Near-Optimal Distributed Tree Embedding

  • Mohsen Ghaffari
  • Christoph Lenzen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8784)

Abstract

Tree embeddings are a powerful tool in the area of graph approximation algorithms. Essentially, they transform problems on general graphs into much easier ones on trees. Fakcharoenphol, Rao, and Talwar (FRT) [STOC’04] present a probabilistic tree embedding that transforms n-node metrics into (probability distributions over) trees, while stretching each pairwise distance by at most an O(logn) factor in expectation. This O(logn) stretch is optimal.

Khan et al. [PODC’08] present a distributed algorithm that implements FRT in O(SPD logn) rounds, where SPD is the shortest-path-diameter of the weighted graph, and they explain how to use this embedding for various distributed approximation problems. Note that SPD can be as large as Θ(n), even in graphs where the hop-diameter D is a constant. Khan et al. noted that it would be interesting to improve this complexity. We show that this is indeed possible.

More precisely, we present a distributed algorithm that constructs a tree embedding that is essentially as good as FRT in \(\tilde{O}(\min\{n^{0.5+\varepsilon },\operatorname{SPD}\}+D)\) rounds, for any constant ε > 0. A lower bound of \(\tilde{\Omega}(\min\{n^{0.5},\operatorname{SPD}\}+D)\) rounds follows from Das Sarma et al. [STOC’11], rendering our round complexity near-optimal.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alon, N., Karp, R.M., Peleg, D., West, D.: A graph-theoretic game and its application to the k-server problem. SIAM J. Comput. 24(1), 78–100 (1995)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: Proc. of the Symp. on Found. of Comp. Sci. (FOCS), pp. 184–193 (1996)Google Scholar
  3. 3.
    Bartal, Y.: On approximating arbitrary metrices by tree metrics. In: Proc. of the Symp. on Theory of Comp. (STOC), pp. 161–168 (1998)Google Scholar
  4. 4.
    Baswana, S., Sen, S.: A simple and linear time randomized algorithm for computing sparse spanners in weighted graphs. Random Structures and Algorithms 30(4), 532–563 (2007)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bourgain, J.: On Lipschitz embedding of finite metric spaces in Hilbert space. Israel Journal of Mathematics 52(1-2), 46–52 (1985)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Cohen, E.: Size-estimation framework with applications to transitive closure and reachability. Journal of Computer and System Sciences 55(3), 441–453 (1997)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    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. of the Symp. on Theory of Comp. (STOC), pp. 363–372 (2011)Google Scholar
  8. 8.
    Elkin, M.: Unconditional lower bounds on the time-approximation tradeoffs for the distributed minimum spanning tree problem. In: Proc. of the Symp. on Theory of Comp. (STOC), pp. 331–340 (2004)Google Scholar
  9. 9.
    Fakcharoenphol, J., Rao, S., Talwar, K.: A tight bound on approximating arbitrary metrics by tree metrics. In: Proc. of the Symp. on Theory of Comp. (STOC), pp. 448–455 (2003)Google Scholar
  10. 10.
    Fakcharoenphol, J., Rao, S., Talwar, K.: A tight bound on approximating arbitrary metrics by tree metrics. Journal of Computer and System Sciences 69(3), 485–497 (2004)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Ghaffari, M.: Near-optimal distributed approximation of minimum-weight connected dominating set. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014, Part II. LNCS, vol. 8573, pp. 483–494. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  12. 12.
    Ghaffari, M., Kuhn, F.: Distributed minimum cut approximation. In: Afek, Y. (ed.) DISC 2013. LNCS, vol. 8205, pp. 1–15. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  13. 13.
    Holzer, S., Wattenhofer, R.: Optimal distributed all pairs shortest paths and applications. In: The Proc. of the Int’l Symp. on Princ. of Dist. Comp. (PODC), pp. 355–364 (2012)Google Scholar
  14. 14.
    Indyk, P., Matousek, J.: Low-distortion embeddings of finite metric spaces. In: Handbook of Discrete and Computational Geometry, vol. 37, p. 46 (2004)Google Scholar
  15. 15.
    Khan, M., Kuhn, F., Malkhi, D., Pandurangan, G., Talwar, K.: Efficient distributed approximation algorithms via probabilistic tree embeddings. In: The Proc. of the Int’l Symp. on Princ. of Dist. Comp. (PODC), pp. 263–272 (2008)Google Scholar
  16. 16.
    Lenzen, C., Patt-Shamir, B.: Fast routing table construction using small messages: Extended abstract. In: Proc. of the Symp. on Theory of Comp. (STOC), pp. 381–390 (2013)Google Scholar
  17. 17.
    Lenzen, C., Patt-Shamir, B.: Improved distributed steiner forest construction. In: The Proc. of the Int’l Symp. on Princ. of Dist. Comp. (PODC) (2014)Google Scholar
  18. 18.
    Matoušek, J.: Lectures on discrete geometry, vol. 212. Springer (2002)Google Scholar
  19. 19.
    Nanongkai, D.: Distributed approximation algorithms for weighted shortest paths. In: Proc. of the Symp. on Theory of Comp. (STOC) (to appear, 2014)Google Scholar
  20. 20.
    Peleg, D.: Distributed Computing: A Locality-sensitive Approach. Society for Industrial and Applied Mathematics, Philadelphia (2000)Google Scholar
  21. 21.
    Peleg, D., Rubinovich, V.: A near-tight lower bound on the time complexity of distributed MST construction. In: Proc. of the Symp. on Found. of Comp. Sci. (FOCS), pp. 253–261 (1999)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Mohsen Ghaffari
    • 1
  • Christoph Lenzen
    • 2
  1. 1.Massachusetts Institute of TechnologyUSA
  2. 2.MPI for InformaticsSaarbrückenGermany

Personalised recommendations