<p>You are given a 2D integer array <code>tiles</code> where <code>tiles[i] = [l<sub>i</sub>, r<sub>i</sub>]</code> represents that every tile <code>j</code> in the range <code>l<sub>i</sub> &lt;= j &lt;= r<sub>i</sub></code> is colored white.</p>

<p>You are also given an integer <code>carpetLen</code>, the length of a single carpet that can be placed <strong>anywhere</strong>.</p>

<p>Return <em>the <strong>maximum</strong> number of white tiles that can be covered by the carpet</em>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2022/03/25/example1drawio3.png" style="width: 644px; height: 158px;" />
<pre>
<strong>Input:</strong> tiles = [[1,5],[10,11],[12,18],[20,25],[30,32]], carpetLen = 10
<strong>Output:</strong> 9
<strong>Explanation:</strong> Place the carpet starting on tile 10. 
It covers 9 white tiles, so we return 9.
Note that there may be other places where the carpet covers 9 white tiles.
It can be shown that the carpet cannot cover more than 9 white tiles.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2022/03/24/example2drawio.png" style="width: 231px; height: 168px;" />
<pre>
<strong>Input:</strong> tiles = [[10,11],[1,1]], carpetLen = 2
<strong>Output:</strong> 2
<strong>Explanation:</strong> Place the carpet starting on tile 10. 
It covers 2 white tiles, so we return 2.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= tiles.length &lt;= 5 * 10<sup>4</sup></code></li>
	<li><code>tiles[i].length == 2</code></li>
	<li><code>1 &lt;= l<sub>i</sub> &lt;= r<sub>i</sub> &lt;= 10<sup>9</sup></code></li>
	<li><code>1 &lt;= carpetLen &lt;= 10<sup>9</sup></code></li>
	<li>The <code>tiles</code> are <strong>non-overlapping</strong>.</li>
</ul>

from typing import List

class Solution:
    def maximumWhiteTiles(self, tiles: List[List[int]], carpetLen: int) -> int:
        # Step 1: Sort the tiles
        tiles.sort(key=lambda x: x[0])
        
        n = len(tiles)
        max_cover = 0
        
        # Step 2: Sliding window approach
        right = 0
        current_cover = 0
        
        for left in range(n):
            while right < n and tiles[right][1] - tiles[left][0] + 1 <= carpetLen:
                current_cover += tiles[right][1] - tiles[right][0] + 1
                right += 1
            
            # Step 3: Calculate the coverage for the current position
            if right < n and tiles[right][0] <= tiles[left][0] + carpetLen - 1:
                max_cover = max(max_cover, current_cover + (tiles[left][0] + carpetLen - 1 - tiles[right][0] + 1))
            else:
                max_cover = max(max_cover, current_cover)
            
            # Step 4: Update the current cover for the next iteration
            current_cover -= tiles[left][1] - tiles[left][0] + 1
            
        return max_cover

# Example 1
solution = Solution()
tiles = [[1,5],[10,11],[12,18],[20,25],[30,32]]
carpetLen = 10
print(solution.maximumWhiteTiles(tiles, carpetLen))  # Output: 9

# Example 2
tiles = [[10,11],[1,1]]
carpetLen = 2
print(solution.maximumWhiteTiles(tiles, carpetLen))  # Output: 2
41900