Skip to main content
SpringerLink
Log in

Efficient distributed approximation algorithms via probabilistic tree embeddings

  • Published:
Distributed Computing Aims and scope Submit manuscript

Abstract

We present a uniform approach to design efficient distributed approximation algorithms for various fundamental network optimization problems. Our approach is randomized and based on a probabilistic tree embedding due to Fakcharoenphol et al. (J Comput Syst Sci 69(3):485–497, 2004) (FRT embedding). We show how to efficiently compute an (implicit) FRT embedding in a decentralized manner and how to use the embedding to obtain efficient expected O(log n)-approximate distributed algorithms for various problems, in particular the generalized Steiner forest problem (including the minimum Steiner tree problem), the minimum routing cost spanning tree problem, and the k-source shortest paths problem. The distributed construction of the FRT embedding is based on the computation of least elements (LE) lists, a distributed data structure that is of independent interest. Assuming a global order on the nodes of a network, the LE-list of a node stores the smallest node (w.r.t. the given order) within every distance d (cf. Cohen in J Comput Syst Sci 55(3):441–453, 1997, Cohen and Kaplan in J Comput Syst Sci 73(3):265–288, 2007). Assuming a random order on the nodes, we give a distributed algorithm for computing LE-lists on a weighted graph with time complexity O(S log n), where S is a graph parameter called the shortest path diameter which can be considered the weighted counterpart of the diameter D of the graph. For unweighted graphs, our LE-lists computation has asymptotically optimal time complexity of O(D). As a byproduct, we get an improved synchronous leader election algorithm for general networks that is both time-optimal and almost message-optimal with high probability.

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(2), 316–328 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  2. Awerbuch, B.: Optimal distributed algorithms for minimum weight spanning tree, counting, leader election, and related problems. In: Proceedings of the 19th ACM Symposium on Theory of Computing (STOC), pp. 230–240. ACM (1987)

  3. Bartal, Y.: Probabilistic approximations of metric spaces and its algorithmic applications. In: Proceedings of the 37th Annual Symposium on Foundations of Computer Science (FOCS), pp. 184–193. IEEE Computer Society (1996)

  4. Bartal, Y.: On approximating arbitrary metrics by tree metrics. In: Proceedings of the 30th ACM Symposium on Theory of Computing (STOC), pp. 161–168. ACM (1998)

  5. Bartal, Y., Byers, J., Raz, D.: Global optimization using local information with applications to flow control. In: Proceedings of the 38th Annual Symposium on Foundations of Computer Science (FOCS), pp. 303–312. IEEE Computer Society (1997)

  6. Chalermsook, P., Fakcharoenphol, J.: Simple distributed algorithms for approximating minimum Steiner trees. In: Proceedings of the 11th Annual International Conference on Computing and Combinatorics (COCOON), pp. 380–389. Springer (2005)

  7. Charikar, M., Chekuri, C., Goel, A., Guha, S., Plotkin, S.: Approximating a finite metric by a small number of tree metrics. In: Proceedings of the 39th Annual Symposium on Foundations of Computer Science (FOCS), pp. 379–388. IEEE Computer Society (1998)

  8. Cohen E.: Size-estimation framework with applications to transitive closure and reachability. J. Comput. Syst. Sci. 55(3), 441–453 (1997)

    Article  MATH  Google Scholar 

  9. Cohen E., Kaplan H.: Spatially-decaying aggregation over a network. J. Comput. Syst. Sci. 73(3), 265–288 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Sarma, A., Holzer, S., Kor, L., Korman, A., Nanongkai, D., Pandurangan, G., Peleg, D., Wattenhofer, R.: Distributed verification and hardness of distributed approximation. In: Proceedings of the 43rd ACM Symposium on Theory of Computing (STOC), pp. 363–372. ACM (2011)

  11. Dubhashi, D., Grandioni, F., Panconesi, A.: Distributed approximation algorithms via LP duality and randomization. In: Gonzalez T. (ed.) Handbook of Approximation Algorithms and Metaheuristics, CRC Press, Boca Raton (2007)

  12. Elkin M.: Computing almost shortest paths. ACM Trans. Algorithms 1(2), 283–322 (2005)

    Article  MathSciNet  Google Scholar 

  13. Elkin M.: Distributed approximation: a survey. ACM SIGACT News 35(4), 40–57 (2004)

    Article  Google Scholar 

  14. Elkin M.: A faster distributed protocol for constructing a minimum spanning tree. J. Comput. Syst. Sci. 72(8), 1282–1308 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Elkin M.: An unconditional lower bound on the time-approximation tradeoff for the minimum spanning tree problem. SIAM J. Comput. 36(2), 433–456 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fakcharoenphol J., Rao S., Talwar K.: A tight bound on approximating arbitrary metrics by tree metrics. J. Comput. Syst. Sci. 69(3), 485–497 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gallager R., Humblet P., Spira P.: A distributed algorithm for minimum-weight spanning trees. ACM Trans. Program. Lang. Syst. 5(1), 66–77 (1983)

    Article  MATH  Google Scholar 

  18. Grandoni, F., Könemann, J., Panconesi, A., Sozio, M.: Primal-dual based distributed algorithms for vertex cover with semi-hard capacities. In: Proceedings of the 24th ACM symposium on Principles of distributed computing (PODC), pp. 118-125. ACM (2005)

  19. Jia L., Rajaraman R., Suel R.: An efficient distributed algorithm for constructing small dominating sets. Distrib. Comput. 15(4), 193–205 (2002)

    Article  Google Scholar 

  20. Khan M., Pandurangan G.: A fast distributed approximation algorithm for minimum spanning trees. Distrib. Comput. 20, 391–402 (2008)

    Article  Google Scholar 

  21. Kuhn, F., Moscibroda, T., Wattenhofer, R.: What cannot be computed locally! In: Proceedings of the 23rd ACM symposium on Principles of distributed computing (PODC), pp. 300–309. ACM (2004)

  22. Kuhn, F., Moscibroda, T., Wattenhofer, R.: The price of being near-sighted. In: Proceedings of the 17th ACM-SIAM Symposium on Discrete Algorithms (SODA). pp. 980–989. SIAM (2006)

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Nutov, Z., Sadeh, A.: Distributed primal-dual approximation algorithms for network design problems. Manuscript, (2009). http://www.openu.ac.il/home/nutov/distributed.pdf

  25. Panconesi A., Srinivasan A.: Randomized distributed edge coloring via an extension of the Chernoff-Hoeffding bounds. SIAM J. Comput. 26, 350–368 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  26. Pandurangan G., Khan M.: Theory of communication networks. In: Atallah, M.J., Blanton, M. (eds) Algorithms and Theory of Computation Handbook., CRC Press, Boca Raton (2009)

    Google Scholar 

  27. Papadimitriou, C., Yannakakis, M.: Linear programming without matrix. In: Proceedings of the 25th ACM Symposium on Theory of Computing (STOC), pp. 121-129, ACM (1993)

  28. Peleg D.: A time optimal leader election algorithm in general networks. J. Parallel Distrib. Comput. 8, 96–99 (1990)

    Article  Google Scholar 

  29. Peleg D.: Distributed Computing: A Locality-Sensitive Approach. SIAM, (2000)

  30. Peleg, D., Rabinovich, V.: A near-tight lower bound on the time complexity of distributed mst construction. In: Proceedings of the 40th Annual Symposium on Foundations of Computer Science (FOCS), pp. 379–388. IEEE Computer Society (1999)

  31. Tel G.: Introduction to Distributed Algorithms. Cambridge University Press, Cambridge (1994)

    Book  MATH  Google Scholar 

  32. Vazirani V.: Approximation Algorithms. Springer, New York (2004)

    Google Scholar 

  33. Wattenhofer, M., Wattenhofer, R.: Distributed weighted matching. In: Proceedings of the 18th Conference on Distributed Computing (DISC), pp. 335–348. Springer, New York (2004)

  34. Wu B., Chao K.: Spanning Trees and Optimization Problems. Chapman and Hall/CRC, Boca Raton (2004)

    MATH  Google Scholar 

  35. Wu, B., Lancia, G., Bafna, V., Chao, K., Ravi, R., Tang, C.: A polynomial time approximation scheme for minimum routing cost spanning trees. In: Proceedings of the 9th ACM-SIAM Symposium on Discrete Algorithms (SODA). pp. 21–32. SIAM (1998)

Download references

Author information

Authors and Affiliations

  1. Network Dynamics and Simulation Science Laboratory, Virginia Bioinformatics Institute, Virginia Tech, Blacksburg, VA, 24061, USA

    Maleq Khan

  2. Faculty of Informatics, University of Lugano, 6904, Lugano, Switzerland

    Fabian Kuhn

  3. Microsoft Research, Mountain View, CA, 94043, USA

    Dahlia Malkhi & Kunal Talwar

  4. Division of Mathematical Sciences, Nanyang Technological University, Nanyang Avenue, 648477, Singapore, Singapore

    Gopal Pandurangan

Authors
  1. Maleq Khan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fabian Kuhn
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dahlia Malkhi
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Gopal Pandurangan
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Kunal Talwar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maleq Khan.

Additional information

M. Khan was supported in part by NSF Grants CNS-0626964 and CCF-1011769.

G. Pandurangan was Supported in part by Nanyang Technological University grant M58110000. Part of the work done when the author was at Purdue University and supported in part by NSF grant CCF-0830476.

A preliminary version of this paper appeared in Proceedings of the 27th Annual ACM Symposium on Principles of Distributed Computing (PODC), pp. 263–272. ACM (2008).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Khan, M., Kuhn, F., Malkhi, D. et al. Efficient distributed approximation algorithms via probabilistic tree embeddings. Distrib. Comput. 25, 189–205 (2012). https://doi.org/10.1007/s00446-012-0157-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00446-012-0157-9

Keywords

Search

Navigation