I knew prefix sums for 1D arrays but extending it to 2D took me a minute to work through.
Explain that you can preprocess the matrix by building a 2D prefix sum array where each cell (i, j) stores the sum of all elements in the sub-matrix from (0,0) to (i,j). Then, any sub-matrix sum can be computed in O(1) using the inclusion-exclusion principle: sum = prefix[r2][c2] - prefix[r1-1][c2] - prefix[r2][c1-1] + prefix[r1-1][c1-1].
Pro tip: Mention edge cases such as empty sub-matrix or indices out of bounds, and clarify that the preprocessing takes O(m*n) time and space, which is optimal for this problem.
Confirm that the matrix is static (no updates) and that multiple queries will be made. Ask about the range of indices and whether the sub-matrix is inclusive of both corners.
Create a 2D array 'prefix' of size (m+1) x (n+1) to handle boundaries easily. prefix[i][j] represents the sum of the sub-matrix from (0,0) to (i-1,j-1) in the original matrix.
Fill the prefix array using the recurrence: prefix[i][j] = matrix[i-1][j-1] + prefix[i-1][j] + prefix[i][j-1] - prefix[i-1][j-1].
For a query (r1, c1, r2, c2), compute the sum as prefix[r2+1][c2+1] - prefix[r1][c2+1] - prefix[r2+1][c1] + prefix[r1][c1].
State that preprocessing takes O(m*n) time and space, and each query takes O(1) time. This is optimal for static matrices.
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