Let $W$ be a Coxeter group with set of simple reflections $S$. Suppose that I have chosen a preferred reduced decomposition for every element of $W$. Given an arbitrary word in the alphabet $S$, is there an algorithm for reducing this word to my chosen decomposition using the Coxeter relations? That is to say, an algorithm which at each step replaces a subword of the form $s_i s_k s_i \dots$ by $s_k s_i s_k \dots$, or which removes an occurrence of $s_k^2$.
I would be happy with an answer in the following situation: $W=\Sigma_n$ is the symmetric group on $n$ letters and the preferred decomposition is given by taking the lexicographically smallest (or largest) reduced decomposition.