<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 binary_closest(self, arr: List[int], target: int) -> int:
        left = 0
        right = len(arr) - 1
        closest = arr[0]
        while left <= right:
            mid = (left + right) // 2
            if abs(arr[mid] - target) < abs(closest - target):
                closest = arr[mid]
            if arr[mid] < target:
                left = mid + 1
            elif arr[mid] > target:
                right = mid - 1
            else:
                return arr[mid]
        return closest

    def minimumDifference(self, nums: List[int]) -> int:
        n = len(nums) // 2  # Size of each partition
        total_sum = sum(nums)  # Total sum of all elements
        half_sum = total_sum // 2  # Half of the total sum, target for balanced partition
        #print(f"half_sum = {half_sum}:")
        # Initial difference using naive split
        left = sum(nums[:n])
        right = sum(nums[n:])
        min_difference = abs(left - right)
        #print(f"original min diff = {min_difference}:")
        
        # Iterate over all possible sizes of subsets for the first half (except empty and full)
        for subset_size in range(1, n):
            left_sums = [sum(comb) for comb in combinations(nums[:n], subset_size)]
            right_sums = [sum(comb) for comb in combinations(nums[n:], n - subset_size)]
            right_sums.sort()
            
            #print(f"subset_size = {subset_size}:")
            #print(f"subset_size = {subset_size}, left_sums = {left_sums}, right_sums = {right_sums}")
            for left_sum in left_sums:
                target = half_sum - left_sum
                closest_right_sum = self.binary_closest(right_sums, target)
                min_difference = min(abs(total_sum - 2 * (left_sum + closest_right_sum)), min_difference)
                #print(f"min_difference = {min_difference}")
        
        return min_difference

class Solution_exceed_time_lmt:
    def minimumDifference(self, nums: List[int]) -> int:
        n = len(nums) // 2  # Size of each partition
        total_sum = sum(nums)  # Total sum of all elements
        half_sum = total_sum // 2  # Half of the total sum, target for balanced partition
        #print(f"half_sum = {half_sum}:")
        # Initial difference using naive split
        left = sum(nums[:n])
        right = sum(nums[n:])
        min_difference = abs(left - right)
        #print(f"original min diff = {min_difference}:")
        
        # Iterate over all possible sizes of subsets for the first half (except empty and full)
        for subset_size in range(1, n):
            left_sums = [sum(comb) for comb in combinations(nums[:n], subset_size)]
            right_sums = [sum(comb) for comb in combinations(nums[n:], n - subset_size)]
            
            #print(f"subset_size = {subset_size}:")
            #print(f"subset_size = {subset_size}, left_sums = {left_sums}, right_sums = {right_sums}")
            for left_sum in left_sums:
                target = half_sum - left_sum
                closest_right_sum = min(right_sums, key=lambda x: abs(x-target))
                min_difference = min(abs(total_sum - 2 * (left_sum + closest_right_sum)), min_difference)
                #print(f"min_difference = {min_difference}")
        
        return min_difference
4333