[LeetCode] Random Point in Non-overlapping Rectangles 非重叠矩形中的随机点

Given a list of non-overlapping axis-aligned rectangles rects, write a function pick which randomly and uniformily picks an integer point in the space covered by the rectangles.

Note:

An integer point is a point that has integer coordinates.
A point on the perimeter of a rectangle is included in the space covered by the rectangles.
ith rectangle = rects[i] = [x1,y1,x2,y2], where [x1, y1] are the integer coordinates of the bottom-left corner, and [x2, y2] are the integer coordinates of the top-right corner.
length and width of each rectangle does not exceed 2000.
1 <= rects.length <= 100
pick return a point as an array of integer coordinates [p_x, p_y]
pick is called at most 10000 times.

Example 1:

Input: 
["Solution","pick","pick","pick"]
[[[[1,1,5,5]]],[],[],[]]
Output: 
[null,[4,1],[4,1],[3,3]]

Example 2:

Input: 
["Solution","pick","pick","pick","pick","pick"]
[[[[-2,-2,-1,-1],[1,0,3,0]]],[],[],[],[],[]]
Output: 
[null,[-1,-2],[2,0],[-2,-1],[3,0],[-2,-2]]

Explanation of Input Syntax:

The input is two lists: the subroutines called and their arguments. Solution's constructor has one argument, the array of rectangles rects. pick has no arguments. Arguments are always wrapped with a list, even if there aren't any.

这道题给了我们一些非重叠的矩形，让我们返回一个这些矩形中的一个随机的点。那么博主的第一直觉就是首先要在这些矩形中随机挑出来一个，然后在这个随机的矩形中再随机生成一个点，通过随机生成一个长和宽即可。博主最开始想到的方法是用rand随机生成一个 [0, n) 范围内的数字，n为输入矩形的个数，这样就得到了一个随机的矩形。但是这种方法貌似行不通，会跪在一个很长的输入测试数据。这使得博主比较困惑了，没有想出原因是为何，有哪位看官大神们知道的，麻烦留言告知博主哈！哈，已经知道了，参见评论区二楼留言～论坛上的解法有一种是用水塘抽样Reservoir Sampling的方法的，LeetCode之前有过几道需要用这种方法的题目 Random Pick Index，Shuffle an Array 和 Linked List Random Node。这里我们使用其来随机出一个矩形，做法是遍历所有的矩形，用变量sumArea来计算当前遍历过的所有矩形面积之和，然后变量area是当前遍历的矩形的面积（注意这里不是严格意义上的面积，而是该区域内整数坐标的点的个数，所以长宽相乘的时候都要加1），然后我们在当前所有矩形面积之和内随机生成一个值，如果这个值小于area，那么选择当前的矩阵为随机矩形。这里相当于一个大小为area的水塘，在这个值之内的话，就更换selected。这个方法是没啥问题，但是博主还是没想通为啥不能直接随机生成矩形的index。当我们拿到随机矩形后，之后就随机出宽和高返回即可，参见代码如下：

解法一：

class Solution {
public:
    Solution(vector<vector<int>> rects) {
        _rects = rects;
    }
    
    vector<int> pick() {
        int sumArea = 0;
        vector<int> selected;
        for (auto rect : _rects) {
            int area = (rect[2] - rect[0] + 1) * (rect[3] - rect[1] + 1);
            sumArea += area;
            if (rand() % sumArea < area) selected = rect;
        }
        int x = rand() % (selected[2] - selected[0] + 1) + selected[0];
        int y = rand() % (selected[3] - selected[1] + 1) + selected[1];
        return {x, y};
    }

private:
    vector<vector<int>> _rects;
};

这道题在论坛上的主流解法其实是这个，我们用TreeMap来建立当前遍历过的矩形面积之和跟该矩形位置之间的映射。然后当我们求出所有的矩形面积之和后，我们随机生成一个值，然后在TreeMap中找到第一个大于这个值的矩形，这里博主还是有疑问，为啥不能直接随机矩形的位置，而是非要跟面积扯上关系。之后的步骤就跟上面的没啥区别了，参见代码如下：

解法二：

class Solution {
public:
    Solution(vector<vector<int>> rects) {
        _rects = rects;
        _totalArea = 0;
        for (auto rect : rects) {
            _totalArea += (rect[2] - rect[0] + 1) * (rect[3] - rect[1] + 1);
            _areaToIdx.insert({_totalArea, _areaToIdx.size()});
        }
    }
    
    vector<int> pick() {
        int val = rand() % _totalArea;
        int idx = _areaToIdx.upper_bound(val)->second;
        int width = _rects[idx][2] - _rects[idx][0] + 1;
        int height = _rects[idx][3] - _rects[idx][1] + 1;
        return {rand() % width + _rects[idx][0], rand() % height + _rects[idx][1]};
    }

private:
    vector<vector<int>> _rects;
    int _totalArea;
    map<int, int> _areaToIdx;
};