<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:
        m = len(grid)
        n = len(grid[0])
        s = 0
        for i in range(m):
            for j in range(n):
                s += grid[i][j]
        if s == m*n:return True

        if stampHeight > m or stampWidth > n:
            return False
            
        mark = [[0 for i in range(n)] for j in range(m)]
        psum = [[0 for i in range(n+1)] for j in range(m+1)]
        for i in range(m):
            for j in range(n):
                psum[i+1][j+1] = grid[i][j] + psum[i][j+1]+psum[i+1][j] - psum[i][j]
                possible = False
                if i+1 >= stampHeight and j+1 >= stampWidth:
                    if psum[i+1][j+1] - psum[i+1][j+1-stampWidth] - psum[i+1 -stampHeight][j+1] + psum[i+1 -stampHeight][j+1-stampWidth] == 0:
                        possible = True
                mark[i][j] = int(possible)
        psum2 = [[0 for i in range(n+1)] for j in range(m+1)]
        for i in range(m-1,-1,-1):
            for j in range(n-1,-1,-1):
                psum2[i][j] = mark[i][j] + psum2[i][j+1]+psum2[i+1][j] - psum2[i+1][j+1]
                if grid[i][j] == 1:
                    continue
                possible = False
                seen = psum2[i][j]
                if i + stampHeight <= m:
                    seen -= psum2[i+stampHeight][j]
                if j + stampWidth <= n:
                    seen -= psum2[i][j+stampWidth]
                if  i + stampHeight <= m and j + stampWidth <= n:
                    seen += psum2[i+stampHeight][j+stampWidth]
                if seen < 1:return False


        return True
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