<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:
    def tallestBillboard(self, rods: List[int]) -> int:
        # Initialize the dp dictionary with a base case of 0 difference and 0 height
        dp = {0: 0}
        
        for rod in rods:
            # Create a copy of the current dp state to avoid overwriting values we still need
            new_dp = dp.copy()
            
            # Iterate through all current differences and their corresponding heights
            for diff, height in dp.items():
                # Case 1: Add the current rod to the taller support
                new_dp[diff + rod] = max(new_dp.get(diff + rod, 0), height)
                
                # Case 2: Add the current rod to the shorter support
                if diff >= rod:
                    new_dp[diff - rod] = max(new_dp.get(diff - rod, 0), height + rod)
                else:
                    new_dp[rod - diff] = max(new_dp.get(rod - diff, 0), height + diff)
            
            # Update the dp dictionary with the new state
            dp = new_dp
        
        # The answer is the height when the difference is 0
        return dp[0]

# Example usage:
solution = Solution()
rods = [1, 2, 3, 6]
print(solution.tallestBillboard(rods))  # Output: 6
16800