<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:
    pair<int,int> aliceCount(int i,int j,int turn,vector<int>& pre,vector<vector<vector<pair<int,int>>>>& dp)
    {
        if(i >= j)
            return {0,0};
        
        if(dp[i][j][turn].first != -1 && dp[i][j][turn].second != -1) 
            return dp[i][j][turn];

        int sum1 = pre[j] - pre[i]; 
        int sum2 = pre[j-1] - ((i-1 >=0) ? pre[i-1] : 0); 
        pair<int,int> ret;

        if(turn == 0) 
        {
            auto ans1 = aliceCount(i+1,j,1,pre,dp);
            auto ans2 = aliceCount(i,j-1,1,pre,dp);
            if(sum1 + ans1.first - ans1.second > sum2 + ans2.first - ans2.second) {
                ret = {sum1 + ans1.first, ans1.second};
            } else {
                ret = {sum2 + ans2.first, ans2.second};
            }
        }
        else 
        {
            auto ans1 = aliceCount(i+1,j,0,pre,dp);
            auto ans2 = aliceCount(i,j-1,0,pre,dp);
            if(ans1.first - ans1.second - sum1 < ans2.first - ans2.second - sum2) {
                ret = {ans1.first, ans1.second + sum1};
            } else {
                ret = {ans2.first, ans2.second + sum2};
            }
        }

        return dp[i][j][turn] = ret;
    }

    int stoneGameVII(vector<int>& stones) {
        int n = stones.size();
        vector<int> pre(n,0);
        pre[0] = stones[0];
        for(int i=1;i<n;i++) {
            pre[i] = pre[i-1] + stones[i];
        }
        vector<vector<vector<pair<int,int>>>> dp(n,vector<vector<pair<int,int>>>(n,vector<pair<int,int>>(2,{-1,-1})));
        auto alice = aliceCount(0,n-1,0,pre,dp);
        return alice.first - alice.second;
    }
};

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