Broadcasting in hypercubes with randomly distributed Byzantine faults

  • Feng Bao
  • Yoshihide Igarashi
  • Keiko Katano
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 972)

Abstract

We study all-to-all broadcasting in hypercubes with randomly distributed Byzantine faults. We construct an efficient broadcasting scheme BC1-n-cube running on the n-dimensional hypercube (n-cube for short) in 2n rounds, where for communication by each node of the n-cube, only one of its links is used in each round. The scheme BC1-n-cube can tolerate ⌊(n −1)/2⌋ Byzantine node and/or link faults in the worst case. If there are at most f Byzantine faulty nodes randomly distributed in the n-cube, BC1-n-cube succeeds with a probability higher than 1-(64nf/2n)n/2⌉. In other words, if 1/(64nk) of all the nodes (i.e., 2n/(64nk) nodes) fail in Byzantine manner randomly in the n-cube, then the scheme succeeds with a probability higher than 1 −k−⌈ n/2 ⌉. We also consider the case where all nodes are faultless but links may fail randomly in the n-cube. Scheme BC1-n-cube succeeds with a probability higher than 1 −k−⌈ n/2 ⌉ provided that not more than l/(64(n + 1)k) of all the links in the n-cube fail in Byzantine manner randomly. For the case where only links may fail, we give another broadcasting scheme BC2-n-cube which runs in 2n2 rounds. Broadcasting by BC2-n-cube is successful with a high probability if the number of Byzantine faulty links randomly distributed in the n-cube is not more than a constant fraction of the total number of links. That is, it succeeds with a probability higher than 1−n·k−⌈ n/2 ⌉ if l/(48k) of all the links in the n-cube fail in Byzantine manner randomly.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alam, M. S. and Melhem, R. G., “ How to use an incomplete hypercube for fault tolerance,” Proc. of the 1st European Workshop on Hypercubes and Distributed Computers, pp. 329–341, 1989.Google Scholar
  2. 2.
    Alon, N., Barak, A. and Mauber, U., “On disseminating information reliably without broadcasting,” the 7th International Conference on Distributed Computing Systems, pp. 74–81, 1987.Google Scholar
  3. 3.
    Bienstock, D., “Broadcasting with random faults,” Discret Applied Mathematics, Vol. 20, pp. 1–7, 1988.CrossRefGoogle Scholar
  4. 4.
    Blough, D. B. and Pelc, A., “Optimal communication in networks with randomly distributed faults” Networks, vol. 23, pp. 691–701, 1993.Google Scholar
  5. 5.
    Carlsson, S., Igarashi, Y., Kanai, K., Lingas, A., Miura, K. and Petersson, O., “Information disseminating schemes for fault tolerance in hypercubes,” IEICE Trans. on Fundamentals of Electronics, Communications and Computer Sciences, E75-A, pp. 255–260, 1992.Google Scholar
  6. 6.
    Chlebus, B. S., Diks, K. and Pelc, A., “Optimal broadcasting in faulty hypercubes,” Proc. of the 21st International Symposium on Fault-Tolerant Computing, pp. 266–273, 1991.Google Scholar
  7. 7.
    Chlebus, B. S., Diks, K and Pelc, A., “Sparse networks supporting efficient reliable broadcasting,” Proc. of ICALP'93, LNCS, Vol. 710, Springer-Verlag, pp. 388–397, 1993.Google Scholar
  8. 8.
    Fraigniaud, P. and Peyrat, C., “Broadcasting in a hypercubes when some calls fail,” Information Processing Letters, Vol. 39, pp. 115–119, 1991.CrossRefMathSciNetGoogle Scholar
  9. 9.
    Gargano, L. and Vaccaro, U., “Minimum time broadcasting network tolerating a logarithmic number of faults,” SIAM J. of Discrete Math., Vol. 5, pp. 178–198, 1992.CrossRefGoogle Scholar
  10. 10.
    Gargano, L., Rescigno, A. and Vaccaro, U., “Fault tolerant hypercubes broadcasting via information dispersal,” Networks, Vol. 23, pp. 271–282, 1993.Google Scholar
  11. 11.
    Hastad, J., Leighton, T. and Newman, M., “Fast computation using faulty hypercubes,” Proc. of the 21st ACM Symposium on Theory of Computing, pp. 251–263, 1989.Google Scholar
  12. 12.
    Hedetniemi, S. M., Hedetniemi, S. T. and Liestman, A. L., “A survey of gossiping and broadcasting in communication networks,” Networks, Vol. 18, pp. 1249–1268, 1988.Google Scholar
  13. 13.
    Igarashi, Y., Kanai, K, Miura, K. and Osawa, S., “Optimal schemes for disseminating information and their fault tolerance,” IEICE Trans. on Information and Systems, pp. 22–29, 1992.Google Scholar
  14. 14.
    Pelc, A., “Reliable communication in networks with Byzantine link failures,” Networks, pp. 441–459, 1992.Google Scholar
  15. 15.
    Ramanathan, P. and Shin, K. G., “Reliable broadcasting in hypercube multiprocessors,” IEEE Trans. on Computers, Vol. 37, pp. 1654–1657, 1988.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Feng Bao
    • 1
  • Yoshihide Igarashi
    • 1
  • Keiko Katano
    • 1
  1. 1.Department of Computer ScienceGunma UniversityKiryuJapan

Personalised recommendations