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Select Cells in Grid With Maximum Score

Difficulty: Hard

Problem Description

You are given a 2D matrix grid consisting of positive integers. You have to select one or more cells from the matrix such that the following conditions are satisfied:

  • No two selected cells are in the same row of the matrix.
  • The values in the set of selected cells are unique.

Your score will be the sum of the values of the selected cells. Return the maximum score you can achieve.

Key Insights

  • The constraints allow for a maximum of 10 rows and 10 columns, making it feasible to explore combinations of cells.
  • The uniqueness constraint means that we must keep track of the selected values to avoid duplicates.
  • We can utilize a backtracking approach to explore all possible combinations of selecting cells from different rows.

Space and Time Complexity

Time Complexity: O(2^(m*n)), where m is the number of rows and n is the number of columns, since we may explore all combinations of rows and columns. Space Complexity: O(n), where n is the maximum number of unique values we can store in our selection.

Solution

To solve the problem, we can use a backtracking approach:

  1. Iterate through each row and for each cell in the row, check if the value is already selected.
  2. If the value is unique, select it and move to the next row recursively.
  3. Keep track of the current score and update the maximum score whenever we reach the end of the rows.
  4. Backtrack by deselecting the value and trying the next cell in the current row.

Data Structures Used:

  • A set to track the selected values.
  • A variable to keep track of the current score.

Code Solutions

def maxScore(grid):
    def backtrack(row, selected_values):
        if row == len(grid):
            return sum(selected_values)

        max_score = 0
        for col in range(len(grid[row])):
            value = grid[row][col]
            if value not in selected_values:
                selected_values.add(value)
                max_score = max(max_score, backtrack(row + 1, selected_values))
                selected_values.remove(value)

        # Also consider not selecting anything in this row
        max_score = max(max_score, backtrack(row + 1, selected_values))
        return max_score

    return backtrack(0, set())
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