<p>You are installing a billboard and want it to have the largest height. The billboard will have two steel supports, one on each side. Each steel support must be an equal height.</p>

<p>You are given a collection of <code>rods</code> that can be welded together. For example, if you have rods of lengths <code>1</code>, <code>2</code>, and <code>3</code>, you can weld them together to make a support of length <code>6</code>.</p>

<p>Return <em>the largest possible height of your billboard installation</em>. If you cannot support the billboard, return <code>0</code>.</p>

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

<pre>
<strong>Input:</strong> rods = [1,2,3,6]
<strong>Output:</strong> 6
<strong>Explanation:</strong> We have two disjoint subsets {1,2,3} and {6}, which have the same sum = 6.
</pre>

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

<pre>
<strong>Input:</strong> rods = [1,2,3,4,5,6]
<strong>Output:</strong> 10
<strong>Explanation:</strong> We have two disjoint subsets {2,3,5} and {4,6}, which have the same sum = 10.
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> rods = [1,2]
<strong>Output:</strong> 0
<strong>Explanation:</strong> The billboard cannot be supported, so we return 0.
</pre>

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

<ul>
	<li><code>1 &lt;= rods.length &lt;= 20</code></li>
	<li><code>1 &lt;= rods[i] &lt;= 1000</code></li>
	<li><code>sum(rods[i]) &lt;= 5000</code></li>
</ul>

class Solution {
public:
    vector<vector<int>> dif2hi;//dif2hi[i][dif]=hi
    int n;
    int calHi(vector<int>& rods, int i, int dif) {
        if (i == n) {
            if (dif == 0) return 0;
            else return INT_MIN;
        }
        
        if (dif2hi[i][dif]!=-1)            
            return dif2hi[i][dif];
       
        int hi = calHi(rods, i+1, dif);
        int rod=rods[i];
        //Place the rod on the tailer side
        hi = max(hi, calHi(rods, i+1, dif+rod) + rod);

        //Place the rod on the shorter side
        hi = max(hi, calHi(rods, i+1, abs(dif-rod)) + max(0, rod-dif));  
        return dif2hi[i][dif] = hi;
    }

    int tallestBillboard(vector<int>& rods) { 
        n=rods.size(); 
        //Use Constraint sum(rods[i]) <= 5000
        dif2hi.assign( rods.size(), vector(5001, -1));   
        return calHi(rods, 0, 0);
    }  
};
43284