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Algorithmica

, Volume 50, Issue 2, pp 236–243 | Cite as

All-Pairs Shortest Paths with Real Weights in O(n 3/log n) Time

  • Timothy M. Chan
Article

Abstract

We describe an O(n 3/log n)-time algorithm for the all-pairs-shortest-paths problem for a real-weighted directed graph with n vertices. This slightly improves a series of previous, slightly subcubic algorithms by Fredman (SIAM J. Comput. 5:49–60, 1976), Takaoka (Inform. Process. Lett. 43:195–199, 1992), Dobosiewicz (Int. J. Comput. Math. 32:49–60, 1990), Han (Inform. Process. Lett. 91:245–250, 2004), Takaoka (Proc. 10th Int. Conf. Comput. Comb., Lect. Notes Comput. Sci., vol. 3106, pp. 278–289, Springer, 2004), and Zwick (Proc. 15th Int. Sympos. Algorithms and Computation, Lect. Notes Comput. Sci., vol. 3341, pp. 921–932, Springer, 2004). The new algorithm is surprisingly simple and different from previous ones.

Keywords

Shortest paths Graph algorithms 

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References

  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.
    Alon, N., Galil, Z., Margalit, O.: On the exponent of the all pairs shortest path problem. J. Comput. Sys. Sci. 54, 255–262 (1997) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Arlazarov, V.L., Dinic, E.C., Kronrod, M.A., Faradzev, I.A.: On economical construction of the transitive closure of a directed graph. Sov. Math. Dokl. 11, 1209–1210 (1970) MATHGoogle Scholar
  4. 4.
    Buchsbaum, A.L., Kaplan, H., Rogers, A., Westbrook, J.R.: Linear-time pointer-machine algorithms for least common ancestors, MST verification, and dominators. In Proc. 30th ACM Sympos. Theory Comput., pp. 279–288 (1998) Google Scholar
  5. 5.
    Chan, T.M.: All-pairs shortest paths for unweighted undirected graphs in o(mn) time. In Proc. 17th ACM-SIAM Sympos. Discrete Algorithms, pp. 514–523 (2006) Google Scholar
  6. 6.
    Coppersmith, D., Winograd, S.: Matrix multiplication via arithmetic progressions. J. Symb. Comput. 9, 251–280 (1990) MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn. McGraw–Hill, New York (2001) MATHGoogle Scholar
  8. 8.
    Dobosiewicz, W.: A more efficient algorithm for the min-plus multiplication. Int. J. Comput. Math. 32, 49–60 (1990) CrossRefMATHGoogle Scholar
  9. 9.
    Eppstein, D.: Quasiconvex analysis of backtracking algorithms. In Proc. 15th ACM-SIAM Sympos. Discrete Algorithms, pp. 788–797 (2004) Google Scholar
  10. 10.
    Feder, T., Motwani, R.: Clique partitions, graph compression and speeding-up algorithms. J. Comput. Syst. Sci. 51, 261–272 (1995) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Fredman, M.L.: New bounds on the complexity of the shortest path problem. SIAM J. Comput. 5, 49–60 (1976) CrossRefMathSciNetGoogle Scholar
  12. 12.
    Galil, Z., Margalit, O.: All pairs shortest paths for graphs with small integer length edges. J. Comput. Syst. Sci. 54, 243–254 (1997) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Han, Y.: Improved algorithm for all pairs shortest paths. Inform. Process. Lett. 91, 245–250 (2004) CrossRefMathSciNetGoogle Scholar
  14. 14.
    Matoušek, J.: Computing dominances in E n. Inform. Process. Lett. 38, 277–278 (1991) CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Pettie, S.: A new approach to all-pairs shortest paths on real-weighted graphs. Theor. Comput. Sci. 312, 47–74 (2004) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Pettie, S., Ramachandran, V.: A shortest path algorithm for real-weighted undirected graphs. SIAM J. Comput. 34, 1398–1431 (2005) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Preparata, F.P., Shamos, M.I.: Computational Geometry: An Introduction. Springer, New York (1985) Google Scholar
  18. 18.
    Seidel, R.: On the all-pairs-shortest-path problem in unweighted undirected graphs. J. Comput. Syst. Sci. 51, 400–403 (1995) CrossRefMathSciNetGoogle Scholar
  19. 19.
    Shoshan, A., Zwick, U.: All pairs shortest paths in undirected graphs with integer weights. In Proc. 40th IEEE Sympos. Found. Comput. Sci., pp. 605–614 (1999) Google Scholar
  20. 20.
    Strassen, V.: Gaussian elimination is not optimal. Numer. Math. 13, 354–356 (1969) MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Takaoka, T.: A new upper bound on the complexity of the all pairs shortest path problem. Inform. Process. Lett. 43, 195–199 (1992) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Takaoka, T.: A faster algorithm for the all-pairs shortest path problem and its application. In: Proc. 10th Int. Conf. Comput. Comb. Lect. Notes Comput. Sci., vol. 3106, pp. 278–289. Springer, Berlin (2004) Google Scholar
  23. 23.
    Takaoka, T.: An O(n 3log log n/log n) time algorithm for the all-pairs shortest path problem. Inform. Process. Lett. 96, 155–161 (2005) CrossRefMathSciNetGoogle Scholar
  24. 24.
    Yuster, R., Zwick, U.: Fast sparse matrix multiplication. In: Proc. 12th European Sympos. Algorithms. Lect. Notes Comput. Sci., vol. 3221, pp. 604–615. Springer, Berlin (2004) Google Scholar
  25. 25.
    Zwick, U.: All-pairs shortest paths using bridging sets and rectangular matrix multiplication. J. ACM 49, 289–317 (2002) MathSciNetGoogle Scholar
  26. 26.
    Zwick, U.: A slightly improved sub-cubic algorithm for the all pairs shortest paths problem with real edge lengths. In: Proc. 15th Int. Sympos. Algorithms and Computation. Lect. Notes Comput. Sci., vol. 3341, pp. 921–932. Springer, Berlin (2004) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.School of Computer ScienceUniversity of WaterlooWaterlooCanada

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