3148. Maximum Difference Score in a Grid
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.
Example 1:
Input: grid = [[9,5,7,3],[8,9,6,1],[6,7,14,3],[2,5,3,1]]
Output: 9
Explanation: We start at the cell (0, 1)
, and we perform the following moves:
- Move from the cell (0, 1)
to (2, 1)
with a score of 7 - 5 = 2
.
- Move from the cell (2, 1)
to (2, 2)
with a score of 14 - 7 = 7
.
The total score is 2 + 7 = 9
.
Example 2:
Input: grid = [[4,3,2],[3,2,1]]
Output: -1
Explanation: We start at the cell (0, 0)
, and we perform one move: (0, 0)
to (0, 1)
. The score is 3 - 4 = -1
.
Constraints:
m == grid.length
n == grid[i].length
2 <= m, n <= 1000
4 <= m * n <= 105
1 <= grid[i][j] <= 105
Solutions
Solution 1: Dynamic Programming
According to the problem description, if the values of the cells we pass through are $c_1, c_2, \cdots, c_k$, then our score is $c_2 - c_1 + c_3 - c_2 + \cdots + c_k - c_{k-1} = c_k - c_1$. Therefore, the problem is transformed into: for each cell $(i, j)$ of the matrix, if we take it as the endpoint, what is the minimum value of the starting point.
We can use dynamic programming to solve this problem. We define $f[i][j]$ as the minimum value of the path with $(i, j)$ as the endpoint. Then we can get the state transition equation:
$$ f[i][j] = \min(f[i-1][j], f[i][j-1], grid[i][j]) $$
So the answer is the maximum value of $\textit{grid}[i][j] - \min(f[i-1][j], f[i][j-1])$.
The time complexity is $O(m \times n)$, and the space complexity is $O(m \times n)$. Where $m$ and $n$ are the number of rows and columns of the matrix, respectively.
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