// Source : https://leetcode.com/problems/range-sum-query-2d-immutable/
// Author : Hao Chen
// Date   : 2015-11-14

/*************************************************************************************** 
 *
 * Given a 2D matrix matrix, find the sum of the elements inside the rectangle defined 
 * by its upper left corner (row1, col1) and lower right corner (row2, col2).
 * 
 * The above rectangle (with the red border) is defined by (row1, col1) = (2, 1) and 
 * (row2, col2) = (4, 3), which contains sum = 8.
 * 
 * Example:
 * 
 * Given matrix = [
 *   [3, 0, 1, 4, 2],
 *   [5, 6, 3, 2, 1],
 *   [1, 2, 0, 1, 5],
 *   [4, 1, 0, 1, 7],
 *   [1, 0, 3, 0, 5]
 * ]
 * 
 * sumRegion(2, 1, 4, 3) -> 8
 * sumRegion(1, 1, 2, 2) -> 11
 * sumRegion(1, 2, 2, 4) -> 12
 * 
 * Note:
 * 
 * You may assume that the matrix does not change.
 * There are many calls to sumRegion function.
 * You may assume that row1 ≤ row2 and col1 ≤ col2.
 * 
 ***************************************************************************************/


/*
 * 
 * Construct a 2D array `sums[row+1][col+1]`
 * 
 * (**notice**: we add additional blank row `sums[0][col+1]={0}` and blank column `sums[row+1][0]={0}` 
 * to remove the edge case checking), so, we can have the following definition 
 * 
 *     `sums[i+1][j+1]` represents the sum of area from `matrix[0][0]` to `matrix[i][j]`
 * 
 * To calculate sums, the ideas as below
 * 
 *     +-----+-+-------+     +--------+-----+     +-----+---------+     +-----+--------+
 *     |     | |       |     |        |     |     |     |         |     |     |        |
 *     |     | |       |     |        |     |     |     |         |     |     |        |
 *     +-----+-+       |     +--------+     |     |     |         |     +-----+        |
 *     |     | |       |  =  |              |  +  |     |         |  -  |              |
 *     +-----+-+       |     |              |     +-----+         |     |              |
 *     |               |     |              |     |               |     |              |
 *     |               |     |              |     |               |     |              |
 *     +---------------+     +--------------+     +---------------+     +--------------+
 *     
 *        sums[i][j]      =    sums[i-1][j]    +     sums[i][j-1]    -   sums[i-1][j-1]   +  
 * 
 *                             matrix[i-1][j-1]
 * 
 * So, we use the same idea to find the specific area's sum.
 * 
 * 
 * 
 *     +---------------+   +--------------+   +---------------+   +--------------+   +--------------+
 *     |               |   |         |    |   |   |           |   |         |    |   |   |          |
 *     |   (r1,c1)     |   |         |    |   |   |           |   |         |    |   |   |          |
 *     |   +------+    |   |         |    |   |   |           |   +---------+    |   +---+          |
 *     |   |      |    | = |         |    | - |   |           | - |      (r1,c2) | + |   (r1,c1)    |
 *     |   |      |    |   |         |    |   |   |           |   |              |   |              |
 *     |   +------+    |   +---------+    |   +---+           |   |              |   |              |
 *     |        (r2,c2)|   |       (r2,c2)|   |   (r2,c1)     |   |              |   |              |
 *     +---------------+   +--------------+   +---------------+   +--------------+   +--------------+
 * 
 * 
 */

class NumMatrix {
private:
    int row, col;
    vector<vector<int>> sums;
public:
    NumMatrix(vector<vector<int>> &matrix) {
        row = matrix.size();
        col = row>0 ? matrix[0].size() : 0;
        sums = vector<vector<int>>(row+1, vector<int>(col+1, 0));
        for(int i=1; i<=row; i++) {
            for(int j=1; j<=col; j++) {
                sums[i][j] = sums[i-1][j] + sums[i][j-1] - sums[i-1][j-1] + matrix[i-1][j-1];
            }
        }
    }

    int sumRegion(int row1, int col1, int row2, int col2) {
        return sums[row2+1][col2+1] - sums[row2+1][col1] - sums[row1][col2+1] + sums[row1][col1];
    }
};


// Your NumMatrix object will be instantiated and called as such:
// NumMatrix numMatrix(matrix);
// numMatrix.sumRegion(0, 1, 2, 3);
// numMatrix.sumRegion(1, 2, 3, 4);