Sub-cubic cost algorithms for the all pairs shortest path problem

  • Tadao Takaoka
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1017)


In this paper we give three sub-cubic cost algorithms for the all pairs shortest distance (APSD) and path (APSP) problems. The first is a parallel algorithm that solves the APSD problem for a directed graph with unit edge costs in O(log2n) time with \(O({{n^\mu } \mathord{\left/{\vphantom {{n^\mu } {\sqrt {\log n} }}} \right.\kern-\nulldelimiterspace} {\sqrt {\log n} }})\) processors where Μ=2.688 on an EREW-PRAM. The second parallel algorithm solves the APSP, and consequently APSD, problem for a directed graph with non-negative general costs (real numbers) in O(log2n) time with o(n3) subcubic cost. Previously this cost was greater than O(n3). Finally we improve with respect to M the complexity O((Mn)μ) of a sequential algorithm for a graph with edge costs up to M into O(M1/3n6+1/3/3(log n)2/3(log log n)1/3) in the APSD problem.


Short Path Directed Graph Matrix Multiplication Parallel Algorithm Short Path Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    N. Alon, Z. Galil and O. Margalit, On the exponent of the all pairs shortest path problem, Proc. 32th IEEE FOCS (1991), pp 569–575.Google Scholar
  2. 2.
    N. Alon, Z. Galil, and O. Margalit and M. Naor, Witnesses for Boolean matrix multiplication and for shortest paths, Proc. 33th IEEE FOCS (1992), pp. 417–426.Google Scholar
  3. 3.
    D. Coppersmith and S. Winograd, Matrix multiplication via arithmetic progressions, Journal of Symbolic Computation 9 (1990), pp. 251–280.Google Scholar
  4. 4.
    E. Dekel, D. Nassimi and S. Sahni, Parallel matix and graph algorithms, SIAM Jour. on Comp. 10 (1981), pp. 657–675.Google Scholar
  5. 5.
    M. L. Fredman, New bounds on the complexity of the shortest path problem, SIAM Jour. Comput. 5 (1976), pp. 49–60.Google Scholar
  6. 6.
    H. Gazit and G. Miller, An improved parallel algorithm that computes the bis numbering of a directed graph, Info. Proc. Lett. 28 (1988) pp. 61–65.Google Scholar
  7. 7.
    A. Gibbons and W. Rytter, Efficient Parallel Algorithms, Cambridge Univ. Press (1988).Google Scholar
  8. 8.
    Y. Han, V. Pan and J. Reif, Efficient parallel algorithms for computing all pairs shortest paths in directed graphs. Proc. 4th ACM SPAA (1992), pp. 353–362.Google Scholar
  9. 9.
    F. Romani, Shortest-path problem is not harder than matrix multiplications, Info. Proc. Lett. 11 (1980) pp.134–136.Google Scholar
  10. 10.
    R. Scidel, On the all-pairs-shortest-path problem, Proc. 24th ACM STOC (1990), pp. 213–223.Google Scholar
  11. 11.
    A. Schönhage and V. Strassen, Schnelle Multiplikation Gro\er Zahlen, Computing 7 (1971) pp. 281–292.Google Scholar
  12. 12.
    T. Takaoka, A new upperbound on the complexity of the all pairs shortest path problem, Info. Proc. Lett. 43 (1992), pp. 195–199.Google Scholar
  13. 13.
    T. Takaoka, An efficient parallel algorithm for the all pairs shortest path problem, WG 88, Lecture Notes in Computer Science 344 (Springer, Berlin, 1988) pp. 276–287.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Tadao Takaoka
    • 1
  1. 1.Department of Computer ScienceIbaraki UniversityIbarakiJapan

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