Brute Force Approach

Algorithm

We are given a string that can contain three characters:

'(' which represents an opening bracket
')' which represents a closing bracket
'*' which can represent an opening bracket, a closing bracket, or be an empty string

The idea is to explore every possible way to interpret the '*' characters. At each step, we try all interpretations for '*', and check whether the resulting string is a valid parentheses string.This approach uses recursion (backtracking) to explore all combinations.

Define a recursive function that takes the current index and the number of open brackets.
If the number of open brackets becomes negative at any point, return false (invalid string).
If we reach the end of the string:
- If the number of open brackets is zero, return true (valid string).
- Otherwise, return false.
If the current character is '(', increase the open bracket count and call the function recursively.
If the character is ')', decrease the open bracket count and call the function recursively.
If the character is '*', try all three possibilities:
- Ignore it: call the function without changing the open count.
- Treat it as '(': increase open count and call recursively.
- Treat it as ')': decrease open count and call recursively.
If any of the recursive calls return true, return true.
If none return true, return false.