String of minimum length in $\{a, b\}^*$ not in a regular expression
First, the string of minimum length might not be defined properly since it might not be unique.

Here is a way to find a string of minimum length:


Convert the regular expression to a nondeterministic finite automaton.
Convert the nondeterministic automaton into a deterministic one.
Use a breadth first search until you encounter the first nonfinal state (if there is one at all).