304. 二维区域和检索 – 矩阵不可变

给定一个二维矩阵matrix，以下类型的多个请求：

计算其子矩形范围内元素的总和，该子矩阵的左上角为(row1, col1)，右下角为(row2, col2)。
实现NumMatrix类：

• NumMatrix(int[][] matrix)给定整数矩阵matrix进行初始化
• int sumRegion(int row1, int col1, int row2, int col2)返回左上角(row1, col1)、右下角(row2, col2)所描述的子矩阵的元素总和。

示例 1：

输入: 
["NumMatrix","sumRegion","sumRegion","sumRegion"]
[[[[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]]],[2,1,4,3],[1,1,2,2],[1,2,2,4]]
输出: 
[null, 8, 11, 12]
解释:
NumMatrix numMatrix = new NumMatrix([[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]]]);
numMatrix.sumRegion(2, 1, 4, 3); // return 8 (红色矩形框的元素总和)
numMatrix.sumRegion(1, 1, 2, 2); // return 11 (绿色矩形框的元素总和)
numMatrix.sumRegion(1, 2, 2, 4); // return 12 (蓝色矩形框的元素总和)

提示：

• m == matrix.length
• n == matrix[i].length
• 1 <= m, n <= 200
• -10^5 <= matrix[i][j] <= 10^5
• 0 <= row1 <= row2 < m
• 0 <= col1 <= col2 < n
• 最多调用10^4次sumRegion方法

1.一维前缀和
Python解答：

class NumMatrix:
    def __init__(self, matrix: List[List[int]]):
        self.value = [[0 for i in range(len(matrix[0])+1)] for j in range(len(matrix))]
        for i in range(len(self.value)):
            for j in range(1, len(self.value[0])):
                self.value[i][j] = self.value[i][j-1] + matrix[i][j-1]
    def sumRegion(self, row1: int, col1: int, row2: int, col2: int) -> int:
        total = 0
        for i in range(row1, row2+1):
            total += self.value[i][col2+1]-self.value[i][col1]
        return total
# Your NumMatrix object will be instantiated and called as such:
# obj = NumMatrix(matrix)
# param_1 = obj.sumRegion(row1,col1,row2,col2)

2.二维前缀和
Python解答：

class NumMatrix:
    def __init__(self, matrix: List[List[int]]):
        self.value = [[0 for i in range(len(matrix[0])+1)] for j in range(len(matrix)+1)]
        for i in range(1, len(matrix)+1):
            for j in range(1, len(matrix[0])+1):
                self.value[i][j] = self.value[i-1][j]+self.value[i][j-1]-self.value[i-1][j-1]+matrix[i-1][j-1]
    def sumRegion(self, row1: int, col1: int, row2: int, col2: int) -> int:
        return self.value[row2+1][col2+1]-self.value[row1][col2+1]-self.value[row2+1][col1]+self.value[row1][col1]

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