<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:
    map<pair<int, int>, long long> mp;
    long long helper(vector <int> &rods, int i, int f) {
        
        if(i >= rods.size()) {
            
            if(f == 0) {
                
                return 0;
            }
            return INT_MIN;
        }
        
        if(mp.find({i, f}) != mp.end()) {
            return mp[{i, f}];
        }
      
     //dont include any
      long a =   helper(rods, i + 1, f);
        //include into first subset
      long b = rods[i] +  helper(rods, i + 1, f + rods[i]);
        //include into second subset
      long c = rods[i] + helper(rods, i + 1, f - rods[i]);
        
      return mp[{i, f}] =  max({a, b, c});
    }
    int tallestBillboard(vector<int>& rods) {
       return helper(rods, 0, 0)/2; 
    }
};
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