Advertisement

Approximation Algorithms for the Gromov Hyperbolicity of Discrete Metric Spaces

  • Ran Duan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8392)

Abstract

This paper discusses new approximation algorithms for computing the Gromov hyperbolicity of an n-point discrete metric space. We give a (1 + ε)-approximation algorithm with running time \(\tilde{O}(\epsilon^{-1}n^{1+\omega})\), where O(n ω ) = O(n 2.373) is the time complexity of matrix multiplications. Here an α-approximation δ′ means δ′ ≤ δ * ≤ αδ′ for the Gromov hyperbolicity δ *. We also give a (2 + ε)-approximation algorithm with running time \(\tilde{O}(\epsilon^{-1}n^{\omega})\). These are faster than the previous O(n (5 + ω)/2)-time algorithm for the exact solution and the O(n (3 + ω)/2)-time algorithm for a 2-approximation [Fournier, Ismail and Vigneron 2012], which directly perform (max, min)-product of matrices.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bonk, M., Schramm, O.: Embeddings of Gromov hyperbolic spaces. Geom. Funct. Anal. 10, 266–306 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Chen, W., Fang, W., Hu, G., Mahoney, M.W.: On the hyperbolicity of small-world and tree-like random graphs. In: Chao, K.-M., Hsu, T.-s., Lee, D.-T. (eds.) ISAAC 2012. LNCS, vol. 7676, pp. 278–288. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  3. 3.
    Chepoi, V., Dragan, F.F., Estellon, B., Habib, M., Vaxès, Y., Xiang, Y.: Additive spanners and distance and routing labeling schemes for hyperbolic graphs. Algorithmica 62(3-4), 713–732 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Coppersmith, D.: Rectangular matrix multiplication revisited. J. Complex. 13(1), 42–49 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Coppersmith, D., Winograd, T.: Matrix multiplication via arithmetic progressions. In: Proc. 19th ACM Symp. on the Theory of Computing (STOC), pp. 1–6 (1987)Google Scholar
  6. 6.
    Duan, R., Pettie, S.: Fast algorithms for (max, min)-matrix multiplication and bottleneck shortest paths. In: SODA 2009: Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 384–391. Society for Industrial and Applied Mathematics, Philadelphia (2009)CrossRefGoogle Scholar
  7. 7.
    Fournier, H., Ismail, A., Vigneron, A.: Computing the Gromov hyperbolicity of a discrete metric space. CoRR, abs/1210.3323 (2012)Google Scholar
  8. 8.
    Gavoille, C., Ly, O.: Distance labeling in hyperbolic graphs. In: Deng, X., Du, D.-Z. (eds.) ISAAC 2005. LNCS, vol. 3827, pp. 1071–1079. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  9. 9.
    Gromov, M.: Hyperbolic groups. In: Gersten, S. (ed.) Essays in Group Theory. Mathematical Sciences Research Institute Publications, vol. 8, pp. 75–263. Springer, New York (1987)CrossRefGoogle Scholar
  10. 10.
    Krioukov, D.V., Papadopoulos, F., Kitsak, M., Vahdat, A., Bogu, M.: Hyperbolic geometry of complex networks. CoRR, abs/1006.5169 (2010)Google Scholar
  11. 11.
    Williams, V.V.: Multiplying matrices faster than Coppersmith-Winograd. In: Proceedings of the 44th Symposium on Theory of Computing, STOC 2012, pp. 887–898. ACM, New York (2012)Google Scholar
  12. 12.
    Zwick, U.: All pairs shortest paths using bridging sets and rectangular matrix multiplication. J. ACM 49(3), 289–317 (2002)MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Ran Duan
    • 1
  1. 1.Max-Planck-Institut für InformatikSaarbrückenGermany

Personalised recommendations