<p>You are given an <code>m x n</code> integer matrix <code>mat</code> and an integer <code>target</code>.</p>

<p>Choose one integer from <strong>each row</strong> in the matrix such that the <strong>absolute difference</strong> between <code>target</code> and the <strong>sum</strong> of the chosen elements is <strong>minimized</strong>.</p>

<p>Return <em>the <strong>minimum absolute difference</strong></em>.</p>

<p>The <strong>absolute difference</strong> between two numbers <code>a</code> and <code>b</code> is the absolute value of <code>a - b</code>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/08/03/matrix1.png" style="width: 181px; height: 181px;" />
<pre>
<strong>Input:</strong> mat = [[1,2,3],[4,5,6],[7,8,9]], target = 13
<strong>Output:</strong> 0
<strong>Explanation:</strong> One possible choice is to:
- Choose 1 from the first row.
- Choose 5 from the second row.
- Choose 7 from the third row.
The sum of the chosen elements is 13, which equals the target, so the absolute difference is 0.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/08/03/matrix1-1.png" style="width: 61px; height: 181px;" />
<pre>
<strong>Input:</strong> mat = [[1],[2],[3]], target = 100
<strong>Output:</strong> 94
<strong>Explanation:</strong> The best possible choice is to:
- Choose 1 from the first row.
- Choose 2 from the second row.
- Choose 3 from the third row.
The sum of the chosen elements is 6, and the absolute difference is 94.
</pre>

<p><strong class="example">Example 3:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/08/03/matrix1-3.png" style="width: 301px; height: 61px;" />
<pre>
<strong>Input:</strong> mat = [[1,2,9,8,7]], target = 6
<strong>Output:</strong> 1
<strong>Explanation:</strong> The best choice is to choose 7 from the first row.
The absolute difference is 1.
</pre>

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

<ul>
	<li><code>m == mat.length</code></li>
	<li><code>n == mat[i].length</code></li>
	<li><code>1 &lt;= m, n &lt;= 70</code></li>
	<li><code>1 &lt;= mat[i][j] &lt;= 70</code></li>
	<li><code>1 &lt;= target &lt;= 800</code></li>
</ul>

class Solution:
    def minimizeTheDifference(self, mat: List[List[int]], target: int) -> int:
        sum_set = set([0])
        for row in mat:
            row = sorted(set(row))
            next_sum_set = set()
            for s in sum_set: 
                for n in row:
                    next_s = n + s
                    next_sum_set.add(next_s)
                    if next_s >= target:
                        break
            sum_set = next_sum_set
        return min(abs(n-target) for n in sum_set)


        # mat = list(map(set, mat))
        # rSet = set(mat.pop())
       
        # for row in mat: 
        #     rSet = {m+n for m, n in product(row, rSet)}
      
        # return min(abs(n-target) for n in rSet)
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