<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:
        n = len(nums) // 2
        total_sum = sum(nums)
        target = total_sum // 2
        
        def generate_sums(arr):
            sums_by_len = [[] for _ in range(len(arr) + 1)]
            
            def helper(i, current_sum, subset_size):
                if i == len(arr):
                    sums_by_len[subset_size].append(current_sum)
                    return
                
                # Include the current element
                helper(i + 1, current_sum + arr[i], subset_size + 1)
                
                # Exclude the current element
                helper(i + 1, current_sum, subset_size)
            
            helper(0, 0, 0)
            return sums_by_len
        
        def binary_search_closest(arr, value):
            low, high = 0, len(arr) - 1
            best_diff = float('inf')
            best_sum = None
            
            while low <= high:
                mid = (low + high) // 2
                mid_value = arr[mid]
                current_diff = abs(value - mid_value)
                
                if current_diff < best_diff:
                    best_diff = current_diff
                    best_sum = mid_value
                
                if mid_value < value:
                    low = mid + 1
                else:
                    high = mid - 1
            
            return best_sum
        
        left_sums = generate_sums(nums[:n])
        right_sums = generate_sums(nums[n:])
        
        min_diff = float('inf')
        
        for i in range(n + 1):
            left_sums[i].sort()
            right_sums[i].sort()
            for left_sum in left_sums[i]:
                closest_sum = binary_search_closest(right_sums[n-i], target - left_sum)
                current_diff = abs(total_sum - 2 * (left_sum + closest_sum))
                min_diff = min(min_diff, current_diff)
        
        return min_diff

        
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