<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: List[List[int]], stampHeight: int, stampWidth: int) -> bool:
        n = len(grid)
        m = len(grid[0])
        sum_occupied = [[0] * (m+1) for _ in range(n+1)]
        ok_stamp = [[0] * (m+1) for _ in range(n+1)]
        for i in range(1, n+1):
            for j in range(1, m+1):
                sum_occupied[i][j] = grid[i-1][j-1] + sum_occupied[i-1][j] + sum_occupied[i][j-1] - sum_occupied[i-1][j-1]
                if i >= stampHeight and j >= stampWidth:
                    ok_stamp[i][j] = 1 if sum_occupied[i][j] - sum_occupied[i-stampHeight][j] - sum_occupied[i][j-stampWidth] + sum_occupied[i-stampHeight][j-stampWidth] == 0 else 0
                    ok_stamp[i][j] = ok_stamp[i][j] + ok_stamp[i-1][j] + ok_stamp[i][j-1] - ok_stamp[i-1][j-1]

        for i in range(1, n+1):
            for j in range(1, m+1):
                if grid[i-1][j-1]:
                    continue
                r = min(m, j + stampWidth - 1)
                d = min(n, i + stampHeight - 1)
                if ok_stamp[d][r] - ok_stamp[i-1][r] - ok_stamp[d][j-1] + ok_stamp[i-1][j-1] == 0:
                    return False
            
        return True

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