Shortest Paths in Matrix Multiplication Time

In this paper we present an ${\tilde O}(W n^{\omega})$ time algorithm solving single source shortest path problem in graphs with integer weights from the set {–W,...,0,...,W}, where ω < 2.376 is the matrix multiplication exponent. For dense graphs with small edge weights, this result improves upon the algorithm of Goldberg that works in ${\tilde O}(mn^{0.5}{\rm log}W)$ time, and the Bellman-Ford algorithm that works in O(nm) time.

[1]  Richard Zippel,et al.  Probabilistic algorithms for sparse polynomials , 1979, EUROSAM.

[2]  E. W. Ng.  Symbolic and Algebraic Computation , 1979, Lecture Notes in Computer Science.

[3]  Arne Storjohann,et al.  High-order lifting and integrality certification , 2003, J. Symb. Comput..

[4]  Piotr Sankowski,et al.  Subquadratic Algorithm for Dynamic Shortest Distances , 2005, COCOON.

[5]  Piotr Sankowski,et al.  Dynamic transitive closure via dynamic matrix inverse: extended abstract , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[6]  Ronald L. Rivest,et al.  Introduction to Algorithms , 1990 .

[7]  Raphael Yuster,et al.  Answering distance queries in directed graphs using fast matrix multiplication , 2005, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05).

[8]  Jacob T. Schwartz,et al.  Fast Probabilistic Algorithms for Verification of Polynomial Identities , 1980, J. ACM.

[9]  Andrew V. Goldberg,et al.  Scaling algorithms for the shortest paths problem , 1995, SODA '93.

[10]  Harold N. Gabow.  Scaling Algorithms for Network Problems , 1985, J. Comput. Syst. Sci..

[11]  L. R. Ford,et al.  NETWORK FLOW THEORY , 1956 .

[12]  Don Coppersmith,et al.  Matrix multiplication via arithmetic progressions , 1987, STOC.

[13]  Robert E. Tarjan,et al.  Faster Scaling Algorithms for Network Problems , 1989, SIAM J. Comput..

[14]  Richard Bellman,et al.  ON A ROUTING PROBLEM , 1958 .

[15]  S. T. Buckland,et al.  An Introduction to the Bootstrap. , 1994 .

[16]  Piotr Sankowski,et al.  Dynamic Transitive Closure via Dynamic Matrix Inverse , 2004 .