题目描述：

Given a 2D matrix, find the sum of the elements inside the rectangle defined by (row1, col1), (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.

题目大意：

给定一个二维矩阵，计算从(row1, col1) 到 (row2, col2)的子矩阵的和。

测试用例见题目描述。

注意：

1. 你可以假设矩阵不会改变
2. sumRegion函数会调用很多次
3. 你可以假设row1 ≤ row2， 并且 col1 ≤ col2。

解题思路：

构造辅助二维数组sums

sums[x][y]表示从0,0到x,y的子矩阵的和

利用容斥原理，可知：

sumRange(row1, col1, row2, col2) = sums[row2][col2] + sums[row1 - 1][col1 - 1] - sums[row1 - 1][col2] - sums[row2][col1 - 1]

将辅助矩阵的行数和列数+1，可以简化对矩阵边界的处理。

Python代码：

class NumMatrix(object):
    def __init__(self, matrix):
        """
        initialize your data structure here.
        :type matrix: List[List[int]]
        """
        m = len(matrix)
        n = len(matrix[0]) if m else 0
        self.sums = [[0] * (n + 1) for x in range(m + 1)]
        for x in range(1, m + 1):
            rowSum = 0
            for y in range(1, n + 1):
                self.sums[x][y] += rowSum + matrix[x - 1][y - 1]
                if x > 1:
                    self.sums[x][y] += self.sums[x - 1][y]
                rowSum += matrix[x - 1][y - 1]

    def sumRegion(self, row1, col1, row2, col2):
        """
        sum of elements matrix[(row1,col1)..(row2,col2)], inclusive.
        :type row1: int
        :type col1: int
        :type row2: int
        :type col2: int
        :rtype: int
        """
        return self.sums[row2 + 1][col2 + 1] + self.sums[row1][col1] \
                 - self.sums[row1][col2 + 1] - self.sums[row2 + 1][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)

 

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