<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>

#include <unordered_set>
#include <vector>
#include <climits>
using namespace std;

class Solution {
public:
    bool isRectangleCover(vector<vector<int>>& rectangles) {
        // To store the set of corners
        unordered_set<string> corners;
        int min_x = INT_MAX, min_y = INT_MAX, max_x = INT_MIN, max_y = INT_MIN;
        long long total_area = 0;  // Use long long for area to avoid overflow
        
        // Helper function to manage corners in a set
        auto toggleCorner = [&](int x, int y) {
            string corner = to_string(x) + "," + to_string(y);
            if (corners.count(corner)) {
                corners.erase(corner);
            } else {
                corners.insert(corner);
            }
        };
        
        // Iterate over all rectangles
        for (const auto& rect : rectangles) {
            int x1 = rect[0], y1 = rect[1], x2 = rect[2], y2 = rect[3];
            
            // Update the bounding rectangle
            min_x = min(min_x, x1);
            min_y = min(min_y, y1);
            max_x = max(max_x, x2);
            max_y = max(max_y, y2);
            
            // Calculate total area and convert to long long to avoid overflow
            total_area += static_cast<long long>(x2 - x1) * static_cast<long long>(y2 - y1);
            
            // Toggle corners (add if new, remove if it already exists)
            toggleCorner(x1, y1);
            toggleCorner(x1, y2);
            toggleCorner(x2, y1);
            toggleCorner(x2, y2);
        }
        
        // Check if the remaining corners are exactly the four corners of the bounding box
        if (corners.size() != 4 ||
            !corners.count(to_string(min_x) + "," + to_string(min_y)) ||
            !corners.count(to_string(min_x) + "," + to_string(max_y)) ||
            !corners.count(to_string(max_x) + "," + to_string(min_y)) ||
            !corners.count(to_string(max_x) + "," + to_string(max_y))) {
            return false;
        }
        
        // Check if the total area matches the area of the bounding rectangle, using long long
        long long bounding_area = static_cast<long long>(max_x - min_x) * static_cast<long long>(max_y - min_y);
        return total_area == bounding_area;
    }
};

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