## Range Sum Query 2D - Immutable
### 描述
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)`.

**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.
### 分析
思路跟一维的类似,建立一个累加和矩阵。令状态`f[i][j]`表示从(0,0)到(i,j)的子矩阵的和,则状态转移方程为
`f[i][j] = f[i-1][j] + rowSum`
其中 `rowSum` 是矩阵`matrix[i][0]`到`matrix[i][j]`这一行的和。
有了`f[i][j]`, 则
`sumRange(i1,j1,i2,j2) = f[i2][j2] + f[i1-1][j1-1] - f[i1-1][j2]-f[i2][j1-1]`
将辅助矩阵`f[i][j]`的行数和列数增1,可以简化对矩阵边界的处理。
### 代码
{% if book.java %}
{% codesnippet "./code/range-sum-query-2d-immutable."+book.suffix, language=book.suffix %}{% endcodesnippet %}
{% endif %}
### 相关题目
* [Range Sum Query - Immutable](range-sum-query-immutable.md)
* [Range Sum Query - Mmutable](../binary-tree/segment-tree/range-sum-query-mutable.md)