🟩 3394. Check if Grid can be Cut into Sections

Medium | Grid | Geometry | Sorting

📝 Problem Statement

You are given an integer n representing the dimensions of an n x n grid, with the origin at the bottom-left corner of the grid.

You are also given a 2D array of coordinates rectangles, where rectangles[i] is in the form [startx, starty, endx, endy], representing a rectangle on the grid.

Each rectangle is defined as follows:

• (startx, starty): The bottom-left corner of the rectangle.
• (endx, endy): The top-right corner of the rectangle.
• Rectangles do not overlap.

Your task is to determine if it is possible to make either two horizontal or two vertical cuts on the grid such that:

1. Each of the three resulting sections formed by the cuts contains at least one rectangle.
2. Every rectangle belongs to exactly one section.

Return true if such cuts can be made; otherwise, return false.

🔹 Example 1

Input:

n = 5  
rectangles = [[1,0,5,2],[0,2,2,4],[3,2,5,3],[0,4,4,5]]

Output:

true

Explanation:

We can make horizontal cuts at y = 2 and y = 4.

  -------------
5 | █ █ █ █    
  |█       █   
4 |█ █ █ █ █   
  |█       █   
3 |█ █   █ █   
  |█ █ █ █ █   
2 |█ █ █ █ █   
  -------------
1 |█ █ █ █ █   
  -------------
0  1 2 3 4 5  

Hence, output is true.

🔹 Example 2

Input:

n = 4  
rectangles = [[0,0,1,1],[2,0,3,4],[0,2,2,3],[3,0,4,3]]

Output:

true

Explanation:

We can make vertical cuts at x = 2 and x = 3.

🔹 Example 3

Input:

n = 4  
rectangles = [[0,2,2,4],[1,0,3,2],[2,2,3,4],[3,0,4,2],[3,2,4,4]]

Output:

false

Explanation:

We cannot make two horizontal or two vertical cuts that satisfy the conditions. Hence, output is false.

🛠 Constraints

✅ 3 <= n <= 10^9
✅ 3 <= rectangles.length <= 10^5
✅ 0 <= rectangles[i][0] < rectangles[i][2] <= n
✅ 0 <= rectangles[i][1] < rectangles[i][3] <= n
✅ No two rectangles overlap.

💡 Approach & Hints

1. Sort rectangles by x or y coordinates.
2. Find possible cut positions that divide the grid into three valid sections.
3. Check if each section contains at least one rectangle.
4. Verify that rectangles do not cross the cuts.

🔹 If a valid set of two horizontal or two vertical cuts exists, return true, else return false.

⏳ Time Complexity Analysis

• Sorting the rectangles takes O(m log m) where m is the number of rectangles.
• Checking the conditions takes O(m).
• Total Complexity: O(m log m) (which is efficient for m ≤ 10^5).

✍ Author

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📢 Feel free to contribute by improving the solution or adding more test cases! 🚀

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This **README.md** file includes:  
✅ **Clear problem statement**  
✅ **Examples with input/output and diagrams**  
✅ **Constraints and approach hints**  
✅ **Time complexity analysis**  

Let me know if you need any modifications! 😊🔥