<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:
        class Interval:
            def __init__(self):
                self.startDict = {}
                self.endDict = {}
            
            def add(self, start, end):
                if start in self.startDict or end in self.endDict:
                    return False
                if start in self.endDict:
                    prevStart = self.endDict[start]
                    del self.startDict[prevStart]
                    del self.endDict[start]
                    return self.add(prevStart, end)
                if end in self.startDict:
                    nextEnd = self.startDict[end]
                    del self.endDict[nextEnd]
                    del self.startDict[end]
                    return self.add(start, nextEnd)
                else:
                    self.startDict[start] = end
                    self.endDict[end] = start
                    return True
            
            def verify(self, begin, end):
                return self.startDict == {begin: end} and self.endDict == {end: begin}
                 
        rectangles.sort(reverse = True)
        left, bot, right, top = float('inf'), float('inf'), float('-inf'), float('-inf'),
        for l, b, r, t in rectangles:
            left, bot, right, top = min(l, left), min(b, bot), max(r, right), max(t, top)

        sweep = left
        rectExpr = collections.defaultdict(dict)
        while sweep < right:
            currInterval = Interval()
            if sweep in rectExpr: del rectExpr[sweep]
            for key in rectExpr:
                for b, t in rectExpr[key]:
                    succ = currInterval.add(b, t)
                    if not succ: return False
            while rectangles and rectangles[-1][0] == sweep:
                l, b, r, t = rectangles.pop()
                if r not in rectExpr: rectExpr[r] = []
                rectExpr[r].append((b, t))
                succ = currInterval.add(b, t)
                if not succ: return False
            if not currInterval.verify(bot, top): return False
            sweep += 1
        return True

        
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