<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>

class Solution:
    def maximumWhiteTiles(self, tiles: List[List[int]], carpetLen: int) -> int:
        s = max_covered = 0
        prefix_sums = []
        tiles.sort()
        merged_tiles = []
        max_r = -1

        for l, r in tiles:
            if not merged_tiles or l > max_r + 1:
                merged_tiles.append([l, r])
            else:
                merged_tiles[-1][1] = max(max_r, r)
            max_r = max(max_r, r)

        print(merged_tiles)
        for l, r in merged_tiles:
            s += r - l + 1
            prefix_sums.append(s)

        for i, interval in enumerate(merged_tiles):
            l, r = interval
            covered = min(l + carpetLen - 1, r) - l + 1
            index = bisect.bisect_left(merged_tiles, [l + carpetLen, l + carpetLen], lo=i) - 1
            if index > i:
                l_end, r_end = merged_tiles[index]
                end = min(l + carpetLen - 1, r_end)
                covered += end - l_end + 1
                if index > i + 1:
                    covered += prefix_sums[index - 1] - prefix_sums[i]
            max_covered = max(max_covered, covered)
        
        return max_covered


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