<p>You are given a <strong>0-indexed binary</strong> string <code>floor</code>, which represents the colors of tiles on a floor:</p>

<ul>
	<li><code>floor[i] = &#39;0&#39;</code> denotes that the <code>i<sup>th</sup></code> tile of the floor is colored <strong>black</strong>.</li>
	<li>On the other hand, <code>floor[i] = &#39;1&#39;</code> denotes that the <code>i<sup>th</sup></code> tile of the floor is colored <strong>white</strong>.</li>
</ul>

<p>You are also given <code>numCarpets</code> and <code>carpetLen</code>. You have <code>numCarpets</code> <strong>black</strong> carpets, each of length <code>carpetLen</code> tiles. Cover the tiles with the given carpets such that the number of <strong>white</strong> tiles still visible is <strong>minimum</strong>. Carpets may overlap one another.</p>

<p>Return <em>the <strong>minimum</strong> number of white tiles still visible.</em></p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2022/02/10/ex1-1.png" style="width: 400px; height: 73px;" />
<pre>
<strong>Input:</strong> floor = &quot;10110101&quot;, numCarpets = 2, carpetLen = 2
<strong>Output:</strong> 2
<strong>Explanation:</strong> 
The figure above shows one way of covering the tiles with the carpets such that only 2 white tiles are visible.
No other way of covering the tiles with the carpets can leave less than 2 white tiles visible.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2022/02/10/ex2.png" style="width: 353px; height: 123px;" />
<pre>
<strong>Input:</strong> floor = &quot;11111&quot;, numCarpets = 2, carpetLen = 3
<strong>Output:</strong> 0
<strong>Explanation:</strong> 
The figure above shows one way of covering the tiles with the carpets such that no white tiles are visible.
Note that the carpets are able to overlap one another.
</pre>

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

<ul>
	<li><code>1 &lt;= carpetLen &lt;= floor.length &lt;= 1000</code></li>
	<li><code>floor[i]</code> is either <code>&#39;0&#39;</code> or <code>&#39;1&#39;</code>.</li>
	<li><code>1 &lt;= numCarpets &lt;= 1000</code></li>
</ul>

class Solution:
    def minimumWhiteTiles(self, floor: str, numCarpets: int, carpetLen: int):

        @lru_cache(None)
        def rec(current_idx, carpet_left):
            if current_idx >= len(floor):
                return 0
            
            white_kept = 0
            min_this = float("inf")
            
            for start_idx in range(current_idx, len(floor)):
                if floor[start_idx] == "1" :
                    if carpet_left:
                        min_this = min(
                            min_this,
                            white_kept + rec(
                                start_idx+carpetLen,
                                carpet_left - 1
                            )
                        )
                        
                        if floor[start_idx:start_idx+carpetLen] == "1" * carpetLen:
                            return min_this
                    
                    white_kept += 1
            
            return min(white_kept, min_this)  
            
        return rec(0, numCarpets) 
        
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