Notice that we could do  +  and  -  subtraction interval-wise. But Fenwick tree could only update per range and query per element or vice versa. If, for each element, say, ft(x, y) hold the sum of prefix sum of the matrix (1, 1, x, y) (ie. prefix sum of prefix sum of (x, y)), then for each query, ft(x1, y1) - ft(x0 - 1, y1) - ft(x1, y0 - 1) + ft(x0 - 1, y0 - 1) would be the result. As the difference between prefix-sum is the sum of the corresponding interval. And it's easy to xor a v to elements in submatrix (x0, y0, x1, y1). Consider 1 dimension version of this problem. For a sequence a, the prefix sum of x is and the prefix sum of prefix sum of x is . Hence, all we have to do is to maintain an extra sequnce i * ai. Then expand method above to arbitrary dimension.