<p>Alice and Bob play a game with piles of stones. There are an <strong>even</strong> number of piles arranged in a row, and each pile has a <strong>positive</strong> integer number of stones <code>piles[i]</code>.</p>

<p>The objective of the game is to end with the most stones. The <strong>total</strong> number of stones across all the piles is <strong>odd</strong>, so there are no ties.</p>

<p>Alice and Bob take turns, with <strong>Alice starting first</strong>. Each turn, a player takes the entire pile of stones either from the <strong>beginning</strong> or from the <strong>end</strong> of the row. This continues until there are no more piles left, at which point the person with the <strong>most stones wins</strong>.</p>

<p>Assuming Alice and Bob play optimally, return <code>true</code><em> if Alice wins the game, or </em><code>false</code><em> if Bob wins</em>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> piles = [5,3,4,5]
<strong>Output:</strong> true
<strong>Explanation:</strong> 
Alice starts first, and can only take the first 5 or the last 5.
Say she takes the first 5, so that the row becomes [3, 4, 5].
If Bob takes 3, then the board is [4, 5], and Alice takes 5 to win with 10 points.
If Bob takes the last 5, then the board is [3, 4], and Alice takes 4 to win with 9 points.
This demonstrated that taking the first 5 was a winning move for Alice, so we return true.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> piles = [3,7,2,3]
<strong>Output:</strong> true
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>2 &lt;= piles.length &lt;= 500</code></li>
	<li><code>piles.length</code> is <strong>even</strong>.</li>
	<li><code>1 &lt;= piles[i] &lt;= 500</code></li>
	<li><code>sum(piles[i])</code> is <strong>odd</strong>.</li>
</ul>

class Solution {
public:
    bool stoneGame(vector<int>& piles) {
        int n = piles.size();
        unordered_map<string, int> memo;

        int Alice = miniMax(piles, true, 0, n-1, INT_MIN, INT_MAX, memo);
        
        int sum = 0;

        for (int& pile: piles)
            sum += pile;

        if (Alice > sum/2)
            return true;
        else
            return false;
    }

    string createStateKey(int headIndex, int tailIndex, bool isMax) {
        return to_string(headIndex) + "-" + to_string(tailIndex) + "-" + to_string(isMax);
    }

    int miniMax(vector<int>& piles, bool isMax, int headIndex, int tailIndex, int alpha, int beta, unordered_map<string, int>& memo) {
        // Base case: if no more piles are left to choose from
        if (headIndex > tailIndex) return 0;

        // Check memoized results
        string key = createStateKey(headIndex, tailIndex, isMax);
        if (memo.find(key) != memo.end()) return memo[key];

        if (isMax) {
            int scoreFromFront = piles[headIndex] + miniMax(piles, false, headIndex + 1, tailIndex, alpha, beta, memo);
            alpha = max(alpha, scoreFromFront);
            if (beta <= alpha) return scoreFromFront; // Prune

            int scoreFromEnd = piles[tailIndex] + miniMax(piles, false, headIndex, tailIndex - 1, alpha, beta, memo);
            alpha = max(alpha, scoreFromEnd);
            if (beta <= alpha) return scoreFromEnd; // Prune

            int maxPoint = max(scoreFromFront, scoreFromEnd);
            memo[key] = maxPoint;
            return maxPoint;
        } else {
            int scoreFromFront = miniMax(piles, true, headIndex + 1, tailIndex, alpha, beta, memo);
            beta = min(beta, scoreFromFront);
            if (beta <= alpha) return scoreFromFront; // Prune

            int scoreFromEnd = miniMax(piles, true, headIndex, tailIndex - 1, alpha, beta, memo);
            beta = min(beta, scoreFromEnd);
            if (beta <= alpha) return scoreFromEnd; // Prune

            int minPoint = min(scoreFromFront, scoreFromEnd);
            memo[key] = minPoint;
            return minPoint;
        }
    }
};
46703