Skip to main content
SpringerLink
Log in

Distributed MST for constant diameter graphs

  • Original Article
  • Published:
Distributed Computing Aims and scope Submit manuscript

Abstract

This paper considers the problem of distributively constructing a minimum-weight spanning tree (MST) for graphs of constant diameter in the bounded-messages model, where each message can contain at most B bits for some parameter B. It is shown that the number of communication rounds necessary to compute an MST for graphs of diameter 4 or 3 can be as high as \(\Omega(\sqrt[3]n/\sqrt{B})\) and \(\Omega(\sqrt[4]n/\sqrt{B})\), respectively. The asymptotic lower bounds hold for randomized algorithms as well. On the other hand, we observe that O(log n) communication rounds always suffice to compute an MST deterministically for graphs with diameter 2, when B = O(log n). These results complement a previously known lower bound of \(\Omega(\sqrt[2]n/B)\) for graphs of diameter Ω(log n).

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. Attiya, H., Bar-Noy, A., Dolev, D.: Sharing memory robustly inmessage-passing systems. J. ACM 42(1), 124–142 (1995)

    Article  Google Scholar 

  2. Attiya, H., Welch, J.: Distributed Algorithms. McGraw-HillPublishing Company, UK (1998)

  3. Awerbuch, B.: Optimal distributed algorithms for minimum weightspanning tree, counting, leader election and related problems. In: Proceedings of 19th ACM Symposium Theory of Computing, pp.230–240 (May 1987)

  4. Awerbuch, B., Shiloach, Y.: New connectivity and MSF algorithmsfor the shuffle-exchange network and PRAM. IEEE Trans. Comput. C-36, 1258–1263 (1987)

    MathSciNet  Google Scholar 

  5. Elkin, M.: Unconditional lower bounds on the time-approximationtradeoffs for the distributed minimum spanning tree problem. In Proceedings of 36th ACM Symposium on Theory of Computing (May2004)

  6. Faloutsos, M., Molle, M.: Optimal distributed algorithm forminimum spanning trees revisited. In: Proceedings of 14th ACMSymposium on Principles of Distributed Computing, pp. 231–237(1995)

  7. Fortune, S., Wyllie, J.: Parallelism in random access machines. In: Proceedings of 19th ACM Symposium on Theory of Computing, pp.230–240 (May 1987)

  8. Gallager, R.G., Humblet, P.A., Spira, P.M.: A distributedalgorithm for minimum-weight spanning trees. ACM Trans. Prog.Lang. Syst. 5(1), 66–77 (1983)

    Article  Google Scholar 

  9. Garay, J., Kutten, S., Peleg, D.: A sub-linear time distributedalgorithm for minimum-weight spanning trees. SIAM J. Comput. 27, 302–316 (1998)

    Article  MathSciNet  Google Scholar 

  10. Gifford, D.K.: Weighted Voting for Replicated Data. In: Proceedings of 7th Symposium on Operating System Principles, pp.150–162 (1979)

  11. Kutten, S., Peleg, D.: Fast distributed construction of smallk-dominating sets and applications. J. Algorithm.28,40–66 (1998)

    Article  MathSciNet  Google Scholar 

  12. Lawler, E.: Combinatorial Optimization: Networks and Matroids.Holt, Rinehart and Winston, New York (1976)

  13. Linial, N.: Locality in distributed graph algorithms. SIAM J.Comput. 21, 193–201 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  14. Lotker, Z., Patt-Shamir, B., Pavlov, E., Peleg, D.: MST construction in O(log log n) communication rounds. In: Proceedings of 2003 ACM Symposium on Parallelism in Algorithmsand Architecture, pp. 94–100 (2003)

  15. Lynch, N.: Distributed Algorithms. Morgan Kaufmann, San Mateo,CA (1995)

  16. Motwani, R., Raghavan, P.: Randomized Algorithms. CambridgeUniversity Press (1995)

  17. Peleg, D.: Time-optimal leader election in general networks. J.Parallel Distrib. Comput. 8, 96–99 (1990)

    Article  Google Scholar 

  18. Peleg, D.: Distributed Computing: A Locality-Sensitive Approach.Society for Industrial and Applied Mathematics, Philadelphia, PA,USA (2000)

  19. Peleg, D., Rubinovich, V.: Near-tight lower bound on the timecomplexity of distributed MST construction. SIAM J. Comput. 30, 1427–1442 (2000)

    Article  MathSciNet  Google Scholar 

  20. Upfal, E., Wigderson, A.: How to share memory in a distributedsystem. J. ACM 34(1), 116–127 (1987)

    Article  MathSciNet  Google Scholar 

  21. Yao, A.C.: Probabilistic computations: Toward a unified measure ofcomplexity. In: Proceedings of 18th ACM Symposium on Theory of Computing, pp. 222–227 (1977)

Download references

Author information

Authors and Affiliations

  1. Department of Electrical Engineering, Tel Aviv University, Tel Aviv, 69978, Israel

    Zvi Lotker & Boaz Patt-Shamir

  2. Department of Computer Science, Weizmann Institute, Rehovot, 76100, Israel

    David Peleg

Authors
  1. Zvi Lotker
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Boaz Patt-Shamir
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. David Peleg
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zvi Lotker.

Additional information

An extended abstract of this work appears in Proceedings of 20th ACM Symposium on Principles of Distributed Computing, August 2001.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lotker, Z., Patt-Shamir, B. & Peleg, D. Distributed MST for constant diameter graphs. Distrib. Comput. 18, 453–460 (2006). https://doi.org/10.1007/s00446-005-0127-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00446-005-0127-6

Keywords

Search

Navigation