Count Submatrices with Top-Left Element and Sum Less Than k - Solution & Explanation

This approach leverages the cumulative prefix sum technique to quickly calculate the sum of submatrices starting from the top-left to any (i, j). The idea is to iterate through all possible submatrices starting at the top-left, using a prefix sum matrix to obtain the total sum efficiently. If the sum is less than or equal to k, we count it.

Initially, we calculate the prefix sum matrix prefix using a nested loop to preprocess submatrix sums efficiently. Here, prefix[i][j] holds the sum of elements from top-left (0,0) to the current cell (i, j). Further we iterate through each cell, (r, c), forming submatrices starting from (0,0) until that cell and if their sum is less than or equal to k, we increment the counter.

Instead of a full 2D prefix sum array, this approach uses a 1D prefix sum and iterative row expansion to calculate submatrix sums and maintain under k leveraging a sliding window or two-pointer technique.

This implementation calculates prefix sums on the fly as rows are included in the submatrices incrementally and checks if the sum remains within the desired limit k. The reset prefixRow array aids in accumulating these values efficiently.