An O(n3 loglogn/log2n) Time Algorithm for All Pairs Shortest Paths

  • Yijie Han
  • Tadao Takaoka
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7357)


We present an O(n 3 loglogn/log2 n) time algorithm for all pairs shortest paths. This algorithm improves on the best previous result of O(n 3 (loglogn)3/log2 n ) time.


Algorithms all pairs shortest paths graph algorithms upper bounds 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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.
    Aho, A.V., Hopcroft, J.E., Ullman, J.D.: Data Structures and Algorithms. Addison-Wesley, Reading (1983)MATHGoogle Scholar
  3. 3.
    Albers, S., Hagerup, T.: Improved parallel integer sorting without concurrent writing. Information and Computation 136, 25–51 (1997)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Batcher, K.E.: Sorting networks and their applications. In: Proc. 1968 AFIPS Spring Joint Summer Computer Conference, pp. 307–314 (1968)Google Scholar
  5. 5.
    Chan, T.M.: More algorithms for all-pairs shortest paths in weighted graphs. In: Proc. 2007 ACM Symp. Theory of Computing, pp. 590–598 (2007)Google Scholar
  6. 6.
    Chan, T.M.: All-Pairs Shortest Paths with Real Weights in O(n 3/logn) Time. In: Dehne, F., López-Ortiz, A., Sack, J.-R. (eds.) WADS 2005. LNCS, vol. 3608, pp. 318–324. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  7. 7.
    Dobosiewicz, W.: A more efficient algorithm for min-plus multiplication. Inter. J. Comput. Math. 32, 49–60 (1990)MATHCrossRefGoogle Scholar
  8. 8.
    Fredman, M.L.: New bounds on the complexity of the shortest path problem. SIAM J. Computing 5, 83–89 (1976)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Fredman, M.L., Tarjan, R.: Fibonacci heaps and their uses in improved network optimization algorithms. Journal of the ACM 34, 596–615 (1987)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Galil, Z., Margalit, O.: All pairs shortest distances for graphs with small integer length edges. Information and Computation 134, 103–139 (1997)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Han, Y.: An O(n 3(loglogn/logn)5/4) time algorithm for all pairs shortest paths. Algorithmica 51(4), 428–434 (2008)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Han, Y.: Improved algorithms for all pairs shortest paths. Information Processing Letters 91, 245–250 (2004)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Han, Y.: A note of an O(n 3/logn) time algorithm for all pairs shortest paths. Information Processing Letters 105, 114–116 (2008)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Pettie, S.: A Faster All-Pairs Shortest Path Algorithm for Real-Weighted Sparse Graphs. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 85–97. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  15. 15.
    Pettie, S., Ramachandran, V.: A shortest path algorithm for real-weighted undirected graphs. SIAM J. Comput. 34(6), 1398–1431 (2005)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Sankowski, P.: Shortest Paths in Matrix Multiplication Time. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 770–778. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  17. 17.
    Seidel, R.: On the all-pairs-shortest-path problem in unweighted undirected graphs. J. Comput. Syst. Sci. 51, 400–403 (1995)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Strassen, V.: Gaussian elimination is not optimal. Numerische Mathematik 14(3), 354–356 (1969)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Takaoka, T.: A new upper bound on the complexity of the all pairs shortest path problem. Information Processing Letters 43, 195–199 (1992)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Takaoka, T.: An O(n 3 loglogn/logn) time algorithm for the all-pairs shortest path problem. Information Processing Letters 96, 155–161 (2005)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Thorup, M.: Undirected single source shortest paths with positive integer weights in linear time. Journal of ACM 46(3), 362–394 (1999)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Yuster, R., Zwick, U.: Answering distance queries in directed graphs using fast matrix multiplication. In: 46th Annual IEEE Symposium on Foundations of Computer Science, pp. 389–396. IEEE Comput. Soc., Los Alamitos (2005)CrossRefGoogle Scholar
  23. 23.
    Zwick, U.: All pairs shortest paths using bridging sets and rectangular matrix multiplication. Journal of the ACM 49(3), 289–317 (2002)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Zwick, U.: A Slightly Improved Sub-cubic Algorithm for the All Pairs Shortest Paths Problem with Real Edge Lengths. In: Fleischer, R., Trippen, G. (eds.) ISAAC 2004. LNCS, vol. 3341, pp. 921–932. Springer, Heidelberg (2004)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Yijie Han
    • 1
  • Tadao Takaoka
    • 2
  1. 1.School of Computing and EngineeringUniversity of Missouri at Kansas CityKansas CityUSA
  2. 2.Department of Computer Science and Software EngineeringUniversity of CanterburyChristchurchNew Zealand

Personalised recommendations