<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:
        tiles.sort()

        n_tiles = len(tiles)
        dp = [0] * (n_tiles + 1)

        for i in range(n_tiles):
            start, end = tiles[i]
            dp[i + 1] = dp[i] + end - start + 1
        
        # print(dp)

        max_cover = 0 
        for left in range(n_tiles):
            start, _ = tiles[left]
            carpet_end = start + carpetLen
            right = self.search_right(tiles, left, carpet_end)

            _, end = tiles[right - 1]
            cover = dp[right] - dp[left] - max(0, end - carpet_end + 1)
            # print(carpet_end, left, right, cover)
            
            max_cover = max(max_cover, cover)
        
        return max_cover

    def search_right(self, tiles: List[List[int]], left: int, carpet_end: int) -> int:
        right = len(tiles)

        while left < right:
            mid = left + (right - left) // 2
            if carpet_end >= tiles[mid][0]:
                left = mid + 1
            else:
                right = mid
        
        return left 
992