2067 Maximum Number Of Points With Cost

Maximum Number of Points with Cost

You are given an m x n integer matrix points (0-indexed). Starting with 0 points, you want to maximize the number of points you can get from the matrix.

To gain points, you must pick 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 (where 0 <= r < m - 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.

abs(x) is defined as:

x for x >= 0.
-x for x < 0.

 

Example 1:** **

**Input:** points = [[1,2,3],[1,5,1],[3,1,1]]
**Output:** 9
**Explanation:**
The blue cells denote the optimal cells to pick, which have coordinates (0, 2), (1, 1), and (2, 0).
You add 3 + 5 + 3 = 11 to your score.
However, you must subtract abs(2 - 1) + abs(1 - 0) = 2 from your score.
Your final score is 11 - 2 = 9.

Example 2:

**Input:** points = [[1,5],[2,3],[4,2]]
**Output:** 11
**Explanation:**
The blue cells denote the optimal cells to pick, which have coordinates (0, 1), (1, 1), and (2, 0).
You add 5 + 3 + 4 = 12 to your score.
However, you must subtract abs(1 - 1) + abs(1 - 0) = 1 from your score.
Your final score is 12 - 1 = 11.

 

Constraints:

m == points.length
n == points[r].length
1 <= m, n <= 105
1 <= m * n <= 105
0 <= points[r][c] <= 105

class Solution:
    def __init__(self):
        self.cache = {}
    def maxPoints(self, points: List[List[int]]) -> int:
        n, m = len(points), len(points[0])
        cache = {}

        def precompute_max_arrays(dp_row):
            max_left = [0] * m
            max_right = [0] * m

            max_left[0] = dp_row[0]
            for c in range(1, m):
                max_left[c] = max(max_left[c - 1], dp_row[c] + c)

            max_right[-1] = dp_row[-1] - (m - 1)
            for c in range(m - 2, -1, -1):
                max_right[c] = max(max_right[c + 1], dp_row[c] - c)

            return max_left, max_right

        def dp(r):
            if r in cache:
                return cache[r]
            if r == n - 1:
                return points[r]

            # Get DP row for the next row
            dp_next_row = dp(r + 1)
            max_left, max_right = precompute_max_arrays(dp_next_row)

            current_dp = [0] * m
            for c in range(m):
                current_dp[c] = points[r][c] + max(max_left[c] - c, max_right[c] + c)

            cache[r] = current_dp
            return current_dp

        # Compute result by taking the maximum of the first row
        return max(dp(0))