<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>

from collections import defaultdict

class Solution:
    def isRectangleCover(self, rectangles: List[List[int]]) -> bool:
        rectangles = [tuple(rect) for rect in rectangles]
        x_min, y_min, x_max, y_max = rectangles[0]
        sum_area = 0
        for rect in rectangles:
            x, y, a, b = rect
            if x < x_min:
                x_min = x
            if y < y_min:
                y_min = y
            if a > x_max:
                x_max = a
            if b > y_max:
                y_max = b
            sum_area += (a - x) * (b - y)
        
        if (x_max - x_min) * (y_max - y_min) != sum_area:
            return False

        # rect_bins = defaultdict(set)
        # for rect in rectangles:
        #     rect_bins[rect[0]].add(tuple(rect))
        
        # for rect_bin in rect_bins.values():


        rectangles.sort(key=lambda x: x[2])
        while rectangles:
            x, y, a, b = rectangles.pop()
            for xi, yi, ai, bi in rectangles:
                if ai < x or b < yi or bi < y:
                    continue
                if (ai - max(x, xi)) > 0 and (min(b, bi) - max(y, yi)) > 0:
                    return False
        return True

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