Given a regular language $L$ and a regular expression $r$ with $L=L(r)$. Is it possible to determine the minimum length of words of $L(r)$ by the structure of $r$?

A straightforward example:

Let's say we have a regular expression $r=aac^\*aa$, then $L(R) = \{aaaa, aacaa, aaccaa, \dots, aac^naa\}$. To determine the minimal length I would erase everything that is postfixed with $\*$, leaving $r'=aaaa$. Now I would count the concatenations and add 1, which would yield in this example, not unsurprisingly, a minimum length of 4.

Is there a general approach to do this for more complex expressions?

**Sidenote:** I need to achieve this without the help of automata.