294. Flip Game II

You are playing the following Flip Game with your friend: Given a string that contains only these two characters: + and -, you and your friend take turns to flip two consecutive "++" into "--". The game ends when a person can no longer make a move and therefore the other person will be the winner.

Write a function to determine if the starting player can guarantee a win.

Example:

Input: s = "++++"
Output: true 
Explanation: The starting player can guarantee a win by flipping the middle "++" to become "+--+".

Follow up: Derive your algorithm's runtime complexity.

// Brute Force Recursion
bool canWin(string s) { // time: O(n!); space: O(n)
    for (int i = 1; i < s.length(); ++i) {
        if (s[i - 1] == '+' && s[i] == '+' && !canWin(s.substr(0, i - 1) + "--" + s.substr(i + 1))) {
            return true;
        }
    }
    return false;
}

// Recursion with Memoization
bool helper(const string& s, unordered_map<string, bool>& memo) {
    if (memo.count(s)) return memo[s];
    for (int i = 1; i < s.length(); ++i) {
        if (s[i - 1] == '+' && s[i] == '+' && !helper(s.substr(0, i - 1) + "--" + s.substr(i + 1), memo)) {
            return memo[s] = true;
        }
    }
    return memo[s] = false;
}
bool canWin(string s) { // time: O(2^n); space: O(2^n)
    unordered_map<string, bool> memo;
    return helper(s, memo);
}

293. Flip Game