Skip to main content
SpringerLink
Log in

Online Embeddings

  • Conference paper
Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (RANDOM 2010, APPROX 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6302))

  • 1530 Accesses

Abstract

We initiate the study of on-line metric embeddings. In such an embedding we are given a sequence of n points X = x 1,...,x n one by one, from a metric space M = (X,D). Our goal is to compute a low-distortion embedding of M into some host space, which has to be constructed in an on-line fashion, so that the image of each x i depends only on x 1,...,x i . We prove several results translating existing embeddings to the on-line setting, for the case of embedding into ℓ p spaces, and into distributions over ultrametrics.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Andoni, A., Onak, K.: Approximating Edit Distance in Near-Linear Time. In: Proceedings of the Symposium on Theory of Computing (STOC), pp. 199–204 (2009)

    Google Scholar 

  2. Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: Proceedings of the Symposium on Foundations of Computer Science (FOCS), pp. 184–193 (1996)

    Google Scholar 

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

    Google Scholar 

  4. Bourgain, J.: On Lipschitz Embedding of Finite Metric Spaces in Hilbert space. Israel J. Math. 52(1-2), 46–52 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  5. Calinescu, G., Karloff, H., Rabani, Y.: Approximation Algorithms for the 0-extension Problem. In: Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 8–16 (2001)

    Google Scholar 

  6. Englert, M., Räcke, H., Westermann, M.: Reordering Buffers for General Metric Spaces. In: Proceedings of the Symposium on Theory of Computing (STOC), pp. 556–564 (2007)

    Google Scholar 

  7. Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 1. John Wiley & Sons, Chichester (January 1968)

    MATH  Google Scholar 

  8. Gupta, A., Krauthgamer, R., Lee, J.R.: Bounded Geometries, Fractals, and Low-Distortion Embeddings. In: Proceedings of the Symposium on Foundations of Computer Science (FOCS), pp. 534–543 (2003)

    Google Scholar 

  9. Gupta, A., Ravi, R.: Lecture Notes of Metric Embeddings and Methods (2003), http://www.cs.cmu.edu/~anupamg/metrics/lectures/lec10.ps

  10. Indyk, P.: High-dimensional Computational Geometry. PhD thesis, Stanford University (2000)

    Google Scholar 

  11. Indyk, P.: Stable Distributions, Pseudorandom Generators, Embeddings and Data Stream Computation. In: Proceedings of the Symposium on Foundations of Computer Science (FOCS), pp. 189–197 (2000)

    Google Scholar 

  12. Indyk, P.: Tutorial: Algorithmic Applications of Low-distortion Geometric Embeddings. In: Proceedings of the Symposium on Foundations of Computer Science (FOCS), pp. 10–33 (2001)

    Google Scholar 

  13. Johnson, W.B., Lindenstrauss, J.: Extensions of Lipschitz Mappings into a Hilbert Space. In: Amer.Math. Soc. (ed.) Conference in Modern Analysis and Probability, pp. 189–206 (1984)

    Google Scholar 

  14. Kirszbraun, M.D.: Uber die Zusammenziehenden und Lipschitzchen Transformationen. Fund. Math. 22, 77–108 (1934)

    Google Scholar 

  15. Krauthgamer, R., Lee, J.R., Mendel, M., Naor, A.: Measured Descent: A new Embedding Method for Finite Metrics. Geometric Aspects of Functional Analysis, 839–858 (2005)

    Google Scholar 

  16. Lee, J.R., Naor, A.: Extending Lipschitz Functions via Random Metric Partitions. Invent. Math. 160(1), 59–95 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  17. Linial, N., London, E., Rabinovich, Y.: The Geometry of Graphs and some of its Algorithmic Applications. Combinatorica 15(2), 215–245 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  18. Matoušek, J.: Bi-Lipschitz Embeddings into Low-dimensional Euclidean Spaces. Comment. Math. Univ. Carolinae 31, 589–600 (1990)

    MATH  Google Scholar 

  19. Matoušek, J.: Lectures on Discrete Geometry. Springer, New York (2002)

    MATH  Google Scholar 

  20. Matoušek, J., Sidiropoulos, A.: Inapproximability for Metric Embeddings into ℝd. In: Proceedings of the Symposium on Foundations of Computer Science (FOCS), pp. 405–413 (2008)

    Google Scholar 

  21. Newman, I., Rabinovich, Y.: Personal communication (2008)

    Google Scholar 

  22. Rao, S.: Small Distortion and Volume Preserving Embeddings for Planar and Euclidean Metrics. In: Proceedings of the ACM Symposium on Computational Geometry (SoCG), pp. 300–306 (1999)

    Google Scholar 

  23. Schechtman, G.: The Random Version of Dvoretzky’s Theorem in ℓ ∞ . In: Zighed, D.A., Komorowski, J., Żytkow, J.M. (eds.) PKDD 2000. LNCS (LNAI), vol. 1910, pp. 265–270. Springer, Heidelberg (2000)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. MIT,  

    Piotr Indyk

  2. University of Toronto,  

    Avner Magen & Anastasios Zouzias

  3. Toyota Technological Institute,  

    Anastasios Sidiropoulos

Authors
  1. Piotr Indyk
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Avner Magen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Anastasios Sidiropoulos
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Anastasios Zouzias
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Universitat Politecnica de Catalunya, Departament de Llenguatges i Sistemes Informatics, Campus Nord - Ed Omega, Desp. 235, Jordi Girona, 1-3, 08034, Barcelona, Spain

    Maria Serna

  2. Department of Computer Science, University of Haifa, Mount Carmel, 31905, Haifa, Israel

    Ronen Shaltiel

  3. Department of Computer Science, University of Kiel, Olshausenstrasse 40, 24098, Kiel, Germany

    Klaus Jansen

  4. University of Geneva, Centre Universitaire d’Informatique, Battelle Bat. A, 7 rte de Drize, 1227, Carouge, Switzerland

    José Rolim

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Indyk, P., Magen, A., Sidiropoulos, A., Zouzias, A. (2010). Online Embeddings. In: Serna, M., Shaltiel, R., Jansen, K., Rolim, J. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. RANDOM APPROX 2010 2010. Lecture Notes in Computer Science, vol 6302. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15369-3_19

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-15369-3_19

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-15368-6

  • Online ISBN: 978-3-642-15369-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation