<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:
        
        # neetcode-x
        
        n = len(nums) // 2
        total_sum = sum(nums)

        # Function to get all possible sums of subsets of length k
        def get_sums(arr):
            n = len(arr)
            sums = [[] for _ in range(n + 1)]
            for size in range(n + 1):
                for comb in combinations(arr, size):
                    sums[size].append(sum(comb))
            return sums

        # Split nums into two halves
        left, right = nums[:n], nums[n:]

        # Get all subset sums for both halves
        left_sums = get_sums(left)
        right_sums = get_sums(right)

        # Sort all the sums in right_sums for efficient binary search later
        for i in range(n + 1):
            right_sums[i].sort()

        result = float('inf')

        # Iterate over all subsets of left side
        for i in range(n + 1):
            for left_sum in left_sums[i]:
                # We need to find the closest subset sum in the right half such that
                # the total sum of left and right is closest to total_sum // 2
                target = (total_sum // 2) - left_sum
                right_list = right_sums[n - i]

                # Binary search for the closest sum in right_list
                lo, hi = 0, len(right_list) - 1
                while lo <= hi:
                    mid = (lo + hi) // 2
                    right_sum = right_list[mid]
                    current_sum = left_sum + right_sum
                    result = min(result, abs(total_sum - 2 * current_sum))
                    if right_sum < target:
                        lo = mid + 1
                    else:
                        hi = mid - 1

        return result
5650