All-Pairs Shortest Paths with a Sublinear Additive Error

  • Liam Roditty
  • Asaf Shapira
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5125)


We show that for every 0 ≤ p ≤ 1 there is an algorithm with running time of O(n2.575 − p/(7.4 − 2.3p)) that given a directed graph with small positive integer weights, estimates the length of the shortest path between every pair of vertices u,v in the graph to within an additive error δp(u,v), where δ(u,v) is the exact length of the shortest path between u and v. This algorithm runs faster than the fastest algorithm for computing exact shortest paths for any 0 < p ≤ 1.

Previously the only way to “bit” the running time of the exact shortest path algorithms was by applying an algorithm of Zwick [FOCS ’98] that approximates the shortest path distances within a multiplicative error of (1 + ε). Our algorithm thus gives a smooth qualitative and quantitative transition between the fastest exact shortest paths algorithm, and the fastest approximation algorithm with a linear additive error. In fact, the main ingredient we need in order to obtain the above result, which is also interesting in its own right, is an algorithm for computing (1 + ε) multiplicative approximations for the shortest paths, whose running time is faster than the running time of Zwick’s approximation algorithm when ε ≪ 1 and the graph has small integer weights.


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  1. 1.
    Aho, A.V., Hopcroft, J.E., Ullman, J.D.: The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading (1974)MATHGoogle Scholar
  2. 2.
    Aingworth, D., Chekuri, C., Indyk, P., Motwani, R.: Fast estimation of diameter and shortest paths (without matrix multiplication). SIAM Journal on Computing 28, 1167–1181 (1999)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Alon, N., Galil, Z., Margalit, O.: On the exponent of the all pairs shortest path problem. Journal of Computer ans System Sciences 54, 255–262 (1997); Also, Proc. of FOCS 1991MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Chan, T.M.: More algorithms for all-pairs shortest paths in weighted graphs. In: Proc. of STOC 2007 (to appear, 2007)Google Scholar
  5. 5.
    Coppersmith, D.: Rectangular matrix multiplication revisited. Journal of Complexity 13, 42–49 (1997)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Coppersmith, D., Winograd, S.: Matrix multiplication via arithmetic progressions. J. Symbol. Comput. 9, 251–280 (1990)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Cormen, T.H., Leisserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn. McGraw-Hill, New York (2001)MATHGoogle Scholar
  8. 8.
    Dijkstra, E.W.: A note on two problems in connexion with graphs. Numerische Mathematik 1, 269–271 (1959)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Dor, D., Halperin, S., Zwick, U.: All pairs almost shortest paths. SIAM Journal on Computing 29, 1740–1759 (2000)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Czumaj, A., Kowaluk, M., Lingas, A.: Faster algorithms for finding lowest common ancestors in directed acyclic graphs (manuscript, 2006)Google Scholar
  11. 11.
    Fischer, M.J., Meyer, A.R.: Boolean matrix multiplication and transitive closure. In: Proc. of the 12th Symposium on Switching and Automata Theory, East Lansing, Mich., pp. 129–131 (1971)Google Scholar
  12. 12.
    Fredman, M.L.: New bounds on the complexity of the shortest path problem. SIAM Journal on Computing 5, 49–60 (1976)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Fredman, M.L., Tarjan, R.E.: Fibonacci heaps and their uses in improved network optimization algorithms. Journal of the ACM 34, 596–615 (1987)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Furman, M.E.: Application of a method of fast multiplication of matrices in the problem of finding the transitive closure of a graph. Dokl. Akad. Nauk SSSR 11(5), 1252 (1970)MATHMathSciNetGoogle Scholar
  15. 15.
    Galil, Z., Margalit, O.: All pairs shortest distances for graphs with small integer length edges. Information and Computation 134, 103–139 (1997)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Galil, Z., Margalit, O.: All pairs shortest paths for graphs with small integer length edges. Journal of Computer and System Sciences 54, 243–254 (1997)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Gabow, H.N., Tarjan, R.E.: Algorithms for two bottleneck optimization problems. Journal of Algorithms 9, 411–417 (1988)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Huang, X., Pan, V.Y.: Fast rectangular matrix multiplications and applications. Journal of Complexity 14, 257–299 (1998)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Munro, I.: Efficient determination of the strongly connected components and the transitive closure of a graph. Univ. of Toronto, Toronto, Canada (1971) (unpublished manuscript)Google Scholar
  20. 20.
    Seidel, R.: On the All-Pairs-Shortest-Path Problem in Unweighted Undirected Graphs. J. Comput. Syst. Sci. 51, 400–403 (1995)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Shoshan, A., Zwick, U.: All pairs shortest paths in undirected graphs with integer weights. In: Proc. of FOCS 1999, pp. 605–614 (1999)Google Scholar
  22. 22.
    Thorup, M., Zwick, U.: Approximate distance oracles. Journal of the ACM 52, 1–24 (2005)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Yuval, G.: An algorithm for finding all shortest paths using N 2.81 infinite-precision multiplications. Information Processing Letters 4, 155–156 (1976)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Zwick, U.: All-pairs shortest paths using bridging sets and rectangular matrix multiplication. Journal of the ACM 49, 289–317 (2002); Also, Proc. of FOCS 1998CrossRefMathSciNetGoogle Scholar
  25. 25.
    Zwick, U.: Exact and approximate distances in graphs - a survey. In: Meyer auf der Heide, F. (ed.) ESA 2001. LNCS, vol. 2161, pp. 33–48. Springer, Heidelberg (2001)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Liam Roditty
    • 1
  • Asaf Shapira
    • 2
  1. 1.Weizmann Institute 
  2. 2.Microsoft Research 

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