Skip to main content
SpringerLink
Log in

An Optimal Algorithm to Solve the All-Pairs Shortest Paths Problem on Permutation Graphs

  • Published:
Journal of Mathematical Modelling and Algorithms

Abstract

In this paper we present an optimal algorithm to solve the all-pairs shortest path problem on permutation graphs with n vertices and m edges which runs in O(n 2) time. Using this algorithm, the average distance of a permutation graph can also be computed in O(n 2) time.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aho, A., Hopcroft, J. and Ullman, J.: The Design and Analysis of Computer Algorithms, Addison-Wesley, Reading, MA, 1974.

    Google Scholar 

  2. Ahuja, R. K., Mehlhorn, K., Orlin, J. B. and Tarjan, R. E.: Faster algorithms for the shortest path problem, J. ACM 37(2) (1990), 213–223.

    Google Scholar 

  3. Alon, N., Galil, Z. and Margalit, O.: On the exponent of the all pairs shortest path problem, In: Proc. 32nd IEEE FOCS, IEEE, 1991, pp. 569–575.

  4. Althofer, I.: Average distance in undirected graphs and the removal of vertices, J. Combin. Theory Ser. B 48 (1990), 140–142.

    Google Scholar 

  5. Barbehenn, M.: A note on the complexity of Dijkstra's algorithm for graphs with weighted vertices, IEEE Trans. Comput. 47(2) (1998), 263.

    Google Scholar 

  6. Bienstock, D. and Gyon, E.: Average distance in graphs with removed elements, J. Graph Theory 12 (1988), 375–390.

    Google Scholar 

  7. Chung, F. R. K.: The average distance and independence number, J. Graph Theory 12 (1988), 229–235.

    Google Scholar 

  8. Dankelmann, P.: Average distance and independence number, Discrete Appl. Math. 51 (1994), 75–83.

    Google Scholar 

  9. Dankelmann, P.: Computing the average distance of an interval graphs, Inform. Process. Lett. 48 (1993), 311–314.

    Google Scholar 

  10. Dijkstra, E. W.: A note on two problems in connexion with graphs, Numer. Math. 1 (1959), 269–271.

    Google Scholar 

  11. Erdos, P., Pach, J. and Spencer, J.: On the mean distance between points of a graph, Congr. Numer. 64 (1988), 121–124.

    Google Scholar 

  12. Favaron, O., Kouider, M. and Maheo, M.: Edge-vulnerability and mean distance, Networks 19 (1989), 493–504.

    Google Scholar 

  13. Fajtlowich, S. and Waller, W. A.: On two conjectures of GRAFFITI, Congr. Numer. 55 (1986), 51–56.

    Google Scholar 

  14. Fredman, M. L. and Tarjan, R. E.: Fibonacci heaps and their uses in improved network optimization algorithms, J. ACM 34 (1987), 596–615.

    Google Scholar 

  15. Floyd, R.: Algorithm 97: Shortest path, Comm. ACM 5(6) (1962), 345.

    Google Scholar 

  16. Galil, Z. and Margalit, O.: All pairs shortest distances for graphs with small integer length edges, Inform. Comput. 134 (1997), 103–139.

    Google Scholar 

  17. Golumbic, M. C.: Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York, 1980.

    Google Scholar 

  18. Hendry, G. R. T.: On mean distance in certain classes of graphs, Networks 19 (1989), 451–557.

    Google Scholar 

  19. Mirchandani, P.: A simple O(n 2 ) algorithm for all pairs shortest path problem on an interval graph, Networks 27 (1996), 215–217.

    Google Scholar 

  20. Mohar, B.: Eigenvalues, diameter and mean distance in graphs, Graphs Combin. 7 (1991), 53–64.

    Google Scholar 

  21. Mondal, S., Pal, M. and Pal, T. K.: An optimal algorithm for solving all-pairs shortest paths on trapezoid graphs, Intern. J. Comput. Engg. Sci. 3(2) (2002), 103–116.

    Google Scholar 

  22. Moore, E. F.: The shortest path through a maze, Proc. Internat. Sympos. Switching Theory, 1957, Harvard Univ. Press, 1959, pp. 285–292.

  23. Muller, J. H. and Spinrad, J.: Incremental modular decomposition, J. ACM 36 (1989), 1–19.

    Google Scholar 

  24. Pal, M. and Bhattacharjee, G. P.: An optimal parallel algorithm for all-pairs shortest paths on unweighted interval graphs, Nordic J. Comput. 4 (1997), 342–356.

    Google Scholar 

  25. Pal, M. and Bhattacharjee, G. P.: A data structure on interval graphs and its applications, J. Circuits Systems Comput. 7(3) (1997), 165–175.

    Google Scholar 

  26. Pnueli, A., Lempel, A. and Even, S.: Transitive orientation of graphs and identification of permutation graphs, Canad. J. Math. 23 (1971), 160–175.

    Google Scholar 

  27. Ravi, R., Marathe, M. V. and Pandu Rangan, C.: An optimal algorithm to solve the all-pair shortest path problem on interval graphs, Networks 22 (1992), 21–35.

    Google Scholar 

  28. Seidel, R.: On the all pairs shortest path problem, In: Proc. 24th ACM STOC, ACM Press, New York, 1992, pp. 745–749.

    Google Scholar 

  29. Tomescu, I. and Melter, R. A.: On distances in chromatic graphs, Quart. J. Math. Oxford (2) 40 (1989), 475–480.

    Google Scholar 

  30. Winkler, P.: Mean distance and the ‘four-thirds conjecture’, Congr. Numer. 54 (1986), 63–72.

    Google Scholar 

  31. Winkler, P.: Mean distance in a tree, Discrete Appl. Math. 27 (1990), 179–185.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore-, 721 102, India

    Sukumar Mondal, Madhumangal Pal & Tapan K. Pal

Authors
  1. Sukumar Mondal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Madhumangal Pal
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tapan K. Pal
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mondal, S., Pal, M. & Pal, T.K. An Optimal Algorithm to Solve the All-Pairs Shortest Paths Problem on Permutation Graphs. Journal of Mathematical Modelling and Algorithms 2, 57–65 (2003). https://doi.org/10.1023/A:1023695531209

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1023695531209

Search

Navigation