<p>You are given an array of non-overlapping axis-aligned rectangles <code>rects</code> where <code>rects[i] = [a<sub>i</sub>, b<sub>i</sub>, x<sub>i</sub>, y<sub>i</sub>]</code> indicates that <code>(a<sub>i</sub>, b<sub>i</sub>)</code> is the bottom-left corner point of the <code>i<sup>th</sup></code> rectangle and <code>(x<sub>i</sub>, y<sub>i</sub>)</code> is the top-right corner point of the <code>i<sup>th</sup></code> rectangle. Design an algorithm to pick a random integer point inside the space covered by one of the given rectangles. A point on the perimeter of a rectangle is included in the space covered by the rectangle.</p>

<p>Any integer point inside the space covered by one of the given rectangles should be equally likely to be returned.</p>

<p><strong>Note</strong> that an integer point is a point that has integer coordinates.</p>

<p>Implement the <code>Solution</code> class:</p>

<ul>
	<li><code>Solution(int[][] rects)</code> Initializes the object with the given rectangles <code>rects</code>.</li>
	<li><code>int[] pick()</code> Returns a random integer point <code>[u, v]</code> inside the space covered by one of the given rectangles.</li>
</ul>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/07/24/lc-pickrandomrec.jpg" style="width: 419px; height: 539px;" />
<pre>
<strong>Input</strong>
[&quot;Solution&quot;, &quot;pick&quot;, &quot;pick&quot;, &quot;pick&quot;, &quot;pick&quot;, &quot;pick&quot;]
[[[[-2, -2, 1, 1], [2, 2, 4, 6]]], [], [], [], [], []]
<strong>Output</strong>
[null, [1, -2], [1, -1], [-1, -2], [-2, -2], [0, 0]]

<strong>Explanation</strong>
Solution solution = new Solution([[-2, -2, 1, 1], [2, 2, 4, 6]]);
solution.pick(); // return [1, -2]
solution.pick(); // return [1, -1]
solution.pick(); // return [-1, -2]
solution.pick(); // return [-2, -2]
solution.pick(); // return [0, 0]
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= rects.length &lt;= 100</code></li>
	<li><code>rects[i].length == 4</code></li>
	<li><code>-10<sup>9</sup> &lt;= a<sub>i</sub> &lt; x<sub>i</sub> &lt;= 10<sup>9</sup></code></li>
	<li><code>-10<sup>9</sup> &lt;= b<sub>i</sub> &lt; y<sub>i</sub> &lt;= 10<sup>9</sup></code></li>
	<li><code>x<sub>i</sub> - a<sub>i</sub> &lt;= 2000</code></li>
	<li><code>y<sub>i</sub> - b<sub>i</sub> &lt;= 2000</code></li>
	<li>All the rectangles do not overlap.</li>
	<li>At most <code>10<sup>4</sup></code> calls will be made to <code>pick</code>.</li>
</ul>

class Solution {
public:
    // Solution 1: large rectangle with higher probability according to area
    // Thus, important to have higher chance to pick the larger rectangle
    Solution(vector<vector<int>>& rects) : rects(rects), area(0) {
        for(const auto& g : rects) m[area+=(g[2]-g[0]+1)*(g[3]-g[1]+1)] = g;
    }
    vector<int> pick() {
        auto g = m.upper_bound(rand()%area)->second;
        int a = g[0]+rand()%(g[2]-g[0]+1), b = g[1]+rand()%(g[3]-g[1]+1);
        return {a,b};
    }
private:
    int area;
    vector<vector<int> > rects;
    map<int, vector<int> > m; // <area, rects>
};


// class Solution {
// public:
//     // Solution 1: large rectangle with higher probability according to area
//     // Thus, important to have higher chance to pick the larger rectangle
//     Solution(vector<vector<int>>& rects) : rects(rects), area(0) {
//         for(int i=0; i<rects.size(); ++i) {
//             area += (rects[i][2]-rects[i][0]+1)*(rects[i][3]-rects[i][1]+1);
//             m[area] = i; // pre area sum
//             // m.insert(m.end(), {area, i});
//         }
//     }
//     vector<int> pick() {
//         int i = m.upper_bound(rand()%area)->second;
//         auto g = rects[i];
//         int a = g[0]+rand()%(g[2]-g[0]+1), b = g[1]+rand()%(g[3]-g[1]+1);
//         return {a,b};
//     }
// private:
//     int area;
//     vector<vector<int> > rects;
//     map<int, int> m; // <area, rects_idx>
// };

/**
 * Your Solution object will be instantiated and called as such:
 * Solution* obj = new Solution(rects);
 * vector<int> param_1 = obj->pick();
 */
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