<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:
        # tile[i] = [li, ri] every tile in the range li <= tile <= ri colored white
        # max number of white tiles that can be covered by the carpet
        # what are the gaps, min gaps

        # how do we find 

        # can we bin search it?

        tiles.sort()
        n = len(tiles)
        res = 0
        # we can build a prefix sum to keep track of all sums of the tiles
        # prefix = [0]
        # for start, end in tiles:            
        #     prefix.append(prefix[-1] + end - start + 1)
        

        # for i, (start, end) in enumerate(tiles):
        #     total = 0
        #     # we try and place the carpet on the first part of the 
        #     idx = bisect_right(tiles, start + carpetLen-1, key=lambda x: x[0]) - 1

        #     # fully covered
        #     total += prefix[idx] - prefix[i]

        #     # partial covered
        #     total += min(tiles[idx][1]-tiles[idx][0]+1, start + carpetLen-1 - tiles[idx][0]+1)

        #     res = max(res, total)


        # Approach 2. Sliding window
        total = 0
        j = 0
        for i, (start, end) in enumerate(tiles):
            while j < n and tiles[j][0] < start + carpetLen:
                total += tiles[j][1]-tiles[j][0] + 1
                j += 1

            # we have all tiles that start
            # see if we need to remove part of the last tile
            res = max(res, total - max(0, tiles[j-1][1] - (start + carpetLen - 1)))

            total -= tiles[i][1]-tiles[i][0] + 1
        return res

        
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