1. Factorizing the Brauer monoid in polynomial time
- Author
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Marchei, Daniele, Merelli, Emanuela, and Francis, Andrew
- Subjects
Mathematics - Rings and Algebras ,Computer Science - Data Structures and Algorithms - Abstract
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 $X$.
- Published
- 2024