Distributed Computing

, Volume 25, Issue 3, pp 189–205 | Cite as

Efficient distributed approximation algorithms via probabilistic tree embeddings

  • Maleq Khan
  • Fabian Kuhn
  • Dahlia Malkhi
  • Gopal Pandurangan
  • Kunal Talwar


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.


Generalized steiner forests Shortest paths Optimum routing cost spanning trees Least element lists Metric spaces Leader election Probabilistic tree embeddings 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Afek Y., Ricklin M.: Sparser: a paradigm for running distributed algorithms. J. Algorithms 14(2), 316–328 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 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)Google Scholar
  3. 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)Google Scholar
  4. 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)Google Scholar
  5. 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)Google Scholar
  6. 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)Google Scholar
  7. 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)Google Scholar
  8. 8.
    Cohen E.: Size-estimation framework with applications to transitive closure and reachability. J. Comput. Syst. Sci. 55(3), 441–453 (1997)zbMATHCrossRefGoogle Scholar
  9. 9.
    Cohen E., Kaplan H.: Spatially-decaying aggregation over a network. J. Comput. Syst. Sci. 73(3), 265–288 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 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)Google Scholar
  11. 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)Google Scholar
  12. 12.
    Elkin M.: Computing almost shortest paths. ACM Trans. Algorithms 1(2), 283–322 (2005)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Elkin M.: Distributed approximation: a survey. ACM SIGACT News 35(4), 40–57 (2004)CrossRefGoogle Scholar
  14. 14.
    Elkin M.: A faster distributed protocol for constructing a minimum spanning tree. J. Comput. Syst. Sci. 72(8), 1282–1308 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 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)zbMATHCrossRefGoogle Scholar
  18. 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)Google Scholar
  19. 19.
    Jia L., Rajaraman R., Suel R.: An efficient distributed algorithm for constructing small dominating sets. Distrib. Comput. 15(4), 193–205 (2002)CrossRefGoogle Scholar
  20. 20.
    Khan M., Pandurangan G.: A fast distributed approximation algorithm for minimum spanning trees. Distrib. Comput. 20, 391–402 (2008)CrossRefGoogle Scholar
  21. 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)Google Scholar
  22. 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)Google Scholar
  23. 23.
    Kutten S., Peleg D.: Fast distributed construction of k-dominating sets and applications. J. Algorithms 28, 40–66 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Nutov, Z., Sadeh, A.: Distributed primal-dual approximation algorithms for network design problems. Manuscript, (2009).
  25. 25.
    Panconesi A., Srinivasan A.: Randomized distributed edge coloring via an extension of the Chernoff-Hoeffding bounds. SIAM J. Comput. 26, 350–368 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 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. 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)Google Scholar
  28. 28.
    Peleg D.: A time optimal leader election algorithm in general networks. J. Parallel Distrib. Comput. 8, 96–99 (1990)CrossRefGoogle Scholar
  29. 29.
    Peleg D.: Distributed Computing: A Locality-Sensitive Approach. SIAM, (2000)Google Scholar
  30. 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)Google Scholar
  31. 31.
    Tel G.: Introduction to Distributed Algorithms. Cambridge University Press, Cambridge (1994)zbMATHCrossRefGoogle Scholar
  32. 32.
    Vazirani V.: Approximation Algorithms. Springer, New York (2004)Google Scholar
  33. 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)Google Scholar
  34. 34.
    Wu B., Chao K.: Spanning Trees and Optimization Problems. Chapman and Hall/CRC, Boca Raton (2004)zbMATHGoogle Scholar
  35. 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)Google Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Maleq Khan
    • 1
  • Fabian Kuhn
    • 2
  • Dahlia Malkhi
    • 3
  • Gopal Pandurangan
    • 4
  • Kunal Talwar
    • 3
  1. 1.Network Dynamics and Simulation Science LaboratoryVirginia Bioinformatics Institute, Virginia TechBlacksburgUSA
  2. 2.Faculty of InformaticsUniversity of LuganoLuganoSwitzerland
  3. 3.Microsoft ResearchMountain ViewUSA
  4. 4.Division of Mathematical SciencesNanyang Technological UniversitySingaporeSingapore

Personalised recommendations