# Range Sum Query 2D - Immutable(matrix)

Given a 2D matrixmatrix, find the sum of the elements inside the rectangle defined by its upper left corner (row1,col1) and lower right corner (row2,col2).

![Range Sum Query 2D](https://leetcode.com/static/images/courses/range_sum_query_2d.png)\
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
```

分析

```
+-----+-+-------+     +--------+-----+     +-----+---------+     +-----+--------+
|     | |       |     |        |     |     |     |         |     |     |        |
|     | |       |     |        |     |     |     |         |     |     |        |
+-----+-+       |     +--------+     |     |     |         |     +-----+        |
|     | |       |  =  |              |  +  |     |         |  -  |              |
+-----+-+       |     |              |     +-----+         |     |              |
|               |     |              |     |               |     |              |
|               |     |              |     |               |     |              |
+---------------+     +--------------+     +---------------+     +--------------+

   sums[i][j]      =    sums[i-1][j]    +     sums[i][j-1]    -   sums[i-1][j-1]   +  

                        matrix[i-1][j-1]
```

别忘了最后index 是 r2+1 c2+1

```
class NumMatrix:

    def __init__(self, matrix):
        """
        :type matrix: List[List[int]]
        """
        if matrix:

            n = len(matrix)
            m = len(matrix[0])
            f = [[0]*(m+1) for _ in range(n+1)]
            for i in range(1,n+1):
                for j in range(1,m+1):
                    f[i][j] = f[i][j-1]+f[i-1][j]-f[i-1][j-1]+matrix[i-1][j-1]

            self.f = f
        else:
            self.f= None




    def sumRegion(self, row1, col1, row2, col2):
        """
        :type row1: int
        :type col1: int
        :type row2: int
        :type col2: int
        :rtype: int
        """
        if not self.f:
            return 0
        return self.f[row2+1][col2+1]-self.f[row1][col2+1]-self.f[row2+1][col1]+self.f[row1][col1]

# Your NumMatrix object will be instantiated and called as such:
# obj = NumMatrix(matrix)
# param_1 = obj.sumRegion(row1,col1,row2,col2)
```


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