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Minimize the Difference Between Target and Chosen Elements

Number: 2108

Difficulty: Medium

Paid? No

Companies: Deutsche Bank, Sprinklr, Amazon

Problem Description

Given an m x n integer matrix, choose one element from each row so that the absolute difference between the target and the sum of the chosen elements is minimized. Return this minimum absolute difference.

Key Insights

  • We need to select exactly one element per row.
  • The problem can be solved using dynamic programming by tracking all possible sums after processing each row.
  • Instead of exploring all combinations explicitly, use a set (or bitset) to keep the possible sums from one row to the next.
  • As the matrix dimensions are moderate, we can afford a DP approach though careful pruning (if necessary) can optimize performance.

Space and Time Complexity

Time Complexity: O(m * n * S) in the worst-case where S is the range of possible sums accumulated.
Space Complexity: O(S) for storing the possible sums.

Solution

We use dynamic programming by processing the matrix row by row. Start with the set of possible sums from the first row. Then, for each subsequent row, for every possible sum so far, add each element in the new row to form new sums. Once all rows are processed, iterate through the final set to determine the minimum absolute difference from the target. Key data structures include sets for storing possible sums. This approach efficiently narrows down the sum combinations without examining every potential combination explicitly.

Code Solutions

# Python solution
def minimizeTheDifference(mat, target):
    # Initialize the set with the elements from the first row.
    possible_sums = set(mat[0])
    
    # Process each subsequent row.
    for row in mat[1:]:
        next_possible_sums = set()
        # For each previous sum, add every element from the current row.
        for current_sum in possible_sums:
            for value in row:
                next_possible_sums.add(current_sum + value)
        possible_sums = next_possible_sums  # Update possible sums

    # Find the minimum absolute difference among all possible sums.
    return min(abs(s - target) for s in possible_sums)

# Example usage:
mat = [[1,2,3],[4,5,6],[7,8,9]]
target = 13
print(minimizeTheDifference(mat, target))  # Expected output: 0
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