<p>Given an <code>m x n</code> <code>matrix</code>, return <em>a new matrix </em><code>answer</code><em> where </em><code>answer[row][col]</code><em> is the </em><em><strong>rank</strong> of </em><code>matrix[row][col]</code>.</p>

<p>The <strong>rank</strong> is an <strong>integer</strong> that represents how large an element is compared to other elements. It is calculated using the following rules:</p>

<ul>
	<li>The rank is an integer starting from <code>1</code>.</li>
	<li>If two elements <code>p</code> and <code>q</code> are in the <strong>same row or column</strong>, then:
	<ul>
		<li>If <code>p &lt; q</code> then <code>rank(p) &lt; rank(q)</code></li>
		<li>If <code>p == q</code> then <code>rank(p) == rank(q)</code></li>
		<li>If <code>p &gt; q</code> then <code>rank(p) &gt; rank(q)</code></li>
	</ul>
	</li>
	<li>The <strong>rank</strong> should be as <strong>small</strong> as possible.</li>
</ul>

<p>The test cases are generated so that <code>answer</code> is unique under the given rules.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/10/18/rank1.jpg" style="width: 442px; height: 162px;" />
<pre>
<strong>Input:</strong> matrix = [[1,2],[3,4]]
<strong>Output:</strong> [[1,2],[2,3]]
<strong>Explanation:</strong>
The rank of matrix[0][0] is 1 because it is the smallest integer in its row and column.
The rank of matrix[0][1] is 2 because matrix[0][1] &gt; matrix[0][0] and matrix[0][0] is rank 1.
The rank of matrix[1][0] is 2 because matrix[1][0] &gt; matrix[0][0] and matrix[0][0] is rank 1.
The rank of matrix[1][1] is 3 because matrix[1][1] &gt; matrix[0][1], matrix[1][1] &gt; matrix[1][0], and both matrix[0][1] and matrix[1][0] are rank 2.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/10/18/rank2.jpg" style="width: 442px; height: 162px;" />
<pre>
<strong>Input:</strong> matrix = [[7,7],[7,7]]
<strong>Output:</strong> [[1,1],[1,1]]
</pre>

<p><strong class="example">Example 3:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/10/18/rank3.jpg" style="width: 601px; height: 322px;" />
<pre>
<strong>Input:</strong> matrix = [[20,-21,14],[-19,4,19],[22,-47,24],[-19,4,19]]
<strong>Output:</strong> [[4,2,3],[1,3,4],[5,1,6],[1,3,4]]
</pre>

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

<ul>
	<li><code>m == matrix.length</code></li>
	<li><code>n == matrix[i].length</code></li>
	<li><code>1 &lt;= m, n &lt;= 500</code></li>
	<li><code>-10<sup>9</sup> &lt;= matrix[row][col] &lt;= 10<sup>9</sup></code></li>
</ul>

class Solution:
    def matrixRankTransform(self, matrix: List[List[int]]) -> List[List[int]]:

        def build_graph(matrix):
            graph = {}
            m = len(matrix)
            n = len(matrix[0])
            for i in range(m):
                for j in range(n):
                    val =  matrix[i][j]
                    graph[val] = graph.get(val, [])
                    graph[val].append((i,j))
            return graph

        def find(node, parent):
            if node == parent[node]:
                return node
            parent[node] = find(parent[node],parent)
            return parent[node]
        
        def union(node1, node2, parent):
            parent1, parent2 = find(node1, parent), find(node2, parent)
            if parent1 == parent2:
                return parent1
            parent[parent1] = parent2
            return parent1


        def solve(matrix):
            graph = build_graph(matrix)
            values = sorted(graph.keys())
            print(values)
            m,n = len(matrix), len(matrix[0])
            rank = [0]*(m+n)
            parent = [ i for i in range(m+n)] 
            for val in values:
                rank2 = [i  for i in rank]
                parent = [ i for i in range(m+n)] 
                for row,col in graph[val]:
                    parent_row, parent_col = find(row, parent), find(col+m, parent)
                    union(row, col+m, parent)
                    # print(val, row,col, parent_row, parent_col, rank[parent_row], rank[parent_col])
                    max_rank = max(rank[parent_row], rank[parent_col], rank2[parent_row] ,rank2[parent_col])
                    rank2[parent_row] ,rank2[parent_col]= max_rank, max_rank
                    # print(rank2[parent_row], rank2[parent_col], rank2)
                
                for row, col in graph[val]:
                    # print(val, rank2[find(row, parent)], rank2[find(col+m, parent)])
                    rank[row] = rank[col+m] = matrix[row][col] = max(rank2[find(row, parent)], rank2[find(col+m, parent)]) +1
                # print(val, parent, graph[val] , rank)
            return matrix
        return solve(matrix)

            
[
    [-37,-50,-3,44],
    [-37,46,13,-32],
    [47,-42,-3,-40],
    [-17,-22,-39,24]
    ]
[
    [2,1,4,6],
    [2,6,5,4],
    [5,2,4,3],
    [4,3,1,5]
    ]
[
    [1,1,4,6],
    [1,6,5,4],
    [5,2,4,3],
    [4,3,1,5]
    ]

        
53700