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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.07874 |
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| _version_ | 1866911776079085568 |
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| author | Marchei, Daniele Merelli, Emanuela Francis, Andrew |
| author_facet | Marchei, Daniele Merelli, Emanuela Francis, Andrew |
| contents | 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 $X$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_07874 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Factorizing the Brauer monoid in polynomial time Marchei, Daniele Merelli, Emanuela Francis, Andrew Rings and Algebras Data Structures and Algorithms 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 $X$. |
| title | Factorizing the Brauer monoid in polynomial time |
| topic | Rings and Algebras Data Structures and Algorithms |
| url | https://arxiv.org/abs/2402.07874 |