<p>Alice and Bob take turns playing a game, with <strong>Alice starting first</strong>.</p>

<p>There are <code>n</code> stones arranged in a row. On each player&#39;s turn, they can <strong>remove</strong> either the leftmost stone or the rightmost stone from the row and receive points equal to the <strong>sum</strong> of the remaining stones&#39; values in the row. The winner is the one with the higher score when there are no stones left to remove.</p>

<p>Bob found that he will always lose this game (poor Bob, he always loses), so he decided to <strong>minimize the score&#39;s difference</strong>. Alice&#39;s goal is to <strong>maximize the difference</strong> in the score.</p>

<p>Given an array of integers <code>stones</code> where <code>stones[i]</code> represents the value of the <code>i<sup>th</sup></code> stone <strong>from the left</strong>, return <em>the <strong>difference</strong> in Alice and Bob&#39;s score if they both play <strong>optimally</strong>.</em></p>

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

<pre>
<strong>Input:</strong> stones = [5,3,1,4,2]
<strong>Output:</strong> 6
<strong>Explanation:</strong> 
- Alice removes 2 and gets 5 + 3 + 1 + 4 = 13 points. Alice = 13, Bob = 0, stones = [5,3,1,4].
- Bob removes 5 and gets 3 + 1 + 4 = 8 points. Alice = 13, Bob = 8, stones = [3,1,4].
- Alice removes 3 and gets 1 + 4 = 5 points. Alice = 18, Bob = 8, stones = [1,4].
- Bob removes 1 and gets 4 points. Alice = 18, Bob = 12, stones = [4].
- Alice removes 4 and gets 0 points. Alice = 18, Bob = 12, stones = [].
The score difference is 18 - 12 = 6.
</pre>

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

<pre>
<strong>Input:</strong> stones = [7,90,5,1,100,10,10,2]
<strong>Output:</strong> 122</pre>

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

<ul>
	<li><code>n == stones.length</code></li>
	<li><code>2 &lt;= n &lt;= 1000</code></li>
	<li><code>1 &lt;= stones[i] &lt;= 1000</code></li>
</ul>

class Solution {
public:
    vector<vector<vector<pair<int, int>>>> memo; 

    int getSum(vector<int>& sums, int l, int r) {
        return sums[r] - ((l > 0) ? sums[l - 1] : 0);
    };

    pair<int, int> dfs(int l, int r, int turn, pair<int, int> score, vector<int>& sums) { 
        if (l == r) 
            return {0, 0};
        else if (memo[l][r][turn].first != -1)
            return memo[l][r][turn];

        pair<int, int> ret;

        if (turn == 0) { 
            auto [aliceL, bobL] = dfs(l + 1, r, 1, score, sums);
            aliceL += getSum(sums, l + 1, r);

            auto [aliceR, bobR] = dfs(l, r - 1, 1, score, sums);
            aliceR += getSum(sums, l, r - 1);

            int takeLeft = aliceL - bobL; 
            int takeRight = aliceR - bobR;

            if (takeLeft > takeRight)
                ret = {aliceL, bobL}; 
            else 
                ret = {aliceR, bobR};
        } else { 
            auto [aliceL, bobL] = dfs(l + 1, r, 0, score, sums);
            bobL += getSum(sums, l + 1, r);

            auto [aliceR, bobR] = dfs(l, r - 1, 0, score, sums);
            bobR += getSum(sums, l, r - 1);

            int takeLeft = bobL - aliceL; 
            int takeRight = bobR - aliceR;

            if (takeLeft > takeRight)
                ret = {aliceL, bobL}; 
            else 
                ret = {aliceR, bobR};
        }

        memo[l][r][turn] = ret;
        return ret;
    }

    int stoneGameVII(vector<int>& stones) {
        int n = stones.size();
        memo.resize(n, vector<vector<pair<int, int>>>(n, vector<pair<int, int>>(2, {-1, -1})));

        vector<int> sums(n, 0); 
        sums[0] = stones[0];

        for (int i = 1; i < n; i++) 
            sums[i] = sums[i - 1] + stones[i];

        auto [alice, bob] = dfs(0, n - 1, 0, {0, 0}, sums);
        return abs(alice - bob); 
    }
};
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