<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 getMax(std::unordered_map<int, int>& diff, int index, int val)
	{
		if (!diff.count(index))
		{
			return val;
		}
		return std::max(val, diff[index]);
	}

	int tallestBillboard(std::vector<int>& rods)
	{
		std::unordered_map<int, int> diffs;
		std::unordered_map<int, int> newDiffs;
		diffs[0] = 0;
		for (auto rod : rods)
		{
			for (auto& [diff, higher] : diffs)
			{
				// add to higher
				newDiffs[diff + rod] = getMax(newDiffs, diff + rod, higher + rod);
				// not add
				newDiffs[diff] = getMax(newDiffs, diff, higher);
				// add to shorter
				int newDiff = std::abs(diff - rod);
				newDiffs[newDiff] = getMax(newDiffs, newDiff, std::max(higher, higher - diff + rod));
			}
			std::swap(diffs, newDiffs);
			newDiffs.clear();
		}
		return diffs[0];
	}
};
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