Theory of Computing Systems

, Volume 53, Issue 2, pp 318–340 | Cite as

Tight Bounds for Distributed Minimum-Weight Spanning Tree Verification

  • Liah Kor
  • Amos Korman
  • David Peleg


This paper introduces the notion of distributed verification without preprocessing. It focuses on the Minimum-weight Spanning Tree (MST) verification problem and establishes tight upper and lower bounds for the time and message complexities of this problem. Specifically, we provide an MST verification algorithm that achieves simultaneously \(\tilde{O}(m)\) messages and \(\tilde{O}(\sqrt{n} + D)\) time, where m is the number of edges in the given graph G, n is the number of nodes, and D is G’s diameter. On the other hand, we show that any MST verification algorithm must send \(\tilde{\varOmega}(m)\) messages and incur \(\tilde{\varOmega}(\sqrt{n} + D)\) time in worst case.

Our upper bound result appears to indicate that the verification of an MST may be easier than its construction, since for MST construction, both lower bounds of \(\tilde{\varOmega}(m)\) messages and \(\tilde{\varOmega}(\sqrt{n} + D)\) time hold, but at the moment there is no known distributed algorithm that constructs an MST and achieves simultaneously \(\tilde{O}(m)\) messages and \(\tilde{O}(\sqrt{n} + D)\) time. Specifically, the best known time-optimal algorithm (using \({\tilde{O}}(\sqrt {n} + D)\) time) requires O(m+n 3/2) messages, and the best known message-optimal algorithm (using \({\tilde{O}}(m)\) messages) requires O(n) time. On the other hand, our lower bound results indicate that the verification of an MST is not significantly easier than its construction.


Distributed algorithms Distributed verification Labeling schemes Minimum-weight spanning tree 


  1. 1.
    Afek, Y., Kutten, S., Yung, M.: The local detection paradigm and its applications to self stabilization. Theor. Comput. Sci. 186(1–2), 199–230 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Awerbuch, B.: Optimal distributed algorithms for minimum weight spanning tree, counting, leader election, and related problems. In: Proc. 19th ACM Symp. on Theory of Computing (STOC), NY, pp. 230–240 (1987) Google Scholar
  3. 3.
    Awerbuch, B., Patt-Shamir, B., Varghese, G.: Self-stabilization by local checking and correction. In: Proc. IEEE Symp. on the Foundations of Computer Science, pp. 268–277 (1991) Google Scholar
  4. 4.
    Awerbuch, B., Goldreich, O., Vainish, R., Peleg, D.: A trade-off between information and communication in broadcast protocols. J. ACM 37(2), 238–256 (1990) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Buchsbaum, A.L., Georgiadis, L., Kaplan, H., Rogers, A., Tarjan, R.E., Westbrook, J.R.: Linear-time algorithms for dominators and other path-evaluation problems. SIAM J. Comput. 38(4), 1533–1573 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Burns, J.E.: A formal model for message passing systems. Technical report TR-91, Computer Science Dept., Indiana University, Bloomington (1980) Google Scholar
  7. 7.
    Cidon, I., Gopal, I., Kaplan, M., Kutten, S.: A distributed control architecture of high-speed networks. IEEE Trans. Commun. 43(5), 1950–1960 (1995) CrossRefGoogle Scholar
  8. 8.
    Das Sarma, A., Holzer, S., Kor, L., Korman, A., Nanongkai, D., Pandurangan, G., Peleg, D., Wattenhofer, R.: Distributed verification and hardness of distributed approximation. In: Proc. 43th ACM Symp. on Theory of Computing (STOC) (2011) Google Scholar
  9. 9.
    Das Sarma, A., Nanongkai, D., Pandurangan, G.: A tight unconditional lower bound on distributed random walk computation. In: Proc. 30th ACM SIGACT-SIGOPS Symp. on Principles of Distributed Computing (PODC) (2011) Google Scholar
  10. 10.
    Dixon, B., Rauch, M., Tarjan, R.E.: Verification and sensitivity analysis of minimum spanning trees in linear time. SIAM J. Comput. 21(6), 1184–1192 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Dixon, B., Tarjan, R.E.: Optimal parallel verification of minimum spanning trees in logarithmic time. Algorithmica 17(1), 11–18 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Dolev, S., Gouda, M., Schneider, M.: Requirements for silent stabilization. Acta Inform. 36(6), 447–462 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Fraigniaud, P., Korman, A., Peleg, D.: Local distributed decision. In: FOCS, pp. 708–717 (2011) Google Scholar
  14. 14.
    Fraigniaud, P., Korman, A., Parter, M., Peleg, D.: Randomized distributed decision. In: DISC, pp. 371–385 (2012) Google Scholar
  15. 15.
    Frederickson, G.N., Lynch, N.A.: The impact of synchronous communication on the problem of electing a leader in a ring. In: Proc. 16th ACM Symp. on Theory of Computing (STOC), NY, pp. 493–503 (1984) Google Scholar
  16. 16.
    Fredman, M.L., Willard, D.E.: Trans-dichotomous algorithms for minimum spanning trees and shortest paths. In: Proc. 31st IEEE FOCS, pp. 719–725 (1990) Google Scholar
  17. 17.
    Gallager, R.G., Humblet, P.A., Spira, P.M.: A distributed algorithm for minimum-weight spanning trees. ACM Trans. Program. Lang. Syst. 5(1), 66–77 (1983) zbMATHCrossRefGoogle Scholar
  18. 18.
    Garay, J., Kutten, S.A., Peleg, D.: A sub-linear time distributed algorithm for minimum-weight spanning trees. SIAM J. Comput. 27(1), 302–316 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Graham, R.L., Hell, P.: On the history of the minimum spanning tree problem. Ann. Hist. Comput. 7(1), 43–47 (1985) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Harel, D.: A linear time algorithm for finding dominators in flow graphs and related problems. In: Proc. 17th ACM Symp. on Theory of Computing (STOC), pp. 185–194 (1985) Google Scholar
  21. 21.
    Karger, D.R., Klein, P.N., Tarjan, R.E.: A randomized linear-time algorithm to find minimum spanning trees. J. ACM 42(2), 321–328 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Katz, M., Katz, N.A., Korman, A., Peleg, D.: Labeling schemes for flow and connectivity. SIAM J. Comput. 34(1), 23–40 (2005) MathSciNetCrossRefGoogle Scholar
  23. 23.
    King, V.: A simpler minimum spanning tree verification algorithm. Algorithmica 18, 263–270 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    King, V., Poon, C.K., Ramachandran, V., Sinha, S.: An optimal EREW PRAM algorithm for minimum spanning tree verification. Inf. Process. Lett. 62(3), 153–159 (1997) MathSciNetCrossRefGoogle Scholar
  25. 25.
    Komlös, J.: Linear verification for spanning trees. Combinatorica 5, 57–65 (1985) MathSciNetCrossRefGoogle Scholar
  26. 26.
    Korman, A., Kutten, S.: Distributed verification of minimum spanning trees. Distrib. Comput. 20(4), 253–266 (2007) CrossRefGoogle Scholar
  27. 27.
    Korman, A., Kutten, S., Peleg, D.: Proof labeling schemes. Distrib. Comput. 22(4), 215–233 (2010) CrossRefGoogle Scholar
  28. 28.
    Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge Univ. Press, New York (1997) zbMATHGoogle Scholar
  29. 29.
    Kutten, S., Peleg, D.: Fast distributed construction of small k-dominating sets and applications. J. Algorithms 28(1), 40–66 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Kutten, S., Pandurangan, G., Peleg, D., Robinson, P., Trehan, A.: Universal bounds for leader election. Unpublished manuscript (2012) Google Scholar
  31. 31.
    Lotker, Z., Patt-Shamir, B., Peleg, D.: Distributed mst for constant diameter graphs. In: Proc. 20th ACM Symp. on Principles of Distributed Computing (PODC), NY, pp. 63–71 (2001) Google Scholar
  32. 32.
    Peleg, D., Rubinovich, V.: A near-tight lower bound on the time complexity of distributed minimum-weight spanning tree construction. SIAM J. Comput. 30(5), 1427–1442 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Pettie, S., Ramachandran, V.: An optimal minimum spanning tree algorithm. J. ACM 49(1), 16–34 (2002) MathSciNetCrossRefGoogle Scholar
  34. 34.
    Rubinovich, V.: Distributed minimum spanning tree construction. Master’s thesis, Department of Computer Science and Applied Mathematics, The Weizmann Institute of Science (1999) Google Scholar
  35. 35.
    Tarjan, R.E.: Applications of path compression on balanced trees. J. ACM 26, 690–715 (1979) MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Tarjan, R.E.: Data Structures and Network Algorithms. SIAM, Philadelphia (1983) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Computer Science and Applied MathematicsThe Weizmann Institute of ScienceRehovotIsrael
  2. 2.CNRS and University Paris Diderot – Paris 7ParisFrance

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