DreamGrid has just found an undirected simple graph with $n$ vertices and no edges (that's to say, i

最新推荐文章于 2022-07-06 10:27:49 发布

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本文介绍了一种在无向简单图上进行操作和查询的方法，包括连接顶点和查询添加新边后连通分量的可能最小和最大数量。通过一系列操作，探讨了图的连通性和变化趋势。

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DreamGrid has just found an undirected simple graph with $n$ vertices and no edges (that’s to say, it’s a graph with $n$ isolated vertices) in his right pocket, where the vertices are numbered from 1 to $n$ . Now he would like to perform $q$ operations of the following two types on the graph:

1 a b – Connect the $a$ -th vertex and the $b$ -th vertex by an edge. It’s guaranteed that before this operation, there does not exist an edge which connects vertex $a$ and $b$ directly.
2 k – Find the answer for the query: What’s the minimum and maximum possible number of connected components after adding $k$ new edges to the graph. Note that after adding the $k$ edges, the graph must still be a simple graph, and the query does NOT modify the graph.
Please help DreamGrid find the answer for each operation of the second type. Recall that a simple graph is a graph with no self loops or multiple edges.

Input
There are multiple test cases. The first line of the input is an integer $T$ , indicating the number of test cases. For each test case:

The first line contains two integers $n$ and $q$ ( $\le n \le 10^5$ , $\le q \le 2 \times 10^5$ ), indicating the number of vertices and the number of operations.

For the following $q$ lines, the $i$ -th line first contains an integer $p_i$ ( $pi∈{1,2}p_i \in \{1, 2\}$ ), indicating the type of the $i$ -th operation.

If $p_i = 1$ , two integers $a_i$ and $b_i$ follow ( $\le a_i, b_i \le n$ , $ai≠bia_i \ne b_i$ ), indicating an operation of the first type. It’s guaranteed that before this operation, there does not exist an edge which connects vertex $a$ and $b$ directly.
If $p_i = 2$ , one integer $k_i$ follows ( $\le k_i \le \frac{n(n-1)}{2}$ ), indicating an operation of the second type. It’s guaranteed that after adding $k_i$ edges to the graph, the graph is still possible to be a simple graph.
It’s guaranteed that the sum of $n$ in all test cases will not exceed $10^6$ , and the sum of $q$ in all test cases will not exceed $\times 10^6$ .

Output
For each operation of the second type output one line containing two integers separated by a space, indicating the minimum and maximum possible number of connected components in this query.

Sample Input
1
5 5
1 1 2
2 1
1 1 3
2 1
2 3
Sample Output
3 3
2 3
1 2