<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:
        mineSet = set([(i, j) for i, j in mines])
        
        def horizontalRange(i, j):
            if (i, j) in mineSet:
                return (-1,-1)
            jmax = j
            while jmax + 1 < n and (i, jmax+1) not in mineSet:
                jmax += 1
            jmin = j
            while jmin - 1 >= 0 and (i, jmin-1) not in mineSet:
                jmin -= 1
            return (jmin, jmax)
                


        def verticalRange(i, j):
            if (i, j) in mineSet:
                return (-1,-1)
            imax = i
            while imax + 1 < n and (imax+1, j) not in mineSet:
                imax += 1
            imin = i
            while imin - 1 >= 0 and (imin-1, j) not in mineSet:
                imin -= 1
            return (imin, imax)

        HRange = dict()
        VRange = dict()

        for i in range(n):
            for j in range(n):
                if (i,j) not in HRange:
                    h = horizontalRange(i,j)
                    for jj in range(h[0], h[1]+1):
                        HRange[(i,jj)] = h
                if (i,j) not in VRange:
                    v = verticalRange(i,j)
                    for ii in range(v[0], v[1]+1):
                        VRange[(ii,j)] = v

        res = 0
        for i in range(n):
            for j in range(n):
                if (i,j) in mineSet:
                    continue
                jmin, jmax = HRange[(i,j)]
                imin, imax = VRange[(i,j)]
                cross = 1 + min(imax-i, i-imin, j-jmin, jmax-j)
                # minimum length of 4 arms, exclude center point
                # add one is the cross size
                res = max(res, cross)

        return res




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