Skip to main content
SpringerLink
Log in

Sublinear Fully Distributed Partition with Applications

  • Published:
Theory of Computing Systems Aims and scope Submit manuscript

Abstract

We present new efficient deterministic and randomized distributed algorithms for decomposing a graph with n nodes into a disjoint set of connected clusters with radius at most k−1 and having O(n 1+1/k) intercluster edges. We show how to implement our algorithms in the distributed \(\mathcal{CONGEST}\) model of computation, i.e., limited message size, which improves the time complexity of previous algorithms (Moran and Snir in Theor. Comput. Sci. 243(1–2):217–241, 2000; Awerbuch in J. ACM 32:804–823, 1985; Peleg in Distributed Computing: A Locality-Sensitive Approach, 2000) from O(n) to O(n 1−1/k). We apply our algorithms for constructing low stretch graph spanners and network synchronizers in sublinear deterministic time in the \(\mathcal{CONGEST}\) model.

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., Ricklin, M.: Sparser: a paradigm for running distributed algorithms. J. Algorithms 14, 316–328 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  2. Althofer, I., Das, G., Dobkin, D., Joseph, D., Soares, J.: On sparse spanners of weighted graphs. Discrete Comput. Geom. 9, 81–100 (1993)

    Article  MathSciNet  Google Scholar 

  3. Awerbuch, B.: Complexity of network synchronization. J. ACM 32, 804–823 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  4. Awerbuch, B., Goldberg, A.V., Luby, M., Poltkin, S.A.: Network decomposition and locality in distributed computation. In: 30th IEEE Symposium on Foundation of Computer Science (FOCS89), pp. 364–369 (1989)

  5. Awerbuch, B., Peleg, D.: Sparse partitions. In 31th Symposium on Foundations of Computer Science (FOCS90), pp. 503–513. IEEE Comput. Soc., Los Alamitos (1990)

    Google Scholar 

  6. Awerbuch, B., Peleg, D.: Network synchronization with polylogarithmic overhead. In: 31st IEEE Symposium on Foundations of Computer Science (FOCS90), vol. 2, pp. 503–513 (1990)

  7. Awerbuch, B., Peleg, D.: Routing with polynomial communication-space trade-off. SIAM J. Discrete Math. 5, 151–162 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  8. Awerbuch, B., Peleg, D.: Online tracking of mobile users. J. ACM 42, 1021–1058 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  9. Awerbuch, B., Berger, B., Cowen, L.J., Peleg, D.: Fast distributed network decompositions and covers. J. Parallel Distrib. Comput. 39, 105–114 (1996)

    Article  MATH  Google Scholar 

  10. Awerbuch, B., Berger, B., Cowen, L.J., Peleg, D.: Near-linear time construction of sparse neighborhood covers. SIAM J. Comput. 28(1), 263–277 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. Baswana, S., Sen, S.: A simple linear time algorithm for computing a (2k−1)-spanner of O(n 1+1/k) size in weighted graphs. In: 30th International Colloquium on Automata, Languages and Programming (ICALP03). Lecture Notes in Computer Science, vol. 2719, pp. 384–396. Springer, Berlin (2003)

    Chapter  Google Scholar 

  12. Baswana, S., Kavitha, T., Mehlhorn, K., Pettie, S.: New constructions of (α,β)-spanners and purely additive spanners. In: 16th Symposium on Discrete Algorithms (SODA05), pp. 672–681. ACM, New York (2005)

    Google Scholar 

  13. Belkouch, F., Bui, M., Chen, L., Datta, A.K.: Self-stabilizing deterministic network decomposition. J. Parralel Distrib. Comput. 62, 696–714 (2002)

    Article  MATH  Google Scholar 

  14. Chalopin, J., Métivier, Y.: A bridge between the asynchronous message passing model and local computations in graphs. In: Mathematical Foundations of Computer Science (MFCS05). Lecture Notes in Computer Science, vol. 3618, pp. 212–223. Springer, Berlin (2005)

    Chapter  Google Scholar 

  15. Cohen, E.: Fast algorithms for constructing t-spanners and paths with stretch t. SIAM J. Comput. 28(1), 210–236 (1998)

    Article  MATH  Google Scholar 

  16. Comellas, F., Ozón, J., Peters, J.G.: Deterministic small-world communication networks. Inf. Process. Lett. 76(1–2), 83–90 (2000)

    Article  Google Scholar 

  17. Cowen, L.J.: On local representations of graphs and networks. Ph.D. Thesis, MIT (1993)

  18. Derbel, B., Mosbah, M.: A fully distributed linear time algorithm for cluster network decomposition. In: 16th IASTED International Conference on Parallel and Distributed Computing and Systems (PDCS04), pp. 548–553 (2004)

  19. Derbel, B., Mosbah, M., Zemmari, A.: Fast distributed graph partition and application. In: 20th IEEE International Parallel & Distributed Processing Symposium (IPDPS06). IEEE Comput. Soc., Los Alamitos (2006)

    Google Scholar 

  20. Derbel, B., Gavoille, C., Peleg, D., Viennot, L.: On the locality of distributed sparse spanner construction. In: 27th Symposium on Principle of Distributed Computing (PODC), pp. 273–282 (2008)

  21. Elkin, M.: Computing almost shortest paths. In: 20th ACM Symposium on Principles of Distributed Computing (PODC01), pp. 53–62. ACM, New York (2001)

    Google Scholar 

  22. Elkin, M., Peleg, D.: (1+ε,β)-spanner constructions for general graphs. SIAM J. Comput. 33(3), 608–631 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. Elkin, M., Zhang, J.: Efficient algorithms for constructing (1+ε,β)-spanners in the distributed and streaming models. In: 23rd ACM Symposium on Principles of Distributed Computing (PODC04), pp. 160–168. ACM, New York (2004)

    Google Scholar 

  24. Gaber, I., Mansour, Y.: Centralized broadcast in multihop radio networks. J. Algorithms 46, 1–20 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  25. Garay, J.A., Kutten, S., Peleg, D.: A sublinear time distributed algorithm for minimum-weight spanning trees. SIAM J. Comput. 27, 302–316 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  26. Godard, E., Métivier, Y., Muscholl, A.: Characterizations of classes of graphs recognizable by local computations. Theory Comput. Syst. 37(2), 249–293 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  27. Halperin, S., Zwick, U.: Unpublished result (1996)

  28. 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 

  29. Linial, N.: Distributive graph algorithms—global solutions from local data. In: 28th IEEE Symposium on Foundations of Computer Science (FOCS87), pp. 331–335. IEEE Comput. Soc., Los Alamitos (1987)

    Google Scholar 

  30. Linial, N.: Locality in distributed graphs algorithms. SIAM J. Comput. 21(1), 193–201 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  31. Litovsky, I., Métivier, Y., Sopena, E.: Different local controls for graph relabelling systems. Math. Syst. Theory 28, 41–65 (1995)

    Article  MATH  Google Scholar 

  32. Litovsky, I., Métivier, Y., Sopena, E.: Graph relabelling systems and distributed algorithms. In: Handbook of Graph Grammars and Computing by Graph Transformation, vol. 3, pp. 1–56. World Scientific, Singapore (1999)

    Google Scholar 

  33. Métivier, Y., Saheb, N., Zemmari, A.: Randomized local elections. Inf. Process. Lett. 82, 313–120 (2002)

    Article  MATH  Google Scholar 

  34. Moran, S., Snir, S.: Simple and efficient network decomposition and synchronization. Theor. Comput. Sci. 243(1–2), 217–241 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  35. Panconesi, A., Srinivasan, A.: Improved distributed algorithms for coloring and network decomposition. In: 24th ACM Symposium on Theory of Computing (STOC92), pp. 581–592 (1992)

  36. Peleg, D.: Distributed Computing: A Locality-Sensitive Approach. SIAM Monographs on Discrete Mathematics and Applications (2000)

  37. 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 

  38. Pettie, S.: Low distortion spanners. In: 34th International Colloquium on Automata, Languages and Programming (ICALP), pp. 78–89 (2007)

  39. Pettie, S.: Distributed algorithms for ultrasparse spanners and linear size skeletons. In: 27th Symposium on Principle of Distributed Computing (PODC), pp. 253–262 (2008)

  40. Roditty, L., Thorup, M., Zwick, U.: Deterministic constructions of approximate distance oracles and spanners. In: 32nd International Colloquium on Automata, Languages and Programming (ICALP), Lecture Notes in Computer Science (2005)

  41. Shabtay, L., Segall, A.: Low complexity network synchronization. In: 8th International Workshop on Distributed Algorithms, pp. 223–237 (1994)

  42. Thorup, M., Zwick, U.: Spanners and emulators with sublinear distance errors. In: 17th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 802–809 (2006)

  43. Williams, D.: Probability with Martingals. Cambridge University Press, Cambridge (1993)

    Google Scholar 

  44. Woodruff, D.P.: Lower bounds for additive spanners, emulators, and more. In: 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 389–398 (2006)

Download references

Author information

Authors and Affiliations

  1. LIFL, INRIA, Université des Sciences et Technologies de Lille Bâtiment M3, 59655, Villeneuve d’Ascq Cedex, France

    Bilel Derbel

  2. LaBRI, Université Bordeaux I, ENSEIRB, 351 Cours de la Libération, 33405, Talence, France

    Mohamed Mosbah & Akka Zemmari

Authors
  1. Bilel Derbel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mohamed Mosbah
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Akka Zemmari
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mohamed Mosbah.

Additional information

Some materials presented in this paper appeared in two extended abstracts published in the proceedings of IPDPS06 (20th IEEE International Parallel & Distributed Processing Symposium) [19] and PDCS04 (16th IASTED International Conference on Parallel and Distributed Computing and Systems) [18].

Rights and permissions

Reprints and permissions

About this article

Cite this article

Derbel, B., Mosbah, M. & Zemmari, A. Sublinear Fully Distributed Partition with Applications. Theory Comput Syst 47, 368–404 (2010). https://doi.org/10.1007/s00224-009-9190-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-009-9190-x

Keywords

Search

Navigation