Skip to main content
SpringerLink
Log in

Distributed verification of minimum spanning trees

  • SPECIAL ISSUE PODC 06
  • Published:
Distributed Computing Aims and scope Submit manuscript

Abstract

The problem of verifying a Minimum Spanning Tree (MST) was introduced by Tarjan in a sequential setting. Given a graph and a tree that spans it, the algorithm is required to check whether this tree is an MST. This paper investigates the problem in the distributed setting, where the input is given in a distributed manner, i.e., every node “knows” which of its own emanating edges belong to the tree. Informally, the distributed MST verification problem is the following. Label the vertices of the graph in such a way that for every node, given (its own state and label and) the labels of its neighbors only, the node can detect whether these edges are indeed its MST edges. In this paper, we present such a verification scheme with a maximum label size of O(log n log W), where n is the number of nodes and W is the largest weight of an edge. We also give a matching lower bound of Ω(log n log W) (as long as W > (log n)1+ε for some fixed ε > 0). Both our bounds improve previously known bounds for the problem.

For the related problem of tree sensitivity also presented by Tarjan, our method yields rather efficient schemes for both the distributed and the sequential settings.

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. Afek Y., Kutten S. and Yung M. (1997). The local detection paradigm and its applications to self stabilization. Theor. Comput. Sci. 186: 199–230

    Article  MATH  MathSciNet  Google Scholar 

  2. Arora A. and Gouda M.G. (1994). Distributed reset. IEEE Trans. Comput. 43(9): 1026–1038

    Article  MATH  Google Scholar 

  3. Awerbuch, B., Varghese, G.: Distributed program checking: a paradigm for building self-stabilizing distributed protocols. In: Proceedings of the 32nd annual symposium on foundations of computer science, pp. 258–267 (1991)

  4. Booth H. and Westbrook J. (1994). Linear algorithms for analysis of minimum spanning and shortest paths trees in planar graphs. Algorithmica 11(4): 341–352

    Article  MATH  MathSciNet  Google Scholar 

  5. Cidon I., Gopal I., Kaplan M. and Kutten S. (1995). A distributed control architecture of high-speed networks. IEEE Trans. Commun. 43(5): 1950–1960

    Article  Google Scholar 

  6. Deering, S., Estrin, D., Farinacci, D., Jacobson, V., Liu, C., Wei, L.: Protocol independent multicast (PIM): motivation and architecture. Internet Draft draft-ietf-pim-arch-01.ps (1995)

  7. Dixon B., Rauch M. and Tarjan R.E. (1992). Verification and sensitivity analysis of minimum spanning trees in linear time. SIAM J. Comput. 21(6): 1184–1192

    Article  MATH  MathSciNet  Google Scholar 

  8. Dixon B. and Tarjan R.E. (1997). Optimal parallel verification of minimum spanning trees in logarithmic time. Algorithmica 17(1): 11–18

    Article  MATH  MathSciNet  Google Scholar 

  9. Dolev S. and Gouda, M.G., Schneider M. (1999). Memory requirements for silent stabilization. Acta Informatica 36(6): 447–462

    Article  MATH  MathSciNet  Google Scholar 

  10. Dolev S., Israeli A. and Moran S. (1997). Uniform dynamic self-stabilizing leader election. IEEE Trans. Parallel Distrib. Syst. 8(4): 424–440

    Article  Google Scholar 

  11. Fraigniaud, P., Gavoille, C.: Routing in trees. In: Proceedings 28th international colloq. on automata, languages & prog. LNCS 2076, 757–772 (2001)

  12. Fredman, M.L., Willard, D.E.: Trans-Dichotomous algorithms for minimum spanning trees and shortest paths. In: Proceedings of the 31st IEEE annual symposium on foundations of computer science, Los Alamitos, CA, pp. 719–725 (1990)

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

    Article  MATH  Google Scholar 

  14. Gavoille, C., Katz, M., Katz, N.A., Paul, C., Peleg, D.: Approximate distance labeling schemes. In 9th European symposium on algorithms, Aarhus, Denmark, SV-LNCS 2161, pp. 476–488 (2001)

  15. Gavoille C. and Peleg D. (2003). Compact and localized distributed data structures. Distrib. Comput. 16(2–3): 111–120

    Article  Google Scholar 

  16. Gavoille, C., Peleg, D., Pérennes, S., Raz, R.: Distance labeling in graphs. In: Proceedings of the 12th ACM-SIAM symposiun on discrete algorithms, pp. 210–219 (2001)

  17. Harel, D.: A linear time algorithm for finding dominators in flow graphs and related problems. In: Proceedings of the 17th annual ACM symposium on theory of computing, Salem, MA. pp. 185–194 (1985)

  18. Jayaram, M., Varghese, G.: The complexity of crash failures. In: Proceedings of the ACM symposium on principles of distributed computing, pp. 179–188 (1997)

  19. Jayaram, M., Varghese, G.: Crash failures can drive protocols to arbitrary states. In: Proceedings of the ACM symposium on principles of distributed computing, pp. 247–256 (1996)

  20. Kannan S., Naor M. and Rudich S. (1992). Implicit representation of graphs. SIAM J. Discrete Math 5: 596–603

    Article  MATH  MathSciNet  Google Scholar 

  21. Katz, M., Katz, N.A., Korman, A., Peleg, D.: Labeling schemes for flow and connectivity. In: Proceedings of the 19th ACM-SIAM symposium on discrete algorithms, pp. 927–936 (2002)

  22. Karger D.R., Klein P.N. and Tarjan R.E. (1955). A randomized linear-time algorithm to find minimum spanning trees. J. ACM 42(2): 321–328

    Article  MathSciNet  Google Scholar 

  23. Komlòs J. (1985). Linear verification for spanning trees. Combinatorica 5: 57–65

    Article  MATH  MathSciNet  Google Scholar 

  24. Korman, A., Kutten, S., Peleg, D.: Proof Labeling Schemes. In: Proceedings of the 24th annual ACM symposium on principles of distributed computing, Las Vegas, NV, USA (2005)

  25. Kutten S. and Peleg D. (1998). Fast distributed construction of small k-dominating sets and applications. J. algorithms 28(1): 40–66

    Article  MATH  MathSciNet  Google Scholar 

  26. King V. (1997). A simpler minimum spanning tree verification algorithm. Algorithmica 18: 263–270

    Article  MATH  MathSciNet  Google Scholar 

  27. King V., Poon C.K., Ramachandran V. and Sinha S. (1997). An optimal EREW PRAM algorithm for minimum spanning tree verification. Inform. Process. Lett. 62(3): 153–159

    Article  MathSciNet  Google Scholar 

  28. Kuhn F. and Watenhoffer R. (2005). Constant time distributed dominating set approximation. Distrib. Comput. 17(4): 303–310

    Article  Google Scholar 

  29. Kuhn, F., Moscibroda, T., Watenhoffer, R.: What cannot be computed locally! In: Proceedings of the ACM symposium on principles of distributed computing, pp. 300–309 (2004)

  30. Naor M. and Stockmeyer L.J. (1995). What can be computed locally?. SIAM J. Comput. 24(6): 1259–1277

    Article  MATH  MathSciNet  Google Scholar 

  31. Pettie S. and Ramachandran V. (2002). An optimal minimum spanning tree algorithm. J. ACM 49(1): 16–34

    Article  MathSciNet  Google Scholar 

  32. Tarjan R.E. (1982). Sensitivity analysis of minimum spanning trees and shortest paths trees. Inform. Process. Lett. 14: 30–33

    Article  MathSciNet  Google Scholar 

  33. Tarjan R.E. (1983). Data Structures and Network Algorithms. SIAM, Philadelphia

    Google Scholar 

  34. Tarjan E.E. (1979). Applications of path compression on balanced trees. J. ACM 26: 690–715

    Article  MATH  MathSciNet  Google Scholar 

  35. Yao, A.C.: Some complexity questions related to distributed computing. In: Proceedings of 11th annual ACM symposium on theory of computing, pp. 209–213 (1979)

Download references

Author information

Authors and Affiliations

  1. Information Systems Group, Faculty of IE&M, The Technion, Haifa, 32000, Israel

    Amos Korman & Shay Kutten

Authors
  1. Amos Korman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shay Kutten
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Amos Korman.

Additional information

A preliminary version of this work was presented in ACM PODC 2006.

A. Korman was supported in part at the Technion by an Aly Kaufman fellowship.

S. Kutten was supported in part by a grant from the Israeli Ministry for Science and Technology.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Korman, A., Kutten, S. Distributed verification of minimum spanning trees. Distrib. Comput. 20, 253–266 (2007). https://doi.org/10.1007/s00446-007-0025-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00446-007-0025-1

Keywords

Search

Navigation