304. Range Sum Query 2D - Immutable (Medium)

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:

1. You may assume that the matrix does not change.
2. There are many calls to sumRegion function.
3. You may assume that row1 ≤ row2 and col1 ≤ col2.

Soltuion: DP

Construct a 2D array dp[n+1][m+1]

(notice: we add additional blank row dp[0][n+1]={0} and blank column dp[m+1][0]={0} to remove the edge case checking), so, we can have the following definition

dp[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.

And we can have the following code:

+---------------+   +--------------+   +---------------+   +--------------+   +--------------+
|               |   |         |    |   |   |           |   |         |    |   |   |          |
|   (r1,c1)     |   |         |    |   |   |           |   |         |    |   |   |          |
|   +------+    |   |         |    |   |   |           |   +---------+    |   +---+          |
|   |      |    | = |         |    | - |   |           | - |      (r1,c2) | + |   (r1,c1)    |
|   |      |    |   |         |    |   |   |           |   |              |   |              |
|   +------+    |   +---------+    |   +---+           |   |              |   |              |
|        (r2,c2)|   |       (r2,c2)|   |   (r2,c1)     |   |              |   |              |
+---------------+   +--------------+   +---------------+   +--------------+   +--------------+

class NumMatrix {
public:
    NumMatrix(vector<vector<int>> &matrix) {
        if (matrix.empty() || matrix[0].empty()) return;
        dp.resize(matrix.size()+1, vector<int>(matrix[0].size()+1, 0));
        for (int i = 1; i <= matrix.size(); ++i) {
            for (int j = 1; j <= matrix[0].size(); ++j) {
                dp[i][j] = dp[i-1][j]+dp[i][j-1]-dp[i-1][j-1]+matrix[i-1][j-1];
            }
        }
    }

    int sumRegion(int row1, int col1, int row2, int col2) {
        return dp[row2+1][col2+1]-dp[row1][col2+1]-dp[row2+1][col1]+dp[row1][col1];
    }
private:
    vector<vector<int>> dp;
};


// 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);