<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:
        
        memo = {}
        
        mines = set([(x, y) for x, y in mines])
        
        if len(mines) == n * n:
            return 0
        
        final = 1
        
        for i in range(1, n - 1):
            for j in range(1, n - 1):
                
                if (i, j) not in mines:
                    
                    largest = min( \
                              self.dfs(i, j, 'R', memo, mines, n),
                              self.dfs(i, j, 'D', memo, mines, n),
                              self.dfs(i, j, 'L', memo, mines, n),
                              self.dfs(i, j, 'U', memo, mines, n))
                    final = max(final, largest)
        return final
    
    
    def dfs(self, i, j, dirct, memo, mines, n):
                
        if (i, j, dirct) in memo:
            return memo[(i, j, dirct)]
        
        if not (0 <= i < n and 0 <= j < n):
            return 0
        
        if (i, j) in mines:
            return 0
        
        if dirct == 'R':
            res = 1 + self.dfs(i, j + 1, dirct, memo, mines, n)
        
        elif dirct == 'D':
            res = 1 + self.dfs(i + 1, j, dirct, memo, mines, n)
            
        elif dirct == 'L':
            res = 1 + self.dfs(i, j - 1, dirct, memo, mines, n)
            
        elif dirct == 'U':
            res = 1 + self.dfs(i - 1, j, dirct, memo, mines, n)
            
        memo[(i, j, dirct)] = res
        
        return res
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