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Metric Structures in L1: Dimension, Snowflakes, and Average Distortion

  • James R. Lee
  • Manor Mendel
  • Assaf Naor
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2976)

Abstract

We study the metric properties of finite subsets of L 1. The analysis of such metrics is central to a number of important algorithmic problems involving the cut structure of weighted graphs, including the Sparsest Cut Problem, one of the most compelling open problems in the field of approximation. Additionally, many open questions in geometric non-linear functional analysis involve the properties of finite subsets of L 1.

We present some new observations concerning the relation of L 1 to dimension, topology, and Euclidean distortion. We show that every n-point subset of L 1 embeds into L 2 with average distortion \(O(\sqrt{log n})\), yielding the first evidence that the conjectured worst-case bound of \(O(\sqrt{log n})\) is valid. We also address the issue of dimension reduction in L p for p ∈ (1,2). We resolve a question left open in [1] about the impossibility of linear dimension reduction in the above cases, and we show that the example of [2,3] cannot be used to prove a lower bound for the non-linear case. This is accomplished by exhibiting constant-distortion embeddings of snowflaked planar metrics into Euclidean space.

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References

  1. 1.
    Charikar, M., Sahai, A.: Dimension reduction in the \({\it l_1}\) norm. In: Proceedings of the 43rd Annual IEEE Conference on Foundations of Computer Science. ACM, New York (2002)Google Scholar
  2. 2.
    Charikar, M., Brinkman, B.: On the impossibility of dimension reduction in \({\it l_1}\). In: To appear in Proceedings of the 44th Annual IEEE Conference on Foundations of Computer Science. ACM, New York (2003)Google Scholar
  3. 3.
    Lee, J.R., Naor, A.: Embedding the diamond graph in L p and dimension reduction in L 1 (preprint 2003) Google Scholar
  4. 4.
    Linial, N., London, E., Rabinovich, Y.: The geometry of graphs and some of its algorithmic applications. Combinatorica 15, 215–245 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Aumann, Y., Rabani, Y.: An O(log k) approximate min-cut max-flow theorem and approximation algorithm. SIAM J. Comput. 27, 291–301 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Johnson, W.B., Lindenstrauss, J.: Extensions of Lipschitz mappings into a Hilbert space. In: Conference in modern analysis and probability (New Haven, Conn., 1982), Providence, RI. Contemp. Math. Amer. Math. Soc., vol. 26, pp. 189–206 (1984)Google Scholar
  7. 7.
    Matoušek, J.: Lectures on discrete geometry. Graduate Texts in Mathematics, vol. 212. Springer, New York (2002)zbMATHGoogle Scholar
  8. 8.
    Matoušek, J.: Open problems, workshop on discrete metric spaces and their algorithmic appl ications, Haifa (2002)Google Scholar
  9. 9.
    Linial, N.: Finite metric spaces - combinatorics, geometry and algorithms. In: Proceedings of the International Congress of Mathematicians III, pp. 573–586 (2002)Google Scholar
  10. 10.
    Indyk, P.: Algorithmic applications of low-distortion geometric embeddings. In: Proceedings of the 42nd Annual IEEE Symposium on Foundations of Computer Science, pp. 10–33 (2001)Google Scholar
  11. 11.
    Rabinovich, Y.: On average distorsion of embedding metrics into the line and into \({\it l_1}\). In: Proceedings of the 35th Annual ACM Symposium on Theory of Computing. ACM, New York (2003)Google Scholar
  12. 12.
    Marcus, M.B., Pisier, G.: Characterizations of almost surely continuous p-stable random Fourier series and strongly stationary processes. Acta Math. 152, 245–301 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Gutpa, A., Krauthgamer, R., Lee, J.R.: Bounded geometries, fractals, and lowdistortion embeddings. In: Proceedings of the 44th Annual Symposium on Foundations of Computer Science (2003)Google Scholar
  14. 14.
    Krauthgamer, R., Lee, J.R.: Navigating nets: Simple algorithms for proximity search (submitted 2003)Google Scholar
  15. 15.
    Laakso, T.J.: Ahlfors Q-regular spaces with arbitrary Q > 1 admitting weak Poincaré inequality. Geom. Funct. Anal. 10, 111–123 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Lang, U., Plaut, C.: Bilipschitz embeddings of metric spaces into space forms. Geom. Dedicata 87, 285–307 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Figiel, T., Lindenstrauss, J., Milman, V.D.: The dimension of almost spherical sections of convex bodies. Acta Math. 139, 53–94 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Assouad, P.: Plongements lipschitziens dans Rn. Bull. Soc. Math. France 111, 429–448 (1983)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Rao, S.: Small distortion and volume preserving embeddings for planar and Euclidean metrics. In: Proceedings of the 15th Annual Symposium on Computational Geometry, pp. 300–306. ACM, New York (1999)Google Scholar
  20. 20.
    Widder, D.V.: The Laplace Transform. Princeton Mathematical Series, vol. 6. Princeton University Press, Princeton (1941)Google Scholar
  21. 21.
    Milman, V.D., Schechtman, G.: Asymptotic theory of finite-dimensional normed spaces. Springer, Berlin (1986); With an appendix by M. GromovzbMATHGoogle Scholar
  22. 22.
    Gupta, A., Newman, I., Rabinovich, Y., Sinclair, A.: Cuts, trees and \({\it l_1}\) embeddings. In: Proceedings of the 40th Annual Symposium on Foundations of Computer Science (1999)Google Scholar
  23. 23.
    Matoušek, J.: On embedding expanders into l p spaces. Israel J. Math. 102, 189–197 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Heinonen, J.: Lectures on analysis on metric spaces. Universitext. Springer, New York (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • James R. Lee
    • 1
  • Manor Mendel
    • 3
  • Assaf Naor
    • 2
  1. 1.U.C. Berkeley 
  2. 2.Microsoft Research 
  3. 3.University of Illinois 

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