Distributed Algorithms for Network Diameter and Girth

  • David Peleg
  • Liam Roditty
  • Elad Tal
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7392)

Abstract

This paper considers the problem of computing the diameter D and the girth g of an n-node network in the CONGEST distributed model. In this model, in each synchronous round, each vertex can transmit a different short (say, O(logn) bits) message to each of its neighbors. We present a distributed algorithm that computes the diameter of the network in O(n) rounds. We also present two distributed approximation algorithms. The first computes a 2/3 multiplicative approximation of the diameter in \(O(D\sqrt n \log n)\) rounds. The second computes a 2 − 1/g multiplicative approximation of the girth in \(O(D+\sqrt{gn}\log n)\) rounds. Recently, Frischknecht, Holzer and Wattenhofer [11] considered these problems in the CONGEST model but from the perspective of lower bounds. They showed an \(\tilde{\Omega}(n)\) rounds lower bound for exact diameter computation. For diameter approximation, they showed a lower bound of \(\tilde{\Omega}(\sqrt n)\) rounds for getting a multiplicative approximation of Open image in new window . Both lower bounds hold for networks with constant diameter. For girth approximation, they showed a lower bound of \(\tilde{\Omega}(\sqrt n)\) rounds for getting a multiplicative approximation of Open image in new window on a network with constant girth. Our exact algorithm for computing the diameter matches their lower bound. Our diameter and girth approximation algorithms almost match their lower bounds for constant diameter and for constant girth.

Keywords

Short Path Network Diameter Multiplicative Approximation Close Vertex Congest Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Aingworth, D., Chekuri, C., Indyk, P., Motwani, R.: Fast estimation of diameter and shortest paths (without matrix multiplication). SIAM J. Comput. 28(4), 1167–1181 (1999)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Almeida, P.S., Baquero, C., Cunha, A.: Fast distributed computation of distances in networks. Technical report (2011)Google Scholar
  3. 3.
    Antonio, J.K., Huang, G.M., Tsai, W.K.: A fast distributed shortest path algorithm for a class of hierarchically clustered data networks. IEEE Trans. Computers 41, 710–724 (1992)CrossRefGoogle Scholar
  4. 4.
    Baswana, S., Kavitha, T.: Faster algorithms for approximate distance oracles and all-pairs small stretch paths. In: FOCS, pp. 591–602. IEEE Computer Society (2006)Google Scholar
  5. 5.
    Cicerone, S., D’Angelo, G., Di Stefano, G., Frigioni, D., Petricola, A.: Partially dynamic algorithms for distributed shortest paths and their experimental evaluation. J. Computers 2, 16–26 (2007)Google Scholar
  6. 6.
    Cidon, I., Jaffe, J.M., Sidi, M.: Local distributed deadlock detection by cycle detection and clustering. IEEE Trans. Software Eng. 13(1), 3–14 (1987)MATHCrossRefGoogle Scholar
  7. 7.
    Dijkstra, E.W.: A note on two problems in connection with graphs. Numer. Math., 269–271 (1959)Google Scholar
  8. 8.
    Dor, D., Halperin, S., Zwick, U.: All-pairs almost shortest paths. SIAM J. Comput. 29(5), 1740–1759 (2000)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Elkin, M.: Computing almost shortest paths. ACM Transactions on Algorithms 1(2), 283–323 (2005)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Floyd, R.W.: Algorithm 97: shortest path. Comm. ACM 5, 345 (1962)CrossRefGoogle Scholar
  11. 11.
    Frischknecht, S., Holzer, S., Wattenhofer, R.: Networks cannot compute their diameter in sublinear time. In: Proc. 23rd ACM-SIAM Symp. on Discrete Algorithms, SODA (2012)Google Scholar
  12. 12.
    Haldar, S.: An ’all pairs shortest paths’ distributed algorithm using 2n 2 messages. J. Algorithms, 20–36 (1997)Google Scholar
  13. 13.
    Holzer, S., Wattenhofer, R.: Optimal distributed all pairs shortest paths and applications. In: Proc. 31st Annual ACM SIGACT-SIGOPS Symp. on Principles of Distributed Computing, PODC (2012)Google Scholar
  14. 14.
    Itai, A., Rodeh, M.: Finding a minimum circuit in a graph. SIAM J. Computing 7(4), 413–423 (1978)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Kanchi, S., Vineyard, D.: Time optimal distributed all pairs shortest path problem. Int. J. of Information Theories and Applications, 141–146 (2004)Google Scholar
  16. 16.
    Kavitha, T., Liebchen, C., Mehlhorn, K., Michail, D., Rizzi, R., Ueckerdt, T., Zweig, K.A.: Cycle bases in graphs characterization, algorithms, complexity, and applications. Computer Science Review 3(4), 199–243 (2009)CrossRefGoogle Scholar
  17. 17.
    Krivelevich, M., Nutov, Z., Yuster, R.: Approximation algorithms for cycle packing problems. In: Proc. SODA, pp. 556–561 (2005)Google Scholar
  18. 18.
    Lingas, A., Lundell, E.-M.: Efficient approximation algorithms for shortest cycles in undirected graphs. Inf. Process. Lett. 109(10), 493–498 (2009)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Peleg, D.: Distributed Computing: A Locality-Sensitive Approach. SIAM (2000)Google Scholar
  20. 20.
    Roditty, L., Tov, R.: Approximating the girth. In: Proc. SODA, pp. 1446–1454 (2011)Google Scholar
  21. 21.
    Roditty, L., Vassilevska Williams, V.: Minimum weight cycles and triangles: Equivalences and algorithms. In: Proc. FOCS, pp. 180–189 (2011)Google Scholar
  22. 22.
    Roditty, L., Vassilevska Williams, V.: Subquadratic time approximation algorithms for the girth. In: SODA, pp. 833–845 (2012)Google Scholar
  23. 23.
    Segall, A.: Distributed network protocols. IEEE Trans. Inf. Th. IT-29, 23–35 (1983)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Seidel, R.: On the all-pairs-shortest-path problem in unweighted undirected graphs. JCSS 51, 400–403 (1995)MathSciNetGoogle Scholar
  25. 25.
    Warshall, S.: A theorem on boolean matrices. J. ACM 9(1), 11–12 (1962)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Vassilevska Williams, V.: Private communicationGoogle Scholar
  27. 27.
    Vassilevska Williams, V.: Breaking the coppersmith-winograd barrier. In: STOC (2012)Google Scholar
  28. 28.
    Yuster, R.: Computing the diameter polynomially faster than apsp. CoRR, abs/1011.6181 (2010)Google Scholar
  29. 29.
    Zwick, U.: All pairs shortest paths using bridging sets and rectangular matrix multiplication. JACM 49(3), 289–317 (2002)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • David Peleg
    • 1
  • Liam Roditty
    • 2
  • Elad Tal
    • 2
  1. 1.Department of Computer Science and Applied MathematicsThe Weizmann Institute of ScienceRehovotIsrael
  2. 2.Department of Computer ScienceBar-Ilan UniversityRamat-GanIsrael

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