Additive Spanners in Nearly Quadratic Time

  • David P. Woodruff
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6198)


We consider the problem of efficiently finding an additive C-spanner of an undirected unweighted graph G, that is, a subgraph H so that for all pairs of vertices u,v, δH(u,v) ≤ δG(u,v) + C, where δ denotes shortest path distance. It is known that for every graph G, one can find an additive 6-spanner with O(n4/3) edges in O(mn2/3) time. It is unknown if there exists a constant C and an additive C-spanner with o(n4/3) edges. Moreover, for C ≤ 5 all known constructions require Ω(n3/2) edges.

We give a significantly more efficient construction of an additive 6-spanner. The number of edges in our spanner is n4/3 polylog n, matching what was previously known up to a polylogarithmic factor, but we greatly improve the time for construction, from O(mn2/3) to \(n^2 {\rm polylog} \ n\). Notice that mn2/3 ≤ n2 only if m ≤ n4/3, but in this case G itself is a sparse spanner. We thus provide both the fastest and the sparsest (up to logarithmic factors) known construction of a spanner with constant additive distortion.

We give similar improvements in the construction time of additive spanners under the assumption that the input graph has large girth, or more generally, the input graph has few edges on short cycles.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abraham, I., Gavoille, C., Malkhi, D.: On space-stretch trade-offs: upper bounds. In: SPAA, pp. 217–224 (2006)Google Scholar
  2. 2.
    Aingworth, D., Chekuri, C., Indyk, P., Motwani, R.: Fast estimation of diameter and shortest paths (without matrix multiplication). SIAM J. Comput. 28(4), 1167–1181 (1999)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Baswana, S., Kavitha, T., Mehlhorn, K., Pettie, S.: Additive spanners and (α, β)-spanners. ACM Transactions on Algorithms (2009)Google Scholar
  4. 4.
    Baswana, S., Kavitha, T.: Faster algorithms for approximate distance oracles and all-pairs small stretch paths. In: FOCS, pp. 591–602 (2006)Google Scholar
  5. 5.
    Baswana, S., Kavitha, T., Mehlhorn, K., Pettie, S.: New constructions of (alpha, beta)-spanners and purely additive spanners. In: SODA, pp. 672–681 (2005)Google Scholar
  6. 6.
    Baswana, S., Sen, S.: Approximate distance oracles for unweighted graphs in õ(n\(^{\mbox{2}}\)) time. In: SODA, pp. 271–280 (2004)Google Scholar
  7. 7.
    Cohen, E.: Fast algorithms for constructing t-spanners and paths with stretch. SIAM J. Comput. 28(1), 210–236 (1998)MATHCrossRefGoogle Scholar
  8. 8.
    Cohen, E.: Polylog-time and near-linear work approximation scheme for undirected shortest paths. J. ACM 47(1), 132–166 (2000)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn. The MIT Press & McGraw-Hill Book Company (2001)Google Scholar
  10. 10.
    Cowen, L.: Compact routing with minimum stretch. J. Algorithms 38(1), 170–183 (2001)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Cowen, L., Wagner, C.G.: Compact roundtrip routing in directed networks. J. Algorithms 50(1), 79–95 (2004)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Dor, D., Halperin, S., Zwick, U.: All-pairs almost shortest paths. SIAM J. Comput. 29(5), 1740–1759 (2000)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Elkin, M.: Personal communication (2009)Google Scholar
  14. 14.
    Elkin, M.: Computing almost shortest paths. ACM Transactions on Algorithms 1(2), 283–323 (2005)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Elkin, M., Peleg, D.: (1+epsilon, beta)-spanner constructions for general graphs. SIAM J. Comput. 33(3), 608–631 (2004)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Peleg, D., Ullman, J.D.: An optimal synchronizer for the hypercube. SIAM J. Comput. 18(4), 740–747 (1989)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Peleg, D., Upfal, E.: A trade-off between space and efficiency for routing tables. J. ACM 36(3), 510–530 (1989)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Pettie, S.: Low distortion spanners. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 78–89. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  19. 19.
    Roditty, L., Thorup, M., Zwick, U.: Deterministic constructions of approximate distance oracles and spanners. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 261–272. Springer, Heidelberg (2005)Google Scholar
  20. 20.
    Thorup, M., Zwick, U.: Compact routing schemes. In: SPAA (2001)Google Scholar
  21. 21.
    Thorup, M., Zwick, U.: Approximate distance oracles. J. ACM 52(1), 1–24 (2005)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Thorup, M., Zwick, U.: Spanners and emulators with sublinear distance errors. In: SODA, pp. 802–809 (2006)Google Scholar
  23. 23.
    Woodruff, D.P.: Lower bounds for additive spanners, emulators, and more. In: FOCS, pp. 389–398 (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • David P. Woodruff
    • 1
  1. 1.IBM Almaden 

Personalised recommendations