【前缀和】304.二维区域和检索-矩阵不可变_二维区域和检索-数组不可变前缀和方法-优快云博客

本文链接：https://blog.youkuaiyun.com/Allenlzcoder/article/details/135352373

文章介绍了如何利用前缀和的概念，通过NumMatrix类中的sumRegion方法，计算给定矩阵中田字形区域内(指定行和列范围)的面积。算法通过逐层累加矩阵元素来快速得到结果。

摘要生成于 C知道，由 DeepSeek-R1 满血版支持，前往体验 >

法1：前缀和

小学数学，田字形，已知整体面积，上面面积，左边面积，左上面积，求右下角矩形的面积。右下角矩形的面积=整体面积-上面面积-左边面积+左上面积

Python

class NumMatrix:

    def __init__(self, matrix: List[List[int]]):
        m, n = len(matrix), len(matrix[0])
        tmp_matrix = [[0] * n for _ in range(m)]
        tmp_matrix[0][0] = matrix[0][0]
        for k in range(1, n): # 第1行
            tmp_matrix[0][k] = tmp_matrix[0][k-1] + matrix[0][k]
        for k in range(1, m): # 1列
            tmp_matrix[k][0] = tmp_matrix[k-1][0] + matrix[k][0]
        for i in range(1, m):
            for j in range(1, n):
                tmp_matrix[i][j] = tmp_matrix[i-1][j] + tmp_matrix[i][j-1] - tmp_matrix[i-1][j-1] + matrix[i][j]
        self.matrix = tmp_matrix

    def sumRegion(self, row1: int, col1: int, row2: int, col2: int) -> int:
        origin_sum = self.matrix[row2][col2]
        left_sum = self.matrix[row2][col1-1] if col1 > 0 else 0
        up_sum = self.matrix[row1-1][col2] if row1 > 0 else 0
        left_up_sum = self.matrix[row1-1][col1-1] if row1 > 0 and col1 > 0 else 0
        return origin_sum - left_sum - up_sum + left_up_sum


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

Java

class NumMatrix {
    private int[][] sumMatrix;
    public NumMatrix(int[][] matrix) {
        int m = matrix.length, n = matrix[0].length;
        sumMatrix = new int[m][n];
        sumMatrix[0][0] = matrix[0][0];
        for (int i = 1; i < n; ++i) {
            sumMatrix[0][i] = sumMatrix[0][i - 1] + matrix[0][i];  
        }
        for (int i = 1; i < m; ++i) {
            sumMatrix[i][0] = sumMatrix[i - 1][0] + matrix[i][0];  
        }
        for (int i = 1; i < m; ++i) {
            for (int j = 1; j < n; ++j) {
                sumMatrix[i][j] = sumMatrix[i - 1][j] + sumMatrix[i][j - 1] - sumMatrix[i - 1][j - 1] +  matrix[i][j];
            }
        }
    }
    
    public int sumRegion(int row1, int col1, int row2, int col2) {
        int res = sumMatrix[row2][col2];
        if (row1 - 1 >= 0) {
            res -= sumMatrix[row1 - 1][col2];
        }
        if (col1 - 1 >= 0) {
            res -= sumMatrix[row2][col1 - 1];
        }
        if (row1 - 1 >= 0 && col1 - 1 >= 0) {
            res += sumMatrix[row1 - 1][col1 - 1];
        }

        return res;
    }
}