An O ( n 3 log log n / log n ) Time Algorithm for the All-Pairs Shortest Path Problem

We design a faster algorithm for the all-pairs shortest path problem under the conventional RAM model, based on distance matrix multiplication (DMM). Specifically we improve the best known time complexity of O(n(log log n)/ logn) to O(n log log n/ log n). As an application, we show the k-maximum subarray problem can be solved in O(kn log log n/ log n) time for small k.

[1]  Tadao Takaoka.  A New Upper Bound on the Complexity of the All Pairs Shortest Path Problem , 1991, WG.

[2]  Yijie Han,et al.  Improved algorithm for all pairs shortest paths , 2004, Inf. Process. Lett..

[3]  Tadao Takaoka,et al.  Subcubic Cost Algorithms for the All Pairs Shortest Path Problem , 1998, Algorithmica.

[4]  Noga Alon,et al.  On the Exponent of the All Pairs Shortest Path Problem , 1991, J. Comput. Syst. Sci..

[5]  Tadao Takaoka,et al.  Algorithms for the problem of K maximum sums and a VLSI algorithm for the K maximum subarrays problem , 2004, 7th International Symposium on Parallel Architectures, Algorithms and Networks, 2004. Proceedings..

[6]  Tadao Takaoka.  A Faster Algorithm for the All-Pairs Shortest Path Problem and Its Application , 2004, COCOON.

[7]  Hisao Tamaki,et al.  Algorithms for the maximum subarray problem based on matrix multiplication , 1998, SODA '98.

[8]  Michael L. Fredman,et al.  New Bounds on the Complexity of the Shortest Path Problem , 1976, SIAM J. Comput..

[9]  Alistair Moffat,et al.  An All Pairs Shortest Path Algorithm with Expected Time O(n² log n) , 1987, SIAM J. Comput..

[10]  Tadao Takaoka,et al.  A New Upper Bound on the Complexity of the All Pairs Shortest Path Problem , 1991, Inf. Process. Lett..

[11]  Jon Louis Bentley.  Programming pearls: perspective on performance , 1984, CACM.

[12]  Uri Zwick,et al.  All pairs shortest paths in weighted directed graphs-exact and almost exact algorithms , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).