<p>Given an array <code>arr</code> of positive integers, consider all binary trees such that:</p>

<ul>
	<li>Each node has either <code>0</code> or <code>2</code> children;</li>
	<li>The values of <code>arr</code> correspond to the values of each <strong>leaf</strong> in an in-order traversal of the tree.</li>
	<li>The value of each non-leaf node is equal to the product of the largest leaf value in its left and right subtree, respectively.</li>
</ul>

<p>Among all possible binary trees considered, return <em>the smallest possible sum of the values of each non-leaf node</em>. It is guaranteed this sum fits into a <strong>32-bit</strong> integer.</p>

<p>A node is a <strong>leaf</strong> if and only if it has zero children.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/08/10/tree1.jpg" style="width: 500px; height: 169px;" />
<pre>
<strong>Input:</strong> arr = [6,2,4]
<strong>Output:</strong> 32
<strong>Explanation:</strong> There are two possible trees shown.
The first has a non-leaf node sum 36, and the second has non-leaf node sum 32.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/08/10/tree2.jpg" style="width: 224px; height: 145px;" />
<pre>
<strong>Input:</strong> arr = [4,11]
<strong>Output:</strong> 44
</pre>

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

<ul>
	<li><code>2 &lt;= arr.length &lt;= 40</code></li>
	<li><code>1 &lt;= arr[i] &lt;= 15</code></li>
	<li>It is guaranteed that the answer fits into a <strong>32-bit</strong> signed integer (i.e., it is less than 2<sup>31</sup>).</li>
</ul>

class Solution {
public:
    int dp[41][41];
    int solve(int left, int right, map<pair<int, int>, int> &maxi, vector<int>& arr){
        if(left==right)return 0;
        int &res= dp[left][right];
        if(res!=-1)return res;
        res = INT_MAX;
        for(int k=left; k<right; k++){
            res = min(res, maxi[{left,k}]*maxi[{k+1, right}]+solve(left, k, maxi, arr)+solve(k+1, right, maxi, arr));
        }
        return res;
    }
    int mctFromLeafValues(vector<int>& arr) {
        map<pair<int, int>, int> maxi;

        int n = arr.size();
        memset(dp, -1, sizeof(dp));

        // precompute the maxi upto this element
        for(int i=0; i<n; i++){
            maxi[{i,i}]=arr[i];
            for(int j=i+1; j<n; j++){
                maxi[{i,j}]=max(arr[j], maxi[{i, j-1}]);
            }
        }


        // for(int i=0; i<arr.size(); i++){
        //     for(int j=0; j<arr.size(); j++){
        //         cout << "for range" << i << " " << j << "max is" << maxi[{i, j}];
        //         cout << endl;
        //     }
        // }

        return solve(0, n-1, maxi, arr);


    }
};
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