compute any sum@subMatrix by O(1) hashmap lookup

suppose origin cell is at northwest [1,1] not [0,0]

Pre-processing — We can incrementally compute each the rooted-submatrix SumToOrigin[r,c] in linear time. Save them in a s2o hashmap (shadow matrix is actually better)

Now Consider the submatrix identified by [2,2] and [3,4] enclosing 6 cells. This submatrix has sum =


12 cells + 1 cell  – 4 cells – 3 cells

So any submatrix sum can be computed in O(1). To go through all NNMM of them requires O(NNMM) where N and M are the board dimensions

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