All Pairs Shortest Paths using Bridging Sets and Rectangular Matrix   Multiplication
  We present two new algorithms for solving the {\em All Pairs Shortest Paths} (APSP) problem for weighted directed graphs. Both algorithms use fast matrix multiplication algorithms.   The first algorithm solves the APSP problem for weighted directed graphs in which the edge weights are integers of small absolute value in $\Ot(n^{2+\mu})$ time, where $\mu$ satisfies the equation $\omega(1,\mu,1)=1+2\mu$ and $\omega(1,\mu,1)$ is the exponent of the multiplication of an $n\times n^\mu$ matrix by an $n^\mu \times n$ matrix. Currently, the best available bounds on $\omega(1,\mu,1)$, obtained by Coppersmith, imply that $\mu<0.575$. The running time of our algorithm is therefore $O(n^{2.575})$. Our algorithm improves on the $\Ot(n^{(3+\omega)/2})$ time algorithm, where $\omega=\omega(1,1,1)<2.376$ is the usual exponent of matrix multiplication, obtained by Alon, Galil and Margalit, whose running time is only known to be $O(n^{2.688})$.   The second algorithm solves the APSP problem {\em almost} exactly for directed graphs with {\em arbitrary} non-negative real weights. The algorithm runs in $\Ot((n^\omega/\eps)\log (W/\eps))$ time, where $\eps>0$ is an error parameter and W is the largest edge weight in the graph, after the edge weights are scaled so that the smallest non-zero edge weight in the graph is 1. It returns estimates of all the distances in the graph with a stretch of at most $1+\eps$. Corresponding paths can also be found efficiently. 
Uri Zwick