<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 prefix_sum(self, grid: List[List[int]]) -> List[List[int]]:
            ps = [[grid[row][col] for col in range(len(grid[0]))]for row in range(len(grid))]
            
            for row in range(len(grid)):
                for col in range(1, len(grid[0])):
                    ps[row][col] = ps[row][col-1] + grid[row][col]
            
            for row in range(1, len(grid)):
                for col in range(len(grid[0])):
                    ps[row][col] = ps[row-1][col] + ps[row][col]
            
            return ps
			
    def sumRegion(self, ps, row1: int, col1: int, row2: int, col2: int) -> int:
            ans = 0
            if row1 == 0 and col1 == 0:
                ans = ps[row2][col2]
            elif row1 == 0:
                ans = ps[row2][col2] - ps[row2][col1-1]
            elif col1 == 0:
                ans = ps[row2][col2] - ps[row1-1][col2]
            else:
                ans = ps[row2][col2] - ps[row1-1][col2] - ps[row2][col1-1] + ps[row1-1][col1-1]
            return ans

    def possibleToStamp(self, grid: List[List[int]], stampHeight: int, stampWidth: int) -> bool:
        diff = [[0 for col in range(len(grid[0])+1)]for row in range(len(grid)+1)]
        
        ps = self.prefix_sum(grid)
        cover = 0
        
        for row in range(len(grid)-(stampHeight-1)):
            for col in range(len(grid[0])-(stampWidth-1)):
                sub_sum = self.sumRegion(ps, row, col, row+stampHeight-1, col+stampWidth-1)
                if sub_sum == 0:
                    diff[row][col] += 1
                    diff[row][col+stampWidth] -= 1
                    diff[row+stampHeight][col] -= 1
                    diff[row+stampHeight][col+stampWidth] = 1
        pref_diff = self.prefix_sum(diff)
        m, n = len(grid), len(grid[0])
        
        for row in range(len(grid)):
            for col in range(len(grid[0])):
                if grid[row][col] == 0 and pref_diff[row][col] == 0: return False 
        
        return True
67000