<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:
        bottomLeft = tuple(rectangles[0][:2])
        topRight = tuple(rectangles[0][2:])
        totalArea = 0
        pointFrequency = defaultdict(int) # point -> number of times it appears

        for r in rectangles:
            x1, y1, x2, y2 = r
            totalArea += (x2 - x1) * (y2 - y1)
            bottomLeft = (x1, y1) if x1 <= bottomLeft[0] and y1 <= bottomLeft[1] else bottomLeft
            topRight = (x2, y2) if x2 >= topRight[0] and y2 >= topRight[1] else topRight
            pointFrequency[(x1, y1)] += 1
            pointFrequency[(x2, y1)] += 1
            pointFrequency[(x2, y2)] += 1
            pointFrequency[(x1, y2)] += 1

        if totalArea != (topRight[0] - bottomLeft[0]) * (topRight[1] - bottomLeft[1]):
            return False

        topLeft = (bottomLeft[0], topRight[1])
        bottomRight = (topRight[0], bottomLeft[1])
        corners = {bottomLeft, topRight, topLeft, bottomRight}

        for point, frequency in pointFrequency.items():
            if point in corners and frequency != 1:
                return False
            
            if point not in corners and frequency % 2 != 0:
                return False

        return True

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