<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 {
public:
    bool isRectangleCover(vector<vector<int>>& rectangles) {
        vector<vector<int>> events;
        for(auto& r : rectangles) {
            events.push_back({r[0], 0, r[1], r[3]});
            events.push_back({r[2], 1, r[1], r[3]});
        }
        
        auto cmp = [&](const vector<int>& v1, const vector<int>& v2) {
            return v1[0] == v2[0] ? v1[1] < v2[1] : v1[0] < v2[0];
        };
        sort(events.begin(), events.end(), cmp);
        
        map<int,int> yLine;
        int minYHeight = INT_MIN, maxYHeight = INT_MIN;
        auto checkYAxis = [&]() {
            if(yLine.empty()) {return false;}
            
            int first = yLine.begin()->first;
            int last = yLine.rbegin()->first;
            if(minYHeight == INT_MIN) {
                minYHeight = first;
            }
            if(maxYHeight == INT_MIN) {
                maxYHeight = last;
            }
            if(minYHeight != first || maxYHeight != last) {
                return false;
            }
            long long curr = 0;
            for(auto yPoint = yLine.begin(); yPoint != yLine.end(); yPoint++) {
                curr += yPoint->second;
                //Case 1: curr = 0 and we haven't reached the end
                if(!curr && std::next(yPoint) != yLine.end()) {
                    return false;
                }
                // Case 2:  curr > 1 (meaning we have overlaps)
                if(curr > 1) {
                    return false;
                }
            }
            return true;
        };
        
        int currX = 0, prev = INT_MIN;
        bool currXInitialized = false;
        for(auto& e : events) {
            if(prev != INT_MIN && prev != e[0] && !checkYAxis()) {
                return false;
            }
            if(e[1]) { // Exit event
                yLine[e[2]]--; //Starting y
                yLine[e[3]]++; //Ending y
                if(!yLine[e[2]]) yLine.erase(e[2]);
                if(!yLine[e[3]]) yLine.erase(e[3]);
            }
            
            else {
                yLine[e[2]]++;
                yLine[e[3]]--;
            }
            
            prev = e[0];
        }
        
        return true;
    }
};









43100