<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:
        n = len(rods)
        dp = {}
        def helper(idx,diff):
            if idx == n:
                return (0,0,diff==0)
            
            if (idx,diff) in dp:
                return dp[(idx,diff)]
            
            if (idx,-diff) in dp:
                return (dp[(idx,-diff)][1],dp[(idx,-diff)][0],dp[(idx,-diff)][2])
            
            ans = (0,0)
            o1_h1,o1_h2,o1 = helper(idx+1,diff)
            if o1:
                ans = max(ans,(o1_h1,o1_h2))
            o2_h1,o2_h2,o2 = helper(idx+1,diff+rods[idx])
            if o2:
                ans = max(ans,(rods[idx]+ o2_h1, o2_h2))
            o3_h1,o3_h2,o3 = helper(idx+1,diff-rods[idx]) 
            if o3:
                ans = max(ans,(o3_h1,rods[idx]+o3_h2))
            if o1 or o2 or o3:
                dp[(idx,diff)] = (ans[0],ans[1],True)
            else:
                dp[(idx,diff)] = (0,0,False)

            return dp[(idx,diff)]
        
        return helper(0,0)[0]


            

        
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