  Computing the empirical Wasserstein distance in the
Wasserstein-distance-based independence test is an optimal transport (OT)
problem with a special structure. This observation inspires us to study a
special type of OT problem and propose a modified Hungarian algorithm to solve
it exactly. For the OT problem involving two marginals with $m$ and $n$ atoms
($m\geq n$), respectively, the computational complexity of the proposed
algorithm is $O(m^2n)$. Computing the empirical Wasserstein distance in the
independence test requires solving this special type of OT problem, where
$m=n^2$. The associated computational complexity of the proposed algorithm is
$O(n^5)$, while the order of applying the classic Hungarian algorithm is
$O(n^6)$. In addition to the aforementioned special type of OT problem, it is
shown that the modified Hungarian algorithm could be adopted to solve a wider
range of OT problems. Broader applications of the proposed algorithm are
discussed -- solving the one-to-many assignment problem and the many-to-many
assignment problem. We conduct numerical experiments to validate our
theoretical results. The experiment results demonstrate that the proposed
modified Hungarian algorithm compares favorably with the Hungarian algorithm,
the well-known Sinkhorn algorithm, and the network simplex algorithm.
