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 $X$.

翻译：对于一般半群的最小分解可以通过Froidure-Pin算法实现，但该算法不适用于大规模半群。然而，若将关注点限定在特定半群，则可利用其结构特性设计更高效的算法。具体而言，已知存在$\mathcal{O}(N^2)$算法分别用于分解对称群$S_N$和Temperley-Lieb幺半群$\mathcal{T}\mathcal{L}_N$，但针对其超集Brauer幺半群$\mathcal{B}_{N}$的此类算法尚未见报道。本文因此提出一种$\mathcal{O}(N^4)$算法用于分解$\mathcal{B}_{N}$。该算法在每次迭代中将输入$X \in \mathcal{B}_{N}$重写为$X = X' \circ p_i$，满足$\ell(X') = \ell(X) - 1$，其中$p_i$为$X$的一个因子，$\ell$是返回生成$X$所需最小因子数目的长度函数。