<p>You are given an integer <code>n</code>. You have an <code>n x n</code> binary grid <code>grid</code> with all values initially <code>1</code>&#39;s except for some indices given in the array <code>mines</code>. The <code>i<sup>th</sup></code> element of the array <code>mines</code> is defined as <code>mines[i] = [x<sub>i</sub>, y<sub>i</sub>]</code> where <code>grid[x<sub>i</sub>][y<sub>i</sub>] == 0</code>.</p>

<p>Return <em>the order of the largest <strong>axis-aligned</strong> plus sign of </em>1<em>&#39;s contained in </em><code>grid</code>. If there is none, return <code>0</code>.</p>

<p>An <strong>axis-aligned plus sign</strong> of <code>1</code>&#39;s of order <code>k</code> has some center <code>grid[r][c] == 1</code> along with four arms of length <code>k - 1</code> going up, down, left, and right, and made of <code>1</code>&#39;s. Note that there could be <code>0</code>&#39;s or <code>1</code>&#39;s beyond the arms of the plus sign, only the relevant area of the plus sign is checked for <code>1</code>&#39;s.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/06/13/plus1-grid.jpg" style="width: 404px; height: 405px;" />
<pre>
<strong>Input:</strong> n = 5, mines = [[4,2]]
<strong>Output:</strong> 2
<strong>Explanation:</strong> In the above grid, the largest plus sign can only be of order 2. One of them is shown.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/06/13/plus2-grid.jpg" style="width: 84px; height: 85px;" />
<pre>
<strong>Input:</strong> n = 1, mines = [[0,0]]
<strong>Output:</strong> 0
<strong>Explanation:</strong> There is no plus sign, so return 0.
</pre>

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

<ul>
	<li><code>1 &lt;= n &lt;= 500</code></li>
	<li><code>1 &lt;= mines.length &lt;= 5000</code></li>
	<li><code>0 &lt;= x<sub>i</sub>, y<sub>i</sub> &lt; n</code></li>
	<li>All the pairs <code>(x<sub>i</sub>, y<sub>i</sub>)</code> are <strong>unique</strong>.</li>
</ul>

class Solution:
    def orderOfLargestPlusSign(self, n: int, mines: List[List[int]]) -> int:
        
        csum = [[[0, 0] for _ in range(n)] for _ in range(n)]
        rsum = [[[0, 0] for _ in range(n)] for _ in range(n)]
        mines_set = set()
        for x, y in mines:
            mines_set.add((x, y))

        for r in range(n):
            for c in range(n):
                if (r, c) not in mines_set:
                    csum[r][c][0] = csum[r - 1][c][0] + 1 if r > 0 else 1
                    csum[r][c][1] = csum[r][c - 1][1] + 1 if c > 0 else 1
        for r in range(n - 1, -1 , -1):
            for c in range(n - 1, -1 , -1):
                if (r, c) not in mines_set:
                    rsum[r][c][0] = rsum[r + 1][c][0] + 1 if r < n - 1 else 1
                    rsum[r][c][1] = rsum[r][c + 1][1] + 1 if c < n - 1 else 1
        
        max_size = 0
        for r in range(n):
            for c in range(n):
                if (r, c) in mines_set:
                    continue
                if r == 0 or r ==  n - 1 or c == 0 or c == n - 1:
                    max_size = max(max_size, 1)
                else:
                    size = 1 + min(csum[r -1][c][0], csum[r][c-1][1], rsum[r + 1][c][0], rsum[r][c + 1][1])
                    max_size = max(max_size, size)
        
        return max_size
        
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