Distance Labeling in Hyperbolic Graphs

  • Cyril Gavoille
  • Olivier Ly
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3827)


A graph G is δ-hyperbolic if for any four vertices u,v,x,y of G the two larger of the three distance sums d G (u,v) + d G (x,y), d G (u,x) + d G (v,y), d G (u,y) + d G (v,x) differ by at most δ, and the smallest δ ≥ 0 for which G is δ-hyperbolic is called the hyperbolicity of G.

In this paper, we construct a distance labeling scheme for bounded hyperbolicity graphs, that is a vertex labeling such that the distance between any two vertices of G can be estimated from their labels, without any other source of information. More precisely, our scheme assigns labels of O(log2 n) bits for bounded hyperbolicity graphs with n vertices such that distances can be approximated within an additive error of O(log n). The label length is optimal for every additive error up to n ε . We also show a lower bound of Ω(log log n) on the approximation factor, namely every s-multiplicative approximate distance labeling scheme on bounded hyperbolicity graphs with polylogarithmic labels requires s = Ω(log log n).


Distance queries distance labeling scheme hyperbolic graphs 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Agarwala, R., Bafna, V., Farach, M., Paterson, M., Thorup, M.: On the approximability of numerical taxonomy (fitting distances by tree metrics). In: 7th Symposium on Discrete Algorithms (SODA), January 1996, pp. 365–372. ACM-SIAM (1996)Google Scholar
  2. 2.
    Alstrup, S., Bille, P., Rauhe, T.: Labeling schemes for small distances in trees. In: 14th Symposium on Discrete Algorithms (SODA), January 2003, pp. 689–698. ACM-SIAM (2003)Google Scholar
  3. 3.
    Bandelt, H.-J., Chepoi, V.D.: 1-hyperbolic graphs. SIAM Journal on Discrete Mathematics 16(2), 323–334 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bandelt, H.-J., Henkmann, A., Nicolai, F.: Powers of distance-hereditary graphs. Discrete Mathematics 145, 37–60 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bandelt, H.-J., Mulder, H.M.: Distance-hereditary graphs. Journal of Combinatorial Theory, Series B 41, 182–208 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bazzaro, F., Gavoille, C.: Localized and compact data-structure for comparability graphs. Research Report RR-1343-05, LaBRI, University of Bordeaux 1, 351, cours de la Libération, 33405 Talence Cedex, France (February 2005)Google Scholar
  7. 7.
    Brinkmann, G., Koolen, J.H., Moulton, V.: On the hyperbolicity of chordal graphs. Annals of Combinatorics 5(1), 61–65 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Buneman, P.: The recovery of trees from measures of dissimilarity. In: Mathematics in Archaeological and Historical Sciences, pp. 387–395 (1971)Google Scholar
  9. 9.
    Chepoi, V.D., Dragan, F.F., Vaxes, Y.: Distance and routing labeling schemes for non-positively curved plane graphs. Journal of Algorithms (2005) (to appear)Google Scholar
  10. 10.
    Courcelle, B., Vanicat, R.: Query efficient implementation of graphs of bounded clique-width. Discrete Applied Mathematics 131, 129–150 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Dourisboure, Y., Gavoille, C.: Tree-decomposition of graphs with small diameter bags. In: Fila, J. (ed.) 2nd European Conference on Combinatorics, Graph Theory and Applications (EUROCOMB), September 2003, pp. 100–104 (2003)Google Scholar
  12. 12.
    Dress, A.W.M., Moulton, V., Steel, M.A.: Trees, taxonomy, and strongly compatible multi-state characters. Advances in Applied Mathematics 19, 1–30 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Dress, A.W.M., Moulton, V., Terhalle, W.: T-theory: an overview. European Journal of Combinatorics 17, 161–175 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Erdös, P.: Extremal problems in graph theory. In: Publ. House Cszechoslovak Acad. Sci., Prague, pp. 29–36 (1964)Google Scholar
  15. 15.
    Gavoille, C., Katz, M., Katz, N.A., Paul, C., Peleg, D.: Approximate distance labeling schemes. In: Meyer auf der Heide, F. (ed.) ESA 2001. LNCS, vol. 2161, pp. 476–488. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  16. 16.
    Gavoille, C., Paul, C.: Distance labeling scheme and split decomposition. Discrete Mathematics 273(1-3), 115–130 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Gavoille, C., Paul, C.: Optimal distance labeling schemes for interval and circular-arc graphs. In: Di Battista, G., Zwick, U. (eds.) ESA 2003. LNCS, vol. 2832, pp. 254–265. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  18. 18.
    Gavoille, C., Peleg, D.: Compact and localized distributed data structures. Journal of Distributed Computing 16, 111–120 (2003) PODC 20-Year Special IssueCrossRefGoogle Scholar
  19. 19.
    Gavoille, C., Peleg, D., Pérennès, S., Raz, R.: Distance labeling in graphs. Journal of Algorithms 53(1), 85–112 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Ghys, E., de La Harpe, P.: Sur les Groupes Hyperboliques d’après Mikhael Gromov. Birkhäuser, Basel (1990)zbMATHGoogle Scholar
  21. 21.
    Gromov, M.: Hyperbolic groups. Essays in Group Theory, Mathematical Sciences Research Institute Publications 8, 75–263 (1987)MathSciNetGoogle Scholar
  22. 22.
    Gupta, A., Krauthgamer, R., Lee, J.R.: Bounded geometries, fractals, and low-distortion embeddings. In: 44th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 534–543. IEEE Computer Society Press, Los Alamitos (2003)Google Scholar
  23. 23.
    Gupta, A., Kumar, A., Rastogi, R.: Traveling with a pez dispenser (or, routing issues in mpls). SIAM Journal on Computing 34(2), 453–474 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Howorka, E.: On metric properties of certain clique graphs. Journal of Combinatorial Theory, Series B 27, 67–74 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Kaplan, H., Milo, T.: Parent and ancestor queries using a compact index. In: 20th ACM Symposium on Principles of Database Systems (PODS), May 2001, ACM-SIAM (2001)Google Scholar
  26. 26.
    Koolen, J.H., Moulton, V.: Hyperbolic bridged graphs. European Journal of Combinatorics 23(6), 683–699 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Lazebnik, F., Ustimenko, V.A., Woldar, A.J.: A new series of dense graphs of high girth. Bulletin of the American Mathematical Society (New Series) 32(1), 73–79 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Mendel, M., Har-Peled, S.: Fast construction of nets in low dimensional metrics, and their applications. In: 21st Annual ACM Symposium on Computational Geometry (SoCG), pp. 150–158 (2005)Google Scholar
  29. 29.
    Moulton, V., Steel, M.A.: Retractions of finite distance functions onto tree metrics. Discrete Applied Mathematics 91, 215–233 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Peleg, D.: Proximity-preserving labeling schemes. Journal of Graph Theory 33, 167–176 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Slivkins, A.: Distance estimation and object location via rings of neighbors. In: 24th Annual ACM Symposium on Principles of Distributed Computing (PODC), pp. 41–50. ACM Press, New York (2005)Google Scholar
  32. 32.
    Talwar, K.: Bypassing the embedding: Algorithms for low dimensional metrics. In: 36th Annual ACM Symposium on Theory of Computing (STOC), June 2004, pp. 281–290 (2004)Google Scholar
  33. 33.
    Thorup, M.: Compact oracles for reachability and approximate distances in planar digraphs. Journal of the ACM 51(6), 993–1024 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Thorup, M., Zwick, U.: Approximate distance oracles. Journal of the ACM 52(1), 1–24 (2005)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Cyril Gavoille
    • 1
  • Olivier Ly
    • 1
  1. 1.LaBRIBordeaux University 

Personalised recommendations