<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:
        
        # store the mxn size of the matrix
        m = len(mat)
        n = len(mat[0])
        
        dp = defaultdict(defaultdict)
        
        # Sorting each row of the array for more efficient pruning
        # Note:this purely based on the observation on problem constraints (although interesting :))
        for i in range(m):
            mat[i] = sorted(mat[i])
        
        # returns minimum absolute starting from from row i to n-1 for the target
        # globalMin = float("inf")
        def findMinAbsDiff(i,prevSum):
            # nonlocal globalMin
            if i == m:
                # globalMin = min(globalMin, abs(prevSum-target))
                return abs(prevSum-target)
            
            # pruning step 1
            # because the array is increasing & prevSum & target will always be positive
            # if prevSum-target > globalMin:
            #     return float("inf")
            
            
            if (i in dp) and (prevSum in dp[i]):
                return dp[i][prevSum]
            
            minDiff = float("inf")
            # for each candidate select that and backtrack
            for j in range(n):
                diff = findMinAbsDiff(i+1, prevSum+mat[i][j])
                # pruning step 2 - break if we found minDiff 0 --> VERY CRTICIAL
                if diff == 0:
                    minDiff = 0
                    break
                minDiff = min(minDiff, diff)
            
            dp[i][prevSum] = minDiff
            return minDiff
        
        return findMinAbsDiff(0, 0)
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