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给定一个二维矩阵 matrix
,以下类型的多个请求:
(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
-105 <= matrix[i][j] <= 105
0 <= row1 <= row2 < m
0 <= col1 <= col2 < n
104
次 sumRegion
方法方法一:二维前缀和
我们用 $s[i + 1][j + 1]$ 表示第 $i$ 行第 $j$ 列左上部分所有元素之和,下标 $i$ 和 $j$ 均从 $0$ 开始。可以得到以下前缀和公式:
$$ s[i + 1][j + 1] = s[i + 1][j] + s[i][j + 1] - s[i][j] + nums[i][j] $$
那么分别以 $(x_1, y_1)$ 和 $(x_2, y_2)$ 为左上角和右下角的矩形的元素之和为:
$$ s[x_2 + 1][y_2 + 1] - s[x_2 + 1][y_1] - s[x_1][y_2 + 1] + s[x_1][y_1] $$
我们在初始化方法中预处理出前缀和数组 $s$,在查询方法中直接返回上述公式的结果即可。
初始化的时间复杂度为 $O(m\times n)$,查询的时间复杂度为 $O(1)$。
class NumMatrix:
def __init__(self, matrix: List[List[int]]):
m, n = len(matrix), len(matrix[0])
self.s = [[0] * (n + 1) for _ in range(m + 1)]
for i, row in enumerate(matrix):
for j, v in enumerate(row):
self.s[i + 1][j + 1] = (
self.s[i][j + 1] + self.s[i + 1][j] - self.s[i][j] + v
)
def sumRegion(self, row1: int, col1: int, row2: int, col2: int) -> int:
return (
self.s[row2 + 1][col2 + 1]
- self.s[row2 + 1][col1]
- self.s[row1][col2 + 1]
+ self.s[row1][col1]
)
# Your NumMatrix object will be instantiated and called as such:
# obj = NumMatrix(matrix)
# param_1 = obj.sumRegion(row1,col1,row2,col2)
class NumMatrix {
private int[][] s;
public NumMatrix(int[][] matrix) {
int m = matrix.length, n = matrix[0].length;
s = new int[m + 1][n + 1];
for (int i = 0; i < m; ++i) {
for (int j = 0; j < n; ++j) {
s[i + 1][j + 1] = s[i + 1][j] + s[i][j + 1] - s[i][j] + matrix[i][j];
}
}
}
public int sumRegion(int row1, int col1, int row2, int col2) {
return s[row2 + 1][col2 + 1] - s[row2 + 1][col1] - s[row1][col2 + 1] + s[row1][col1];
}
}
/**
* Your NumMatrix object will be instantiated and called as such:
* NumMatrix obj = new NumMatrix(matrix);
* int param_1 = obj.sumRegion(row1,col1,row2,col2);
*/
class NumMatrix {
public:
vector<vector<int>> s;
NumMatrix(vector<vector<int>>& matrix) {
int m = matrix.size(), n = matrix[0].size();
s.resize(m + 1, vector<int>(n + 1));
for (int i = 0; i < m; ++i) {
for (int j = 0; j < n; ++j) {
s[i + 1][j + 1] = s[i + 1][j] + s[i][j + 1] - s[i][j] + matrix[i][j];
}
}
}
int sumRegion(int row1, int col1, int row2, int col2) {
return s[row2 + 1][col2 + 1] - s[row2 + 1][col1] - s[row1][col2 + 1] + s[row1][col1];
}
};
/**
* Your NumMatrix object will be instantiated and called as such:
* NumMatrix* obj = new NumMatrix(matrix);
* int param_1 = obj->sumRegion(row1,col1,row2,col2);
*/
type NumMatrix struct {
s [][]int
}
func Constructor(matrix [][]int) NumMatrix {
m, n := len(matrix), len(matrix[0])
s := make([][]int, m+1)
for i := range s {
s[i] = make([]int, n+1)
}
for i, row := range matrix {
for j, v := range row {
s[i+1][j+1] = s[i+1][j] + s[i][j+1] - s[i][j] + v
}
}
return NumMatrix{s}
}
func (this *NumMatrix) SumRegion(row1 int, col1 int, row2 int, col2 int) int {
return this.s[row2+1][col2+1] - this.s[row2+1][col1] - this.s[row1][col2+1] + this.s[row1][col1]
}
/**
* Your NumMatrix object will be instantiated and called as such:
* obj := Constructor(matrix);
* param_1 := obj.SumRegion(row1,col1,row2,col2);
*/
/**
* @param {number[][]} matrix
*/
var NumMatrix = function (matrix) {
const m = matrix.length;
const n = matrix[0].length;
this.s = new Array(m + 1).fill(0).map(() => new Array(n + 1).fill(0));
for (let i = 0; i < m; ++i) {
for (let j = 0; j < n; ++j) {
this.s[i + 1][j + 1] =
this.s[i + 1][j] +
this.s[i][j + 1] -
this.s[i][j] +
matrix[i][j];
}
}
};
/**
* @param {number} row1
* @param {number} col1
* @param {number} row2
* @param {number} col2
* @return {number}
*/
NumMatrix.prototype.sumRegion = function (row1, col1, row2, col2) {
return (
this.s[row2 + 1][col2 + 1] -
this.s[row2 + 1][col1] -
this.s[row1][col2 + 1] +
this.s[row1][col1]
);
};
/**
* Your NumMatrix object will be instantiated and called as such:
* var obj = new NumMatrix(matrix)
* var param_1 = obj.sumRegion(row1,col1,row2,col2)
*/
class NumMatrix {
private s: number[][];
constructor(matrix: number[][]) {
const m = matrix.length;
const n = matrix[0].length;
this.s = new Array(m + 1).fill(0).map(() => new Array(n + 1).fill(0));
for (let i = 0; i < m; ++i) {
for (let j = 0; j < n; ++j) {
this.s[i + 1][j + 1] =
this.s[i + 1][j] +
this.s[i][j + 1] -
this.s[i][j] +
matrix[i][j];
}
}
}
sumRegion(row1: number, col1: number, row2: number, col2: number): number {
return (
this.s[row2 + 1][col2 + 1] -
this.s[row2 + 1][col1] -
this.s[row1][col2 + 1] +
this.s[row1][col1]
);
}
}
/**
* Your NumMatrix object will be instantiated and called as such:
* var obj = new NumMatrix(matrix)
* var param_1 = obj.sumRegion(row1,col1,row2,col2)
*/
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