  Finding a minimal factorization for a generic semigroup can be done by using
the Froidure-Pin Algorithm, which is not feasible for semigroups of large
sizes. On the other hand, if we restrict our attention to just a particular
semigroup, we could leverage its structure to obtain a much faster algorithm.
In particular, $\mathcal{O}(N^2)$ algorithms are known for factorizing the
Symmetric group $S_N$ and the Temperley-Lieb monoid $\mathcal{T}\mathcal{L}_N$,
but none for their superset the Brauer monoid $\mathcal{B}_{N}$. In this paper
we hence propose a $\mathcal{O}(N^4)$ factorization algorithm for
$\mathcal{B}_{N}$. At each iteration, the algorithm rewrites the input $X \in
\mathcal{B}_{N}$ as $X = X' \circ p_i$ such that $\ell(X') = \ell(X) - 1$,
where $p_i$ is a factor for $X$ and $\ell$ is a length function that returns
the minimal number of factors needed to generate $X$.
