<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 math import *

class Solution:
    def allposible(self, arr):
        n = len(arr)
        def sum_len(st):
            s = 0
            l = 0
            for i, e in enumerate(st):
                if e == '1':
                    s += arr[i]
                    l += 1
            return (s, l)

        ans = []
        for i in range(2**n):
            take = bin(i)[2:]
            take = '0' * (n - len(take)) + take
            ans.append(sum_len(take))
        
        return ans
    def min_possible(self, a1, a2, target):
        a1.sort()
        a2.sort()
        n1 = len(a1)
        n2 = len(a2)
        pt1 = 0
        pt2 = n2 - 1
        best_diff = inf
        while pt1 < n1 and pt2 > -1:
            best_diff = min(best_diff, abs(a1[pt1] + a2[pt2] - target))
            if a1[pt1] + a2[pt2] < target:
                pt1 += 1
            elif a1[pt1] + a2[pt2] == target:
                return 0
            else:
                pt2 -= 1
        
        return best_diff

    def minimumDifference(self, nums: List[int]) -> int:
        n = len(nums)
        TS = sum(nums)
        a1 = self.allposible(nums[:n//2])
        a2 = self.allposible(nums[n//2:])    

        hm1 = {}
        for i in a1:
            if i[1] in hm1:
                hm1[i[1]].append(i[0])
            else:
                hm1[i[1]] = [i[0]]
        for k in hm1:
            hm1[k] = [2 * i for i in hm1[k]]
            hm1[k].sort()
        hm2 = {}
        for i in a2:
            if i[1] in hm2:
                hm2[i[1]].append(i[0])
            else:
                hm2[i[1]] = [i[0]]
        for k in hm2:
            hm2[k] = [2 * i for i in hm2[k]]
            hm2[k].sort()
        
        ans = inf
        for k in hm1:
            ans = min(ans, self.min_possible(hm1[k], hm2[n//2 - k], TS))
        
        return ans


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