<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>

class DSU {
    vector<int> parent;
    vector<int> size;
    public:
    DSU(int n) {
        parent.clear();
        size.clear();
        parent.resize(n);
        size.resize(n);
        for(int i = 0; i < n; i++) {
            parent[i] = i;
            size[i] = 1;
        }
    }

    int find_parent(int a) {
        if(a == parent[a]) {
            return a;
        }
        return parent[a] = find_parent(parent[a]);
    }

    void union_set(int a, int b) {
        int newA = find_parent(a);
        int newB = find_parent(b);
        if(newA != newB) {
            if(size[newA] < size[newB]) {
                parent[newA] = newB;
                size[newB] += size[newA];
            }
            if(size[newB] <= size[newA]) {
                parent[newB] = newA;
                size[newA] += size[newB];
            }
        }
    }
};

class Solution {
public:
    long long countPairs(int n, vector<vector<int>>& edges) {
        // Count number of connected components 
        DSU dsu(n);
        for(int i = 0; i < edges.size(); i++) {
            dsu.union_set(edges[i][0], edges[i][1]);
        }
        unordered_map<int, int> componentSizeMap;
        for(int i = 0; i < n; i++) {
            componentSizeMap[dsu.find_parent(i)]++;
        }
        vector<int> uniqueSizes;
        long long sum = 0;
        for(auto itr: componentSizeMap) {
            uniqueSizes.push_back(itr.second);
            sum += itr.second;
        }
        long long ans = 0LL;
        for(int i = 0; i < uniqueSizes.size(); i++) {
            ans += (long long)(sum - uniqueSizes[i]) * (uniqueSizes[i]);
            sum -= uniqueSizes[i];
        }
        return ans;
    }
};
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