  We introduce a new problem that combines the well known All Pairs Shortest
Paths (APSP) problem and the All Pairs Bottleneck Paths (APBP) problem to
compute the shortest paths for all pairs of vertices for all possible flow
amounts. We call this new problem the All Pairs Shortest Paths for All Flows
(APSP-AF) problem. We firstly solve the APSP-AF problem on directed graphs with
unit edge costs and real edge capacities in
$\tilde{O}(\sqrt{t}n^{(\omega+9)/4}) = \tilde{O}(\sqrt{t}n^{2.843})$ time,
where $n$ is the number of vertices, $t$ is the number of distinct edge
capacities (flow amounts) and $O(n^{\omega}) < O(n^{2.373})$ is the time taken
to multiply two $n$-by-$n$ matrices over a ring. Secondly we extend the problem
to graphs with positive integer edge costs and present an algorithm with
$\tilde{O}(\sqrt{t}c^{(\omega+5)/4}n^{(\omega+9)/4}) =
\tilde{O}(\sqrt{t}c^{1.843}n^{2.843})$ worst case time complexity, where $c$ is
the upper bound on edge costs.
