<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:
    # Time: O()
    # Space: O(), where n = len(rectangles)
    def isRectangleCover(self, rectangles: List[List[int]]) -> bool:
        
        n = len(rectangles)
        v, h = [], []
        tot_area = 0
        min_x = min_y = float('inf')
        max_x = max_y = float('-inf')
        
        points = set()
        for x1, y1, x2, y2 in rectangles:
            v.append([x1, 0])
            v.append([x2, 1])
            h.append([y1, 0])
            h.append([y2, 1])
            tot_area += (y2-y1)*(x2-x1)
            min_x = min(min_x, x1, x2)
            min_y = min(min_y, y1, y2)
            max_x = max(max_x, x1, x2)
            max_y = max(max_y, y1, y2)
            pts = [(x1, y1), (x1, y2), (x2, y1), (x2, y2)]
            for p in pts:
                if p in points:
                    points.remove(p)
                else:
                    points.add(p)

        if tot_area != (max_y-min_y)*(max_x-min_x):
            return 0

        if len(points) != 4:     
            return False
        
        v.sort(), h.sort()
        count = 0
        for x, type_ in v[:-1]:
            count = (count+1) if type_ == 0 else (count-1)
            if not count:
                return 0
        count = 0
        for x, type_ in h[:-1]:
            count = (count+1) if type_ == 0 else (count-1)
            if not count:
                return 0   
        
        if (min_x, min_y) not in points:
            return 0
        if (min_x, max_y) not in points:
            return 0
        if (max_x, min_y) not in points:
            return 0
        if (max_x, max_y) not in points:
            return 0  

        return 1   

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