You are given an integer n. There is an undirected graph with n nodes, numbered from 0 to n - 1. You are given a 2D integer array edges where edges[i] = [ai, bi] denotes that there exists an undirected edge connecting nodes ai and bi.
Return the number of pairs of different nodes that are unreachable from each other.
Example 1:
Input: n = 3, edges = [[0,1],[0,2],[1,2]]
Output: 0
Explanation: There are no pairs of nodes that are unreachable from each other. Therefore, we return 0.
Example 2:
Input: n = 7, edges = [[0,2],[0,5],[2,4],[1,6],[5,4]]
Output: 14
Explanation: 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.
Constraints:
1 <= n <= 105
0 <= edges.length <= 2 * 105
edges[i].length == 2
0 <= ai, bi < n
ai != bi
There are no repeated edges.
Solutions
Solution 1: DFS
For any two nodes in an undirected graph, if there is a path between them, then they are mutually reachable.
Therefore, we can use depth-first search to find the number of nodes $t$ in each connected component, and then multiply the current number of nodes $t$ in the connected component by the number of nodes $s$ in all previous connected components to obtain the number of unreachable node pairs in the current connected component, which is $s \times t$. Then, we add $t$ to $s$ and continue to search for the next connected component until all connected components have been searched, and we can obtain the final answer.
The time complexity is $O(n + m)$, and the space complexity is $O(n + m)$. Here, $n$ and $m$ are the number of nodes and edges, respectively.