750. Number Of Corner Rectangles

问题描述：

Given a grid where each entry is only 0 or 1, find the number of corner rectangles.

A corner rectangle is 4 distinct 1s on the grid that form an axis-aligned rectangle. Note that only the corners need to have the value 1. Also, all four 1s used must be distinct.

Example 1:

Input: grid = 
[[1, 0, 0, 1, 0],
 [0, 0, 1, 0, 1],
 [0, 0, 0, 1, 0],
 [1, 0, 1, 0, 1]]
Output: 1
Explanation: There is only one corner rectangle, with corners grid[1][2], grid[1][4], grid[3][2], grid[3][4].

Example 2:

Input: grid = 
[[1, 1, 1],
 [1, 1, 1],
 [1, 1, 1]]
Output: 9
Explanation: There are four 2x2 rectangles, four 2x3 and 3x2 rectangles, and one 3x3 rectangle.

Example 3:

Input: grid = 
[[1, 1, 1, 1]]
Output: 0
Explanation: Rectangles must have four distinct corners.

Note:

1. The number of rows and columns of grid will each be in the range [1, 200].
2. Each grid[i][j] will be either 0 or 1.
3. The number of 1s in the grid will be at most 6000.

解题思路：

一开始想到的是暴力破解法：即对每一个点进行检查以它为左上角的矩形的个数。

先向下走，再向右走，最后再向上走，复杂度我认为是O(n⁴)

看到了别人提交的较快的方法，实际上是进行了剪枝，因为若能构成矩形，必定有另一个顶点在同一行。

我们记录改顶点往后的为1的 j 值，对后面的每一行检查对应的两个顶点的值是否为1

代码：

class Solution {
public:
    int countCornerRectangles(vector<vector<int>>& grid) {
        if(grid.size() <= 1 || grid[0].size() <= 1)
            return 0;
        int ret = 0;
        for(int i = 0; i < grid.size() - 1; i++){
            for(int j = 0; j < grid[0].size() - 1; j++){
                if(grid[i][j] == 1)
                    ret += findRec(grid, i, j);
            }
        }
        return ret;
    }
    
    int findRec(vector<vector<int>> &grid, int i, int j){
        int ret = 0;
        for(int x = i+1; x < grid.size(); x++){
            if(grid[x][j] == 1){
                for(int y = j+1; y < grid[0].size(); y++){
                    if(grid[x][y] == 1 && grid[i][y] == 1)
                        ret++;
                }
            }
        }
        return ret;
    }
};

较快的方法：

classclass  SolutionSolution { {
 publicpublic:
    :     intint  countCornerRectanglescountCornerRectangles((vectorvector<<vectorvector<<intint>>& grid)>>& grid)  {
        {         ifif(grid.empty() || grid[(grid.empty() || grid[00].empty()){
            ].empty()){             return 0;
        }
        int res = 0;
        int row = grid.size(), col = grid[0].size();
        for(int i = 0; i < row-1; i++){
            vector<int> ones;
            for(int j = 0; j < col; j++){
                if(grid[i][j] == 1){
                    ones.push_back(j);
                }
            }
            for(int k = i+1; k < row; k++){
                int cur = 0;
                for(auto o:ones){
                    if(grid[k][o] == 1){
                        cur++;
                    }
                }
                res += cur*(cur-1)/2;
            }
        }
        return res;
    }
};