# Represent an element in a free monoid with minimum weight

Consider a rank \(k\) free monoid \((M,\cdot)\) with free generators \(G\). Sometimes there are ways to express them by writing a little less than write the whole string of generators. We can group some generators by powers. For example, \(aababababaaaa = a(ab)^4a^4\).

Find the shortest way to write down an element in a free monoid.

There are problems on how long are the parentheses, exponents etc. Therefore we generalize it to allow weight to those operations.

Formally. For any free monoid \(M\) with free generators \(G\), we can construct another free monoid \((M^*,\cdot)\),

- \(a\in G \implies Atom(a)\in M^*\).
- \(a\in M^*\), \(n\in\N\), then \(Power(a,n) \in M^*\).

Consider a homomorphism \(w:M^*\to \N\). Such that for all \(n\), it satisfy the following criteria:

- \(w(a)\leq w(b) \implies w(Power(a,n))\leq w(Power(b,n))\),
- \(w(a)\leq w(Power(a,1))\).

\(w\) is a weight function.

Let \(f:M^*\to M\), such that

- \(f(ab) = f(a)f(b)\),
- \(f(Atom(a)) = a\),
- \(f(Power(a,n)) = a^n\).

Given \(a\in M\), we want to find \(a'\in M^*\), such that \(f(a') = a\) and \(w(a')\) is minimized.

The input is \(a_1\ldots a_n\).

Let \(D(i,j)\) represent the minimum weight representation for \(a_i\ldots a_j\). Let \(P(i,j)\) represent the set of all possible \(Power(x,k)\), such that \(f(Power(x,k)) = a_i\ldots a_j\) for some \(k\neq 1\).

\begin{align*} D(i,i) &= a_i\\ D(i,j) &= \min(P(i,j)\cup \{ D(i,k)+D(k+1,j)| i\leq k\leq j-1\}) \end{align*}Here \(\min\) return any of the expressions that achieves the minimum weight. This allows a \(O(n^3)\) algorithm if one uses suffix tree for finding \(P(i,j)\). One can naively try all possible \(Power(x,k)\) instead, where \(k|n\).

Here is an Haskell code for it. It is designed to show the algorithm instead of been efficient. This has real life usage to compress regular expressions.