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Minimize the Difference Between Target and Chosen Elements

Difficulty: Medium

Problem Description

You are given an m x n integer matrix mat and an integer target. Choose one integer from each row in the matrix such that the absolute difference between target and the sum of the chosen elements is minimized. Return the minimum absolute difference.

Key Insights

  • The problem can be approached using dynamic programming to explore combinations of sums from selected elements across rows.
  • We need to keep track of possible sums and the minimum difference from the target.
  • The constraints allow for a manageable search space, making a dynamic programming solution feasible.

Space and Time Complexity

Time Complexity: O(m * n * k), where k is the range of possible sums based on the values in the matrix.
Space Complexity: O(k), where k is the range of possible sums we store in the dynamic programming array.

Solution

The solution involves using a dynamic programming approach. We maintain a set of possible sums as we iterate through each row of the matrix. For each row, we generate new possible sums by adding each element of the row to each of the existing sums. After processing all rows, we find the minimum absolute difference between the target and the sums we have collected.

Code Solutions

def minimizeDifference(mat, target):
    possible_sums = {0}
    
    for row in mat:
        new_sums = set()
        for num in row:
            for s in possible_sums:
                new_sums.add(s + num)
        possible_sums = new_sums

    # Find the minimum absolute difference
    min_diff = float('inf')
    for s in possible_sums:
        min_diff = min(min_diff, abs(s - target))
    
    return min_diff
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