<p>You are given an integer <code>n</code>. There is an <strong>undirected</strong> graph with <code>n</code> vertices, numbered from <code>0</code> to <code>n - 1</code>. You are given a 2D integer array <code>edges</code> where <code>edges[i] = [a<sub>i</sub>, b<sub>i</sub>]</code> denotes that there exists an <strong>undirected</strong> edge connecting vertices <code>a<sub>i</sub></code> and <code>b<sub>i</sub></code>.</p>

<p>Return <em>the number of <strong>complete connected components</strong> of the graph</em>.</p>

<p>A <strong>connected component</strong> is a subgraph of a graph in which there exists a path between any two vertices, and no vertex of the subgraph shares an edge with a vertex outside of the subgraph.</p>

<p>A connected component is said to be <b>complete</b> if there exists an edge between every pair of its vertices.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<p><strong class="example"><img alt="" src="https://assets.leetcode.com/uploads/2023/04/11/screenshot-from-2023-04-11-23-31-23.png" style="width: 671px; height: 270px;" /></strong></p>

<pre>
<strong>Input:</strong> n = 6, edges = [[0,1],[0,2],[1,2],[3,4]]
<strong>Output:</strong> 3
<strong>Explanation:</strong> From the picture above, one can see that all of the components of this graph are complete.
</pre>

<p><strong class="example">Example 2:</strong></p>

<p><strong class="example"><img alt="" src="https://assets.leetcode.com/uploads/2023/04/11/screenshot-from-2023-04-11-23-32-00.png" style="width: 671px; height: 270px;" /></strong></p>

<pre>
<strong>Input:</strong> n = 6, edges = [[0,1],[0,2],[1,2],[3,4],[3,5]]
<strong>Output:</strong> 1
<strong>Explanation:</strong> The component containing vertices 0, 1, and 2 is complete since there is an edge between every pair of two vertices. On the other hand, the component containing vertices 3, 4, and 5 is not complete since there is no edge between vertices 4 and 5. Thus, the number of complete components in this graph is 1.
</pre>

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

<ul>
	<li><code>1 &lt;= n &lt;= 50</code></li>
	<li><code>0 &lt;= edges.length &lt;= n * (n - 1) / 2</code></li>
	<li><code>edges[i].length == 2</code></li>
	<li><code>0 &lt;= a<sub>i</sub>, b<sub>i</sub> &lt;= n - 1</code></li>
	<li><code>a<sub>i</sub> != b<sub>i</sub></code></li>
	<li>There are no repeated edges.</li>
</ul>

class Solution:
    def countCompleteComponents(self, n: int, edges: List[List[int]]) -> int:
        parents = [None]*n
        rank = [None]*n # here we define rank[i] as the size of the tree represented by i

        def make_set(j: int):
            assert 0 <= j < n, "j must be bounded by n"
            assert parents[j] == None, "Seems you're `make` on `j` multiple times"
            parents[j] = j
            rank[j] = 1
        
        def find(j: int):
            assert 0 <= j < n, "j must be bounded by n"
            if parents[j] != j:
                parents[j] = find(parents[j])
            return parents[j]
        
        def union(j1: int, j2:  int):
            assert 0 <= j1 < n, "j1 must be bounded by n"
            assert 0 <= j1 < n, "j1 must be bounded by n"

            r1, r2 = find(j1), find(j2)
            if rank[r1] < rank[r2]:
                parents[r1] = r2
                rank[r2] += rank[r1]
                rank[r1] = 0
            else:
                parents[r2] = r1
                rank[r1] += rank[r2]
                rank[r2] = 0
        
        # make the disjoint forest
        for j in range(n):
            make_set(j)

        cc_size = rank
        cc_edges = [0] * n
        
        for u, v in edges:
            if find(u) != find(v):
                union(u, v)
            cc_edges[find(v)] += 1
        
        # a connected components with m nodes is complete iff the number of edges
        # in the components equal to m*(m-1)/2

        count = 0
        for i in range(0, n):
            m, n = cc_size[i], cc_edges[i]
            if (m > 0 or n > 0) and m*(m-1)/2 == n:
                count += 1
        
        return count

"""
Here we used a disjoint union-find data structure!
"""
        
599