easy impossibility proofs for distributed consensus problems. Distributed Computing 1(1), 26–39 (1986)zbMATHCrossRefGoogle Scholar
  18. 18.
    Fischer, M.J., Lynch, N.A., Paterson, M.S.: Impossibility of distributed consensus with one faulty process. J. ACM 32(2) (1985)Google Scholar
  19. 19.
    Fraigniaud, P., Peyrat, C.: Broadcasting in a hypercube when some calls fail. Information Processing Letters 39, 115–119 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Gasienic, L., Pelc, A.: Broadcasting with linearly bounded faults. Discrete Applied Mathematics 83, 121–133 (1998)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Garay, J., Moses, Y.: Fully polynomial Byzantine agreement for n > 3t processors in t + 1 rounds. SIAM J. Computing 27(1), 247–290 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Guerraoui, R., Levy, R.R.: Robust emulation of shared memory in a crash-recovery model. In: Proc. 24th Int. Conf. on Dist. Computing Systems (ICDCS 2004), pp. 400–407 (2004)Google Scholar
  23. 23.
    Hadzilacos, V.: Connectivity requirements for Byzantine agreement under restricted types of failures. Distributed Computing 2, 95–103 (1987)CrossRefGoogle Scholar
  24. 24.
    Kralovic, R., Kralovic, R., Ruzicka, P.: Broadcasting with many faulty links. In: Proc. 10th Coll. on Structural Information and Communication complexity (SIROCCO 2003), pp. 211–222 (2003)Google Scholar
  25. 25.
    Lamport, L., Shostak, R., Pease, M.: The Byzantine generals problem. ACM Trans. Programming Languages and Systems 4(3), 382–401 (1982)zbMATHCrossRefGoogle Scholar
  26. 26.
    Liptak, Z., Nickelsen, A.: Broadcasting in complete networks with dynamic edge faults. In: Proc. 4th Int. Conf. on Principles of Distributed Systems (OPODIS 2000), Paris, pp. 123–142 (2000)Google Scholar
  27. 27.
    Moses, Y., Rajsbaum, S.: A Layered Analysis of Consensus. SIAM J. Computing 31(4), 989–1021 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Perry, K.J., Toueg, S.: Distributed agreement in the presence of processor and communication faults. IEEE Trans. Software Engineering SE-12 (3), 477–482 (1986)Google Scholar
  29. 29.
    Santoro, N., Widmayer, P.: Time is not a healer. In: Proc. 6th Symposium on Theoretical Aspects of Computer Science (STACS 1989), pp. 304–313 (1989)Google Scholar
  30. 30.
    Santoro, N., Widmayer, P.: Distributed function evaluation in the presence of transmission faults. In: Proc. Int. Symposium on Algorithms (SIGAL 1990), pp. 358–367 (1990)Google Scholar
  31. 31.
    Schmid, U., Weiss, B.: Formally verified Byzantine agreement in presence of link faults. In: Proc. 22nd Int. Conf. on Distributed Computing Systems (ICDCS 2002), pp. 608–616 (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Nicola Santoro
    • 1
  • Peter Widmayer
    • 2
  1. 1.School of Computer ScienceCarleton UniversityCanada
  2. 2.Institut for Theoretical InformaticsETH ZurichSwitzerland

Personalised recommendations