<p>You are given an integer <code>n</code>. There is an <strong>undirected</strong> graph with <code>n</code> nodes, 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 nodes <code>a<sub>i</sub></code> and <code>b<sub>i</sub></code>.</p>

<p>Return <em>the <strong>number of pairs</strong> of different nodes that are <strong>unreachable</strong> from each other</em>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2022/05/05/tc-3.png" style="width: 267px; height: 169px;" />
<pre>
<strong>Input:</strong> n = 3, edges = [[0,1],[0,2],[1,2]]
<strong>Output:</strong> 0
<strong>Explanation:</strong> There are no pairs of nodes that are unreachable from each other. Therefore, we return 0.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2022/05/05/tc-2.png" style="width: 295px; height: 269px;" />
<pre>
<strong>Input:</strong> n = 7, edges = [[0,2],[0,5],[2,4],[1,6],[5,4]]
<strong>Output:</strong> 14
<strong>Explanation:</strong> There are 14 pairs of nodes that are unreachable from each other:
[[0,1],[0,3],[0,6],[1,2],[1,3],[1,4],[1,5],[2,3],[2,6],[3,4],[3,5],[3,6],[4,6],[5,6]].
Therefore, we return 14.
</pre>

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

<ul>
	<li><code>1 &lt;= n &lt;= 10<sup>5</sup></code></li>
	<li><code>0 &lt;= edges.length &lt;= 2 * 10<sup>5</sup></code></li>
	<li><code>edges[i].length == 2</code></li>
	<li><code>0 &lt;= a<sub>i</sub>, b<sub>i</sub> &lt; n</code></li>
	<li><code>a<sub>i</sub> != b<sub>i</sub></code></li>
	<li>There are no repeated edges.</li>
</ul>

from collections import defaultdict
class Solution:
    def countPairs(self, n: int, edges: List[List[int]]) -> int:
        # find the number of nodes in each group and at the end multiply each and return sum

        adj_list = defaultdict(list)
        nodes = set()
        for x, y in edges:
            adj_list[x].append(y)
            adj_list[y].append(x)

            nodes.add(x)
            nodes.add(y)

        # determine sizes
        sizes = []
        seen = set()

        def bfs(node):
            q = deque()
            q.append(node)
            count = 0
            while q:
                node = q.popleft()
                if node in seen:
                    continue
                seen.add(node)
                count += 1

                for nei in adj_list[node]:
                    if nei not in seen:
                        q.append(nei)
            return count

        for node in nodes:
            if node not in seen:
                sizes.append(bfs(node))
        
        added = 0
        while len(seen) + added < n:
            sizes.append(1)
            added += 1
        
        total_pairs = n * (n - 1) // 2

        # Subtract the pairs within each connected component
        for size in sizes:
            total_pairs -= size * (size - 1) // 2

        return total_pairs
78200