<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 {
    struct pair_hash {
        template <class T1, class T2>
        std::size_t operator()(const std::pair<T1, T2>& p) const {
            auto h1 = std::hash<T1>{}(p.first);
            auto h2 = std::hash<T2>{}(p.second);
            return h1 ^ h2;
        }
    };
public:
    bool isRectangleCover(vector<vector<int>>& rectangles) {
        if (rectangles.empty()) return false;

        // Set to store the points
        unordered_set<pair<int, int>, pair_hash> points;

        int x1 = INT_MAX, y1 = INT_MAX, x2 = INT_MIN, y2 = INT_MIN;
        long totalArea = 0;

        // Iterate over all rectangles
        for (const auto& rect : rectangles) {
            int xi = rect[0], yi = rect[1], ai = rect[2], bi = rect[3];

            // Calculate the area of the current rectangle
            totalArea += (long)(ai - xi) * (bi - yi);

            // Update the bounding rectangle
            x1 = min(x1, xi);
            y1 = min(y1, yi);
            x2 = max(x2, ai);
            y2 = max(y2, bi);

            // Define the four corners of the current rectangle
            pair<int, int> bl = {xi, yi};
            pair<int, int> br = {ai, yi};
            pair<int, int> tl = {xi, bi};
            pair<int, int> tr = {ai, bi};

            // Add or remove corners from the set
            for (const auto& p : {bl, br, tl, tr}) {
                if (!points.insert(p).second) {
                    points.erase(p);
                }
            }
        }

        // Check the area of the bounding rectangle
        long boundingArea = (long)(x2 - x1) * (y2 - y1);
        if (totalArea != boundingArea) {
            return false;
        }

        // Check if exactly four points remain in the set
        if (points.size() != 4) {
            return false;
        }

        // Check if the remaining points are the corners of the bounding rectangle
        if (!points.count({x1, y1}) || !points.count({x1, y2}) ||
            !points.count({x2, y1}) || !points.count({x2, y2})) {
            return false;
        }

        return true;
    }
};
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