<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:
    int tallestBillboard(vector<int>& rods) {

        using ll = long long;

        unordered_map<ll, int> memo;
        
        std::function<int(int, int)> search = [&](int i, int diff) {
            
            if (i == rods.size()) {
                return diff == 0 ? 0 : INT_MIN;
            }
            
            ll key = i;
            key = (key<<32) + diff;
            
            if (memo.count(key) > 0) return memo[key];
            
            auto longer = search(i+1, diff + rods[i]);
            auto shorter = search(i+1, abs(diff - rods[i])) + std::min(rods[i], diff);
            auto equal = search(i+1, diff);
            memo[key] = std::max(longer, std::max(shorter, equal));
            
            return memo[key];
        };
        
        return search(0, 0);
    }
};
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