Online, Dynamic, and Distributed Embeddings of Approximate Ultrametrics

  • Michael Dinitz
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5218)


The theoretical computer science community has traditionally used embeddings of finite metrics as a tool in designing approximation algorithms. Recently, however, there has been considerable interest in using metric embeddings in the context of networks to allow network nodes to have more knowledge of the pairwise distances between other nodes in the network. There has also been evidence that natural network metrics like latency and bandwidth have some nice structure, and in particular come close to satisfying an ε-three point condition or an ε-four point condition. This empirical observation has motivated the study of these special metrics, including strong results about embeddings into trees and ultrametrics. Unfortunately all of the current embeddings require complete knowledge about the network up front, and so are less useful in real networks which change frequently. We give the first metric embeddings which have both low distortion and require only small changes in the structure of the embedding when the network changes. In particular, we give an embedding of semimetrics satisfying an ε-three point condition into ultrametrics with distortion (1 + ε)logn + 4 and the property that any new node requires only O(n 1/3) amortized edge swaps, where we use the number of edge swaps as a measure of “structural change”. This notion of structural change naturally leads to small update messages in a distributed implementation in which every node has a copy of the embedding. The natural offline embedding has only (1 + ε)logn distortion but can require Ω(n) amortized edge swaps per node addition. This online embedding also leads to a natural dynamic algorithm that can handle node removals as well as insertions.


Point Condition Span Tree Minimum Span Tree Active Node Hyperbolic Group 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abraham, I., Balakrishnan, M., Kuhn, F., Malkhi, D., Ramasubramanian, V., Talwar, K.: Reconstructing approximate tree metrics. In: PODC 2007: Proceedings of the twenty-sixth annual ACM Symposium on Principles of Distributed Computing, pp. 43–52. ACM, New York (2007)Google Scholar
  2. 2.
    Althöfer, I., Das, G., Dobkin, D., Joseph, D., Soares, J.: On sparse spanners of weighted graphs. Discrete Comput. Geom. 9(1), 81–100 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Arora, S., Lovász, L., Newman, I., Rabani, Y., Rabinovich, Y., Vempala, S.: Local versus global properties of metric spaces. In: SODA 2006: Proceedings of the seventeenth annual ACM-SIAM Symposium on Discrete Algorithm, pp. 41–50. ACM, New York (2006)CrossRefGoogle Scholar
  4. 4.
    Bollobás, B.: Extremal graph theory. London Mathematical Society Monographs, vol. 11. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London (1978)Google Scholar
  5. 5.
    Charikar, M., Makarychev, K., Makarychev, Y.: Local global tradeoffs in metric embeddings. In: FOCS 2007: Proceedings of the forty-eighth annual IEEE Symposium on Foundations of Computer Science, pp. 713–723 (2007)Google Scholar
  6. 6.
    Costa, M., Castro, M., Rowstron, A., Key, P.: Pic: Practical internet coordinates for distance estimation. In: ICDCS 2004: Proceedings of the 24th International Conference on Distributed Computing Systems (ICDCS 2004), Washington, DC, USA, pp. 178–187. IEEE Computer Society, Los Alamitos (2004)Google Scholar
  7. 7.
    Dabek, F., Cox, R., Kaashoek, F., Morris, R.: Vivaldi: a decentralized network coordinate system. SIGCOMM Comput. Commun. Rev. 34(4), 15–26 (2004)CrossRefGoogle Scholar
  8. 8.
    Gromov, M.: Hyperbolic groups. In: Essays in group theory. Math. Sci. Res. Inst. Publ, vol. 8, pp. 75–263. Springer, New York (1987)Google Scholar
  9. 9.
    Lim, H., Hou, J.C., Choi, C.-H.: Constructing internet coordinate system based on delay measurement. In: IMC 2003: Proceedings of the 3rd ACM SIGCOMM Conference on Internet Measurement, pp. 129–142. ACM, New York (2003)CrossRefGoogle Scholar
  10. 10.
    Tang, L., Crovella, M.: Virtual landmarks for the internet. In: IMC 2003: Proceedings of the 3rd ACM SIGCOMM Conference on Internet Measurement, pp. 143–152. ACM, New York (2003)CrossRefGoogle Scholar
  11. 11.
    Thorup, M., Zwick, U.: Approximate distance oracles. J. ACM 52(1), 1–24 (2005)CrossRefMathSciNetGoogle Scholar
  12. 12.
    wei Lehman, L., Lerman, S.: Pcoord: Network position estimation using peer-to-peer measurements. In: NCA 2004: Proceedings of the Network Computing and Applications, Third IEEE International Symposium on (NCA 2004), Washington, DC, USA, pp. 15–24. IEEE Computer Society, Los Alamitos (2004)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Michael Dinitz
    • 1
  1. 1.Computer Science DepartmentCarnegie Mellon UniversityPittsburghUSA

Personalised recommendations