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Tight Bounds for Distributed Minimum-Weight Spanning Tree Verification

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Abstract

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.

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Notes

  1. \(\tilde{\varOmega}\) (respectively \(\tilde{O}\)) is a relaxed variant of the Ω (rep., O) notation that ignores polylog factors.

  2. Equivalently, we may consider also disconnected graphs, and require T to be a spanning forest of G.

References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  6. Burns, J.E.: A formal model for message passing systems. Technical report TR-91, Computer Science Dept., Indiana University, Bloomington (1980)

  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)

    Article  Google Scholar 

  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. 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. 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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Dolev, S., Gouda, M., Schneider, M.: Requirements for silent stabilization. Acta Inform. 36(6), 447–462 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fraigniaud, P., Korman, A., Peleg, D.: Local distributed decision. In: FOCS, pp. 708–717 (2011)

    Google Scholar 

  14. Fraigniaud, P., Korman, A., Parter, M., Peleg, D.: Randomized distributed decision. In: DISC, pp. 371–385 (2012)

    Google Scholar 

  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. 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. 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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  19. Graham, R.L., Hell, P.: On the history of the minimum spanning tree problem. Ann. Hist. Comput. 7(1), 43–47 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  22. Katz, M., Katz, N.A., Korman, A., Peleg, D.: Labeling schemes for flow and connectivity. SIAM J. Comput. 34(1), 23–40 (2005)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  25. Komlös, J.: Linear verification for spanning trees. Combinatorica 5, 57–65 (1985)

    Article  MathSciNet  Google Scholar 

  26. Korman, A., Kutten, S.: Distributed verification of minimum spanning trees. Distrib. Comput. 20(4), 253–266 (2007)

    Article  Google Scholar 

  27. Korman, A., Kutten, S., Peleg, D.: Proof labeling schemes. Distrib. Comput. 22(4), 215–233 (2010)

    Article  Google Scholar 

  28. Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge Univ. Press, New York (1997)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  30. Kutten, S., Pandurangan, G., Peleg, D., Robinson, P., Trehan, A.: Universal bounds for leader election. Unpublished manuscript (2012)

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  34. Rubinovich, V.: Distributed minimum spanning tree construction. Master’s thesis, Department of Computer Science and Applied Mathematics, The Weizmann Institute of Science (1999)

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science and Applied Mathematics, The Weizmann Institute of Science, Rehovot, 76100, Israel

    Liah Kor & David Peleg

  2. CNRS and University Paris Diderot – Paris 7, Paris, France

    Amos Korman

Authors
  1. Liah Kor
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  2. Amos Korman
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  3. David Peleg
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Correspondence to Amos Korman.

Additional information

L.K. and D.P. are supported by a grant from the United States-Israel Binational Science Foundation (BSF).

A.K. is supported by the ANR project DISPLEXITY, and by the INRIA project GANG.

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Kor, L., Korman, A. & Peleg, D. Tight Bounds for Distributed Minimum-Weight Spanning Tree Verification. Theory Comput Syst 53, 318–340 (2013). https://doi.org/10.1007/s00224-013-9479-7

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