A Combinatorial Algorithm for All-Pairs Shortest Paths in Directed Vertex-Weighted Graphs with Applications to Disc Graphs

  • Andrzej Lingas
  • Dzmitry Sledneu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7147)


We consider the problem of computing all-pairs shortest paths in a directed graph with non-negative real weights assigned to vertices.

For an n×n 0 − 1 matrix C, let K C be the complete weighted graph on the rows of C where the weight of an edge between two rows is equal to their Hamming distance. Let MWT(C) be the weight of a minimum weight spanning tree of K C .

We show that the all-pairs shortest path problem for a directed graph G on n vertices with non-negative real weights and adjacency matrix A G can be solved by a combinatorial randomized algorithm in time
$$\widetilde{O}(n^{2}\sqrt{n + \min\{MWT(A_G), MWT(A_G^t)\}})$$
As a corollary, we conclude that the transitive closure of a directed graph G can be computed by a combinatorial randomized algorithm in the aforementioned time.

We also conclude that the all-pairs shortest path problem for vertex-weighted uniform disk graphs induced by point sets of bounded density within a unit square can be solved in time \(\widetilde{O}(n^{2.75})\).


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.
    Alon, N., Naor, M.: Derandomization, Witnesses for Boolean Matrix Multiplication and Construction of Perfect hash functions. Algorithmica 16, 434–449 (1996)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Alon, N., Galil, Z., Margalit, O.: On the exponent of all pairs shortest path problem. J. Comput. System Sci. 54, 25–51 (1997)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bansal, N., Williams, R.: Regularity Lemmas and Combinatorial Algorithms. In: Proc. of 50th IEEE Symposium on Foundations on Computer Science, Atlanta (2009)Google Scholar
  5. 5.
    Björklund, A., Lingas, A.: Fast Boolean Matrix Multiplication for Highly Clustered Data. In: Dehne, F., Sack, J.-R., Tamassia, R. (eds.) WADS 2001. LNCS, vol. 2125, pp. 258–263. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  6. 6.
    Borodin, A., Ostrovsky, R., Rabani, Y.: Subquadratic Approximation Algorithms For Clustering Problems in High Dimensional Spaces. In: Proceedings of the 31st ACM Symposium on Theory of Computing (1999)Google Scholar
  7. 7.
    Cameron, P.J.: Combinatorics. Cambridge University Press (1994)Google Scholar
  8. 8.
    Chan, T.M.: More algorithms for all-pairs shortest paths in weighted graphs. SIAM J. Comput. 39(5), 2075–2089; preliminary version in proc. STOC 2007, pp. 590–598 (2007)Google Scholar
  9. 9.
    Chan, T.M.: All-pairs shortest paths with real weights in O(n 3/logn) time. Algorithmica 41, 330–337 (2008)CrossRefGoogle Scholar
  10. 10.
    Coppersmith, D., Winograd, S.: Matrix Multiplication via Arithmetic Progressions. J. of Symbolic Computation 9, 251–280 (1990)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Fürer, M., Kasiviswanathan, S.P.: Approximate Distance Queries in Disk Graphs. In: Erlebach, T., Kaklamanis, C. (eds.) WAOA 2006. LNCS, vol. 4368, pp. 174–187. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  12. 12.
    Galil, Z., Margalit, O.: Witnesses for Boolean Matrix Multiplication and Shortest Paths. Journal of Complexity, 417–426 (1993)Google Scholar
  13. 13.
    Gąsieniec, L., Lingas, A.: An Improved Bound on Boolean Matrix Multiplication for Highly Clustered Data. In: Dehne, F., Sack, J.-R., Smid, M. (eds.) WADS 2003. LNCS, vol. 2748, pp. 329–339. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  14. 14.
    Huang, X., Pan, V.Y.: Fast rectangular matrix multiplications and applications. Journal of Complexity 14(2), 257–299 (1998)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Indyk, P.: High-dimensional computational geometry. PhD dissertation, Standford University (2000)Google Scholar
  16. 16.
    Indyk, P., Motwani, R.: Approximate Nearest Neighbors: Towards Removing the Curse of Dimensionality. In: Proceedings of the 30th ACM Symposium on Theory of Computing (1998)Google Scholar
  17. 17.
    Indyk, P., Schmidt, S.E., Thorup, M.: On reducing approximate mst to closest pair problems in high dimensions (1999) (manuscript)Google Scholar
  18. 18.
    Karp, R.M., Steele, J.M.: Probabilistic analysis of heuristics. In: Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G., Shmoys, D.B. (eds.) The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, ch. 6, pp. 181–205. John Wiley & Sons Ltd. (1985)Google Scholar
  19. 19.
    Kushilevitz, E., Ostrovsky, E., Rabani, Y.: Efficient search for approximate nearest neighbor in high dimensional spaces. SIAM J. Comput. 30(2), 457–474; Preliminary version in Proc. 30th STOC (1989)Google Scholar
  20. 20.
    Lingas, A.: A Geometric Approach to Boolean Matrix Multiplication. In: Bose, P., Morin, P. (eds.) ISAAC 2002. LNCS, vol. 2518, pp. 501–510. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  21. 21.
    Munro, J.I.: Efficient determination of the transitive closure of a directed graph. Information Processing Letters 1(2), 56–58 (1971)CrossRefMATHGoogle Scholar
  22. 22.
    Rytter, W.: Fast recognition of pushdown automaton and context-free languages. Information and Control 67(1-3), 12–22 (1985)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Seidel, R.: On the All-Pairs-Shortest-Path Problem. In: Proc. 24th ACM STOC, pp. 745–749 (1992)Google Scholar
  24. 24.
    Vassilevska, V., Williams, R.: Subcubic Equivalences Between Path, Matrix, and Triangle Problems. In: Proceedings 51st Annual IEEE Symposium on Foundations of Computer Science, FOCS (2010)Google Scholar
  25. 25.
    Yuster, R.: Efficient algorithms on sets of permutations, dominance, and real-weighted APSP. In: Proc. of the 20th ACM-SIAM Symposium on Discrete Algorithms, pp. 950–957 (2009)Google Scholar
  26. 26.
    Zwick, U.: All pairs shortest paths using bridging rectangular matrix multiplication. Journal of the ACM 49(3), 289–317 (2002)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    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 2012

Authors and Affiliations

  • Andrzej Lingas
    • 1
  • Dzmitry Sledneu
    • 2
  1. 1.Department of Computer ScienceLund UniversityLundSweden
  2. 2.The Centre for Mathematical SciencesLund UniversityLundSweden

Personalised recommendations