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TR00-060 | 17th August 2000 00:00

All Pairs Shortest Paths using Bridging Sets and Rectangular Matrix Multiplication


Authors: Uri Zwick
Publication: 18th August 2000 10:43
Downloads: 1658


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.

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