<p>You are given an integer array <code>nums</code> of <code>2 * n</code> integers. You need to partition <code>nums</code> into <strong>two</strong> arrays of length <code>n</code> to <strong>minimize the absolute difference</strong> of the <strong>sums</strong> of the arrays. To partition <code>nums</code>, put each element of <code>nums</code> into <strong>one</strong> of the two arrays.</p>

<p>Return <em>the <strong>minimum</strong> possible absolute difference</em>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="example-1" src="https://assets.leetcode.com/uploads/2021/10/02/ex1.png" style="width: 240px; height: 106px;" />
<pre>
<strong>Input:</strong> nums = [3,9,7,3]
<strong>Output:</strong> 2
<strong>Explanation:</strong> One optimal partition is: [3,9] and [7,3].
The absolute difference between the sums of the arrays is abs((3 + 9) - (7 + 3)) = 2.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> nums = [-36,36]
<strong>Output:</strong> 72
<strong>Explanation:</strong> One optimal partition is: [-36] and [36].
The absolute difference between the sums of the arrays is abs((-36) - (36)) = 72.
</pre>

<p><strong class="example">Example 3:</strong></p>
<img alt="example-3" src="https://assets.leetcode.com/uploads/2021/10/02/ex3.png" style="width: 316px; height: 106px;" />
<pre>
<strong>Input:</strong> nums = [2,-1,0,4,-2,-9]
<strong>Output:</strong> 0
<strong>Explanation:</strong> One optimal partition is: [2,4,-9] and [-1,0,-2].
The absolute difference between the sums of the arrays is abs((2 + 4 + -9) - (-1 + 0 + -2)) = 0.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= n &lt;= 15</code></li>
	<li><code>nums.length == 2 * n</code></li>
	<li><code>-10<sup>7</sup> &lt;= nums[i] &lt;= 10<sup>7</sup></code></li>
</ul>


class Solution:
    def minimumDifference(self, nums: List[int]) -> int:
        sumi = sum(nums)
        n = len(nums)
        half_length = n // 2
        lefti = defaultdict(list)
        righti = defaultdict(list)
        
        # Generate all possible sums for the left and right halves
        for i in range(1 << half_length):
            sz = 0
            l_sum = 0
            r_sum = 0
            for j in range(half_length):
                if i & (1 << j):
                    sz += 1
                    l_sum += nums[j]
                    r_sum += nums[j + half_length]
            lefti[sz].append(l_sum)
            righti[sz].append(r_sum)
        
        # Sort the sums in the right half for efficient binary search
        for i in range(half_length + 1):
            righti[i].sort()
        
        # Initialize the result with a large number
        res = float('inf')
        
        # Iterate over each subset size
        for i in range(half_length + 1):
            for l_sum in lefti[i]:
                r_sumi = (sumi - 2 * l_sum) // 2
                r_index = half_length - i
                index = bisect_left(righti[r_index], r_sumi)
                
                # Check if a valid index was found and update the result
                if index < len(righti[r_index]):
                    res = min(res, abs(sumi - 2 * (l_sum + righti[r_index][index])))
                if index > 0:
                    res = min(res, abs(sumi - 2 * (l_sum + righti[r_index][index - 1])))
        
        return res

3821