<p>You are given an integer array <code>nums</code> and an integer <code>goal</code>.</p>

<p>You want to choose a subsequence of <code>nums</code> such that the sum of its elements is the closest possible to <code>goal</code>. That is, if the sum of the subsequence&#39;s elements is <code>sum</code>, then you want to <strong>minimize the absolute difference</strong> <code>abs(sum - goal)</code>.</p>

<p>Return <em>the <strong>minimum</strong> possible value of</em> <code>abs(sum - goal)</code>.</p>

<p>Note that a subsequence of an array is an array formed by removing some elements <strong>(possibly all or none)</strong> of the original array.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> nums = [5,-7,3,5], goal = 6
<strong>Output:</strong> 0
<strong>Explanation:</strong> Choose the whole array as a subsequence, with a sum of 6.
This is equal to the goal, so the absolute difference is 0.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> nums = [7,-9,15,-2], goal = -5
<strong>Output:</strong> 1
<strong>Explanation:</strong> Choose the subsequence [7,-9,-2], with a sum of -4.
The absolute difference is abs(-4 - (-5)) = abs(1) = 1, which is the minimum.
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> nums = [1,2,3], goal = -7
<strong>Output:</strong> 7
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= nums.length &lt;= 40</code></li>
	<li><code>-10<sup>7</sup> &lt;= nums[i] &lt;= 10<sup>7</sup></code></li>
	<li><code>-10<sup>9</sup> &lt;= goal &lt;= 10<sup>9</sup></code></li>
</ul>

class Solution:
    def minAbsDifference(self, nums: List[int], goal: int) -> int:
        def subsum(start, end, i, current_sum):
            if i == end + 1:
                return [current_sum] 

            include = subsum(start, end, i + 1, current_sum + nums[i])
            exclude = subsum(start, end, i + 1, current_sum)

            return include + exclude
        def lb(target):
            left, right = 0, len(sp) - 1
            while left <= right:
                mid = (left + right) // 2
                if sp[mid] == target:
                    return mid 
                elif sp[mid]>target:
                    right = mid - 1
                else:
                    left=mid+1
            return left 
        mid=(0+len(nums)-1)//2
        fp=subsum(0,mid,0,0)
        sp=subsum(mid+1,len(nums)-1,mid+1,0)
        sp.sort()
        res=float('inf')
        for i in fp:
            lbo = lb(goal - i)
            if lbo > 0:
                res = min(res, abs(goal - sp[lbo - 1] - i))
            if lbo < len(sp):
                res = min(res, abs(goal - sp[lbo] - i))



        return res
            

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