如何有效地查找网格中某个范围内的元素总数？

Question

Given a n by n grid of positive numbers, what is the best complexity that can be achieved for obtaining the sum of elements in a range described by the corners of rectangle of area considered given as (x1,y1) and (x2,y2) ? There will be q such queries.

PS: Considering the naive solution, the complexity is O(q*n^2) .

Answer 1

Suppose you sum along the rows as in m69's comment to produce a matrix where every element is the sum of the corresponding element in the original matrix and all elements to the left of it. Then you do the same thing summing down the columns of that matrix of sums and you get a matrix where every element is the sum of a rectangular sub-array of elements to its left and above it.

Now take four points in this array of sums:

AB

CD

The value D - B - C + A contains the sum of a rectangular region with one corner at D and the other corners just to D's side of A, B, and C, as you can see by working out how many times points in the various regions are added and subtracted. So after O(n^2) pre-processing you can answer queries in time O(1)