Maximum Number of Points with Cost

Problem Description

You are given an m x n integer matrix points. Starting with 0 points, you want to maximize the number of points you can get from the matrix by picking one cell in each row. Picking the cell at coordinates (r, c) will add points[r][c] to your score. However, you will lose points if you pick a cell too far from the cell that you picked in the previous row. For every two adjacent rows r and r + 1, picking cells at coordinates (r, c1) and (r + 1, c2) will subtract abs(c1 - c2) from your score. Return the maximum number of points you can achieve.

Key Insights

• The problem involves maximizing points collected from a matrix while considering penalties for choosing cells from adjacent rows that are far apart.
• A dynamic programming approach is suitable since we need to consider decisions made in previous rows when deciding on the current row.
• The main challenge is efficiently calculating the best possible score at each step while keeping track of penalties incurred from column differences.

Space and Time Complexity

Time Complexity: O(m * n)
Space Complexity: O(n)

Solution

The solution employs a dynamic programming approach where we maintain an array to track the maximum points achievable for each column in the current row. For each cell in the current row, we compute the potential maximum score by considering the maximum scores from the previous row, adjusted for the penalties associated with column differences.

1. Initialize a current array to store the maximum points for the current row.
2. Iterate through each row of the matrix:
   • For each cell in the row, compute the maximum points by considering all cells in the previous row.
   • Keep track of the maximum points from the previous row and calculate the potential scores based on column distances.
3. Return the maximum value from the final row.

Code Solutions