<p>You are given an <code>m x n</code> binary matrix <code>grid</code> where each cell is either <code>0</code> (empty) or <code>1</code> (occupied).</p>

<p>You are then given stamps of size <code>stampHeight x stampWidth</code>. We want to fit the stamps such that they follow the given <strong>restrictions</strong> and <strong>requirements</strong>:</p>

<ol>
	<li>Cover all the <strong>empty</strong> cells.</li>
	<li>Do not cover any of the <strong>occupied</strong> cells.</li>
	<li>We can put as <strong>many</strong> stamps as we want.</li>
	<li>Stamps can <strong>overlap</strong> with each other.</li>
	<li>Stamps are not allowed to be <strong>rotated</strong>.</li>
	<li>Stamps must stay completely <strong>inside</strong> the grid.</li>
</ol>

<p>Return <code>true</code> <em>if it is possible to fit the stamps while following the given restrictions and requirements. Otherwise, return</em> <code>false</code>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/11/03/ex1.png" style="width: 180px; height: 237px;" />
<pre>
<strong>Input:</strong> grid = [[1,0,0,0],[1,0,0,0],[1,0,0,0],[1,0,0,0],[1,0,0,0]], stampHeight = 4, stampWidth = 3
<strong>Output:</strong> true
<strong>Explanation:</strong> We have two overlapping stamps (labeled 1 and 2 in the image) that are able to cover all the empty cells.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/11/03/ex2.png" style="width: 170px; height: 179px;" />
<pre>
<strong>Input:</strong> grid = [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]], stampHeight = 2, stampWidth = 2 
<strong>Output:</strong> false 
<strong>Explanation:</strong> There is no way to fit the stamps onto all the empty cells without the stamps going outside the grid.
</pre>

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

<ul>
	<li><code>m == grid.length</code></li>
	<li><code>n == grid[r].length</code></li>
	<li><code>1 &lt;= m, n &lt;= 10<sup>5</sup></code></li>
	<li><code>1 &lt;= m * n &lt;= 2 * 10<sup>5</sup></code></li>
	<li><code>grid[r][c]</code> is either <code>0</code> or <code>1</code>.</li>
	<li><code>1 &lt;= stampHeight, stampWidth &lt;= 10<sup>5</sup></code></li>
</ul>

class Solution:
    def possibleToStamp(self, grid, stampHeight, stampWidth):
        m, n = len(grid), len(grid[0])
        prefix = [[0] * (n + 1) for _ in range(m + 1)]
        
        # Calculate prefix sum
        for i in range(1, m + 1):
            for j in range(1, n + 1):
                prefix[i][j] = prefix[i-1][j] + prefix[i][j-1] - prefix[i-1][j-1] + grid[i-1][j-1]
        
        def sum_region(r1, c1, r2, c2):
            return prefix[r2][c2] - prefix[r1-1][c2] - prefix[r2][c1-1] + prefix[r1-1][c1-1]
        
        # Check if a stamp can be placed at (i, j)
        can_place = [[False] * n for _ in range(m)]
        for i in range(m - stampHeight + 1):
            for j in range(n - stampWidth + 1):
                if sum_region(i + 1, j + 1, i + stampHeight, j + stampWidth) == 0:
                    can_place[i][j] = True
        
        # Check if all empty cells can be covered
        for i in range(m):
            for j in range(n):
                if grid[i][j] == 0:
                    found = False
                    for di in range(max(0, i - stampHeight + 1), min(m - stampHeight + 1, i + 1)):
                        for dj in range(max(0, j - stampWidth + 1), min(n - stampWidth + 1, j + 1)):
                            if can_place[di][dj]:
                                found = True
                                break
                        if found:
                            break
                    if not found:
                        return False
        return True
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