You are given a 0-indexed integer matrix grid and an integer k.
Return the number of submatrices that contain the top-left element of thegrid, and have a sum less than or equal to k.
Example 1:
Input: grid = [[7,6,3],[6,6,1]], k = 18
Output: 4
Explanation: There are only 4 submatrices, shown in the image above, that contain the top-left element of grid, and have a sum less than or equal to 18.
Example 2:
Input: grid = [[7,2,9],[1,5,0],[2,6,6]], k = 20
Output: 6
Explanation: There are only 6 submatrices, shown in the image above, that contain the top-left element of grid, and have a sum less than or equal to 20.
Constraints:
m == grid.length
n == grid[i].length
1 <= n, m <= 1000
0 <= grid[i][j] <= 1000
1 <= k <= 109
Solutions
Solution 1: Two-Dimensional Prefix Sum
The problem is actually asking for the number of prefix submatrices in a two-dimensional matrix whose sum is less than or equal to $k$.
The calculation formula for the two-dimensional prefix sum is:
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.