<p>Given an array <code>rectangles</code> where <code>rectangles[i] = [x<sub>i</sub>, y<sub>i</sub>, a<sub>i</sub>, b<sub>i</sub>]</code> represents an axis-aligned rectangle. The bottom-left point of the rectangle is <code>(x<sub>i</sub>, y<sub>i</sub>)</code> and the top-right point of it is <code>(a<sub>i</sub>, b<sub>i</sub>)</code>.</p>

<p>Return <code>true</code> <em>if all the rectangles together form an exact cover of a rectangular region</em>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/03/27/perectrec1-plane.jpg" style="width: 300px; height: 294px;" />
<pre>
<strong>Input:</strong> rectangles = [[1,1,3,3],[3,1,4,2],[3,2,4,4],[1,3,2,4],[2,3,3,4]]
<strong>Output:</strong> true
<strong>Explanation:</strong> All 5 rectangles together form an exact cover of a rectangular region.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/03/27/perfectrec2-plane.jpg" style="width: 300px; height: 294px;" />
<pre>
<strong>Input:</strong> rectangles = [[1,1,2,3],[1,3,2,4],[3,1,4,2],[3,2,4,4]]
<strong>Output:</strong> false
<strong>Explanation:</strong> Because there is a gap between the two rectangular regions.
</pre>

<p><strong class="example">Example 3:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/03/27/perfecrrec4-plane.jpg" style="width: 300px; height: 294px;" />
<pre>
<strong>Input:</strong> rectangles = [[1,1,3,3],[3,1,4,2],[1,3,2,4],[2,2,4,4]]
<strong>Output:</strong> false
<strong>Explanation:</strong> Because two of the rectangles overlap with each other.
</pre>

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

<ul>
	<li><code>1 &lt;= rectangles.length &lt;= 2 * 10<sup>4</sup></code></li>
	<li><code>rectangles[i].length == 4</code></li>
	<li><code>-10<sup>5</sup> &lt;= x<sub>i</sub> &lt; a<sub>i</sub> &lt;= 10<sup>5</sup></code></li>
	<li><code>-10<sup>5</sup> &lt;= y<sub>i</sub> &lt; b<sub>i</sub> &lt;= 10<sup>5</sup></code></li>
</ul>

class Solution {
public:
    long long getCombinedArea(vector<vector<int>> &majorCorners) {
        sort(majorCorners.begin(), majorCorners.end());
        vector<int> bottomLeft = majorCorners[0], topRight = majorCorners[3];
        long long area = (topRight[0] - bottomLeft[0]) * (long long)(topRight[1] - bottomLeft[1]);
        return area;
    }
    bool isRectangleCover(vector<vector<int>>& rectangles) {
        // OBSERVATION 1:
        // Every corner must be repeated even times,
        // If any corner occures only once, then it 
        // must the corners of the combined rectangle.

        // OBSERVATION 2;
        // To identify overlaping, total area of all the individual rectangles
        // must be same as the area of the combined rectangle.
        map<vector<int>, int> cornerFreq;
        long long totalArea = 0;
        for(auto &v : rectangles) {
            totalArea += (v[3] - v[1]) * (long long)(v[2] - v[0]);
            cornerFreq[{v[0], v[3]}]++;
            cornerFreq[{v[0], v[1]}]++;
            cornerFreq[{v[2], v[3]}]++;
            cornerFreq[{v[2], v[1]}]++;
        }
        vector<vector<int>> majorCorners;
        for(auto &it : cornerFreq) if(it.second % 2 == 1) majorCorners.push_back(it.first);
        if(majorCorners.size() != 4) return false;
        long long combinedArea = getCombinedArea(majorCorners);
        return totalArea == combinedArea;
    }
};
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