Papers › Factorizing the Brauer monoid in polynomial time
Factorizing the Brauer monoid in polynomial time
Daniele Marchei, Emanuela Merelli, Andrew Francis
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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, 𝒪(N²) algorithms are known for factorizing the Symmetric group S_N and the Temperley-Lieb monoid 𝒯ℒ_N, but none for their superset the Brauer monoid ℬ_N. In this paper we hence propose a 𝒪(N⁴) factorization algorithm for ℬ_N. At each iteration, the algorithm rewrites the input X ∈ℬ_N as X = X′ ∘pᵢ such that ℓ(X′) = ℓ(X) - 1, where pᵢ is a factor for X and ℓ is a length function that returns the minimal number of factors needed to generate X.
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