Advertisement

Abstract

A spanner of an undirected unweighted graph is a subgraph that approximates the distance metric of the original graph with some specified accuracy. Specifically, we say H ⊆ G is an f-spanner of G if any two vertices u,v at distance d in G are at distance at most f(d) in H. There is clearly some tradeoff between the sparsity of H and the distortion function f, though the nature of this tradeoff is still poorly understood.

In this paper we present a simple, modular framework for constructing sparse spanners that is based on interchangable components called connection schemes. By assembling connection schemes in different ways we can recreate the additive 2- and 6-spanners of Aingworth et al. and Baswana et al. and improve on the (1 + ε,β)-spanners of Elkin and Peleg, the sublinear additive spanners of Thorup and Zwick, and the (non constant) additive spanners of Baswana et al. Our constructions rival the simplicity of all comparable algorithms and provide substantially better spanners, in some cases reducing the density doubly exponentially.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aingworth, D., Chekuri, C., Indyk, P., Motwani, R.: Fast estimation of diameter and shortest paths. SIAM J. Comput 28(4), 1167–1181 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Althöfer, I., Das, G., Dobkin, D., Joseph, D., Soares, J.: On sparse spanners of weighted graphs. Discrete and Computational Geometry 9, 81–100 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Awerbuch, B.: Complexity of network synchronization. J. ACM 32, 804–823 (1985)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Baswana, S., Kavitha, T.: Faster algorithms for approximate distance oracles and all-pairs small stretch paths. In: FOCS 2006, (2006)Google Scholar
  5. 5.
    Baswana, S., Kavitha, T., Mehlhorn, K., Pettie, S.: New constructions of (α,β)-spanners and purely additive spanners. In: SODA 2005, (2005)Google Scholar
  6. 6.
    Bollobás, B., Coppersmith, D., Elkin, M.: Sparse subgraphs that preserve long distances and additive spanners. SIAM J. Discr. Math. 9(4), 1029–1055 (2006)Google Scholar
  7. 7.
    Coppersmith, D., Elkin, M.: Sparse source-wise and pair-wise distance preservers. In: SODA 2005 (2005)Google Scholar
  8. 8.
    Coppersmith, D., Elkin, M.: Sparse source-wise and pair-wise preservers. SIAM J. Discrete Math (to appear)Google Scholar
  9. 9.
    Cowen, L.J., Wagner, C.G.: Compact roundtrip routing in directed networks. J. Algor. 50(1), 79–95 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Dor, D., Halperin, S., Zwick, U.: All-pairs almost shortest paths. SIAM J. Comput. 29(5), 1740–1759 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Elkin, M., Peleg, D.: (1 + ε,β)-spanner constructions for general graphs. SIAM J. Comput. 33(3), 608–631 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Elkin, M., Zhang, J.: Efficient algorithms for constructing (1 + ε,β)-spanners in the distributed and streaming models. In: PODC 2004 (2004)Google Scholar
  13. 13.
    Fakcharoenphol, J., Rao, S., Talwar, K.: A tight bound on approximating arbitrary metrics by tree metrics. J. Comput. Syst. Sci. 69(3), 485–497 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Halperin, S., Zwick, U.: Unpublished result (1996)Google Scholar
  15. 15.
    Narasimhan, G., Smid, M.: Geometric Spanner Networks (2007)Google Scholar
  16. 16.
    Peleg, D., Schaffer, A.A.: Graph spanners. J. Graph Theory  13, 99–116 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Peleg, D., Ullman, J.D.: An optimal synchronizer for the hypercube. SIAM J. Comput. 18, 740–747 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Pettie, S.: Low distortion spanners. See, http://www.eecs.umich.edu/~pettie
  19. 19.
    Roditty, L., Thorup, M., Zwick, U.: Roundtrip spanners and roundtrip routing in directed graphs. In: SODA 2002 (2002)Google Scholar
  20. 20.
    Spielman, D.A., Teng, S.-H.: Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems. In: STOC 2004 (2004)Google Scholar
  21. 21.
    Thorup, M., Zwick, U.: Compact routing schemes. In: SPAA 2001 (2001)Google Scholar
  22. 22.
    Thorup, M., Zwick, U.: Spanners and emulators with sublinear distance errors. In: SODA 2006 (2006)Google Scholar
  23. 23.
    Thorup, M., Zwick, U.: Approximate distance oracles. J.ACM 52, 1–24 (2005)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Wenger, R.: Extremal graphs with no C 4’s, C 6’s, or C 10’s. J. Combin. Theory Ser. B 52(1), 113–116 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Woodruff, D.: Lower bounds for additive spanners, emulators, and more. In: FOCS 2006 (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Seth Pettie
    • 1
  1. 1.The University of Michigan 

Personalised recommendations