<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>

from collections import defaultdict
from math import inf

class Solution:
    def minimumDifference(self, nums):
        """
        Calculate the minimum absolute difference between the sum of numbers chosen 
        from the first half and the sum of numbers chosen from the second half.
        """
        # n represents half the length of the nums array
        n = len(nums) // 2
      
        # Initialize defaultdicts to store sums for different counts of selected numbers
        first_half_sums = defaultdict(set)
        second_half_sums = defaultdict(set)
      
        # Iterate over all possible subsets of both halves
        for i in range(1 << n):
            sum_first = count_first = 0
            sum_second = count_second = 0
          
            # Calculate sum and count for the first and second halves
            for j in range(n):
                if (i & (1 << j)) != 0:
                    sum_first += nums[j]
                    count_first += 1
                    sum_second += nums[n + j]
                    count_second += 1
                else:
                    sum_first -= nums[j]
                    sum_second -= nums[n + j]
          
            # Store the sums for the counts
            first_half_sums[count_first].add(sum_first)
            second_half_sums[count_second].add(sum_second)
      
        # Initialize answer to infinity (represents a very large value)
        answer = inf
      
        # Iterate over all possible counts of elements in the subset
        for i in range(n + 1):
            sorted_first_half_sums = sorted(list(first_half_sums[i]))
            sorted_second_half_sums = sorted(list(second_half_sums[n - i]))
          
            # Find the pair with minimum absolute difference
            for a in sorted_first_half_sums:
                left, right = 0, len(sorted_second_half_sums) - 1
                target = -a
              
                # Perform binary search to find the closest sum in the second half to the target
                while left < right:
                    mid = (left + right) // 2
                    if sorted_second_half_sums[mid] >= target:
                        right = mid
                    else:
                        left = mid + 1
              
                # Update the answer with the new minimum if it's smaller
                answer = min(answer, abs(a + sorted_second_half_sums[left]))
              
                # Check the previous element for a better minimum, if possible
                if left > 0:
                    answer = min(answer, abs(a + sorted_second_half_sums[left - 1]))
      
        # Return the final answer
        return answer
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