<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: 
    def isRectangleCover(self, rectangles: List[List[int]]) -> bool: 
        # Use a dictionary to count corners 
        corner_count = {} 

        def add_corner(x, y): 
            if (x, y) in corner_count: 
                corner_count[(x, y)] += 1 
            else: 
                corner_count[(x, y)] = 1 

        def remove_corner(x, y): 
            if (x, y) in corner_count: 
                corner_count[(x, y)] -= 1 
                if corner_count[(x, y)] == 0: 
                    del corner_count[(x, y)] 

        # Variables to determine the bounding box of all rectangles 
        min_x = float('inf') 
        min_y = float('inf') 
        max_x = float('-inf') 
        max_y = float('-inf') 
        total_area = 0 

        for x1, y1, x2, y2 in rectangles: 
            min_x = min(min_x, x1) 
            min_y = min(min_y, y1) 
            max_x = max(max_x, x2) 
            max_y = max(max_y, y2) 
            total_area += (x2 - x1) * (y2 - y1) 
            add_corner(x1, y1) 
            add_corner(x1, y2) 
            add_corner(x2, y1) 
            add_corner(x2, y2) 

        if total_area != (max_x - min_x) * (max_y - min_y): 
            return False 

        expected_corners = {(min_x, min_y), (min_x, max_y), (max_x, min_y), (max_x, max_y)} 

        for corner in expected_corners: 
            if corner not in corner_count or corner_count[corner] != 1: 
                return False 
            del corner_count[corner] 

        for count in corner_count.values(): 
            if count % 2 != 0: 
                return False 

        return True
24500