Maximum Difference Score in a Grid

Problem Description

You are given an m x n matrix grid consisting of positive integers. You can move from a cell in the matrix to any other cell that is either to the bottom or to the right (not necessarily adjacent). The score of a move from a cell with the value c1 to a cell with the value c2 is c2 - c1. You can start at any cell, and you have to make at least one move. Return the maximum total score you can achieve.

Key Insights

• The problem allows movement to any cell below or to the right, providing flexibility in maximizing score.
• The score for a move is always calculated as the difference between the destination cell value and the source cell value.
• The maximum score can be derived by using dynamic programming to track the best possible scores from each cell.

Space and Time Complexity

Time Complexity: O(m * n)
Space Complexity: O(1) (if we modify the grid in place) or O(m * n) for a separate DP array.

Solution

To solve the problem, we can use a dynamic programming approach. We will iterate through the grid while maintaining the minimum value encountered up to that point for each cell. For each cell, the score can be calculated by subtracting the minimum value from the current cell's value.

1. Initialize a variable to keep track of the maximum score.
2. Iterate through each cell in the grid.
3. For each cell, calculate the score based on the minimum value encountered so far.
4. Update the maximum score if the current score is greater.
5. Finally, return the maximum score.

Code Solutions