Discrete & Computational Geometry

, Volume 50, Issue 4, pp 977–1032 | Cite as

Dimension Reduction for Finite Trees in \(\varvec{\ell _1}\)

  • James R. Lee
  • Arnaud de Mesmay
  • Mohammad Moharrami


We show that every \(n\)-point tree metric admits a \((1+\varepsilon )\)-embedding into \(\ell _1^{C(\varepsilon ) \log n}\), for every \(\varepsilon > 0\), where \(C(\varepsilon ) \le O\big ((\frac{1}{\varepsilon })^4 \log \frac{1}{\varepsilon })\big )\). This matches the natural volume lower bound up to a factor depending only on \(\varepsilon \). Previously, it was unknown whether even complete binary trees on \(n\) nodes could be embedded in \(\ell _1^{O(\log n)}\) with \(O(1)\) distortion. For complete \(d\)-ary trees, our construction achieves \(C(\varepsilon ) \le O\big (\frac{1}{\varepsilon ^2}\big )\).


Dimension reduction Metric embeddings Bi-Lipschitz distortion 


  1. 1.
    Andoni, A., Charikar, M., Neiman, O., Nguyen, H.L.: Near linear lower bound for dimension reduction in l1. In: FOCS, pp. 315–323 (2011)Google Scholar
  2. 2.
    Brinkman, B., Charikar, M.: On the impossibility of dimension reduction in \(\ell _1\). J. ACM 52(5), 766–788 (2005)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bourgain, J., Lindenstrauss, J., Milman, V.: Approximation of zonoids by zonotopes. Acta Math. 162(1–2), 73–141 (1989)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Batson, J.D., Spielman, D.A., Srivastava, N.: Twice-Ramanujan sparsifiers. SIAM J. Comput. 41(6), 1704–1721 (2012)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Charikar, M., Sahai, A.: Dimension reduction in the \(\ell _1\) norm. In: FOCS, pp. 551–560 (2002)Google Scholar
  6. 6.
    Erdős, P., Lovász, L.: Problems and results on 3-chromatic hypergraphs and some related questions. In: Infinite and Finite Sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), vol. II, pp. 609–627. Colloquium Mathematical Society János Bolyai, vol. 10. North-Holland, Amsterdam (1975)Google Scholar
  7. 7.
    Gupta, A., Krauthgamer, R., Lee, J.R.: Bounded geometries, fractals, and low-distortion embeddings. In: FOCS, pp. 534–543 (2003)Google Scholar
  8. 8.
    Johnson, W.B., Lindenstrauss, J.: Extensions of Lipschitz mappings into a Hilbert space. In: Conference in Modern Analysis and Probability (New Haven, CT, 1982), Contemporary Mathematics, vol. 26, pp. 189–206. American Mathematical Society, Providence (1984)Google Scholar
  9. 9.
    Lee, J.R., de Mesmay, A., Moharrami, M.: Dimension reduction for finite trees in \(l_{1}\). In: SODA, pp. 43–50 (2012)Google Scholar
  10. 10.
    Lee, J.R., Naor, A.: Embedding the diamond graph in \(L_{p}\) and dimension reduction in \(L_{1}\). Geom. Funct. Anal. 14(4), 745–747 (2004)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Lee, J.R., Naor, A., Peres, Y.: Trees and Markov convexity. Geom. Funct. Anal. 18(5), 1609–1659 (2009)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Matoušek, J.: On embedding trees into uniformly convex Banach spaces. Israel J. Math. 114, 221–237 (1999)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Matoušek, J.: Open problems on embeddings of finite metric spaces. http://kam.mff.cuni.cz/matousek/haifaop.ps (2002)
  14. 14.
    McDiarmid, C.: Concentration. In: Habib, M., McDiarmid, C., Ramirez-Alfonsin, J., Reed, B. (eds.) Probabilistic Methods for Algorithmic Discrete Mathematics, Algorithms and Combinations, vol. 16, pp. 195–248. Springer, Berlin (1998)CrossRefGoogle Scholar
  15. 15.
    Newman, I., Rabinovich, Y.: On cut dimension of \(\ell _1\) metrics and volumes, and related sparsification techniques. CoRR. http://CoRR/abs/1002.3541 (2010)
  16. 16.
    Regev, O.: Entropy-based bounds on dimension reduction in \(L_1\). Israel J. Math. http://arxiv/abs/1108.1283 (2011)
  17. 17.
    Schechtman, G.: More on embedding subspaces of \(L_p\) in \(l^{n}_{r}\). Compos. Math. 61(2), 159–169 (1987)MathSciNetMATHGoogle Scholar
  18. 18.
    Schulman, L.J.: Coding for interactive communication. IEEE Trans. Inform. Theory 42(6, part 1), 1745–1756 (1996). Codes and complexityGoogle Scholar
  19. 19.
    Talagrand, M.: Embedding subspaces of \(L_1\) into \(l^N_1\). Proc. Am. Math. Soc. 108(2), 363–369 (1990)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • James R. Lee
    • 1
  • Arnaud de Mesmay
    • 2
  • Mohammad Moharrami
    • 1
  1. 1.University of WashingtonSeattleUSA
  2. 2.École Normale SupérieureParisFrance

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