Score : $600$ points

问题描述

我们有一个有向图，包含 $N$ 个顶点和 $M$ 条边。顶点从 $1$ 到 $N$ 编号，第 $i$ 条边从顶点 $U_i$ 指向顶点 $V_i$。

你现在位于顶点 $1$。判断你是否能够执行以下操作 $10^{10^{100}}$ 次，并最终回到顶点 $1$：

• 从当前所在的顶点选择一条出发的边，并沿着这条边移动到它所指向的顶点。

给定 $T$ 组测试用例，请分别解决每组问题。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

We have a directed graph with $N$ vertices and $M$ edges. The vertices are numbered from $1$ through $N$, and the $i$-th edge goes from vertex $U_i$ to vertex $V_i$.

You are currently at vertex $1$. Determine if you can make the following move $10^{10^{100}}$ times to end up at vertex $1$:

• choose an edge going from the vertex you are currently at, and move to the vertex that the edge points at.

Given $T$ test cases, solve each of them.

Constraints

• All input values are integers.
• $1\leq T \leq 2\times 10^5$
• $2\leq N \leq 2\times 10^5$
• $1\leq M \leq 2\times 10^5$
• The sum of $N$ over all test cases is at most $2 \times 10^5$.
• The sum of $M$ over all test cases is at most $2 \times 10^5$.
• $1 \leq U_i, V_i \leq N$
• $U_i \neq V_i$
• If $i\neq j$, then $(U_i,V_i) \neq (U_j,V_j)$.

Input

The input is given from Standard Input in the following format. Here, $\text{test}_i$ denotes the $i$-th test case.

$T$

$\text{test}_1$

$\text{test}_2$

$\vdots$

$\text{test}_T$

Each test case is given in the following format.

$N$ $M$

$U_1$ $V_1$

$U_2$ $V_2$

$\vdots$

$U_M$ $V_M$

Output

Print $T$ lines.

The $i$-th $(1\leq i \leq T)$ line should contain Yes if you can make the move described in the problem statement $10^{10^{100}}$ times to end up at vertex $1$, and No otherwise.

Sample Input 1

4
2 2
1 2
2 1
3 3
1 2
2 3
3 1
7 10
1 6
6 3
1 4
5 1
7 1
4 5
2 1
4 7
2 7
4 3
7 11
1 6
6 3
1 4
5 1
7 1
4 5
2 1
4 7
2 7
4 3
3 7

Sample Output 1

Yes
No
No
Yes

For the $1$-st test case,

• You inevitably repeat visiting vertices $1 \rightarrow 2 \rightarrow 1 \rightarrow \dots$. Thus, you will end up at vertex $1$ after making the move $10^{10^{100}}$ times, so the answer is Yes.

For the $2$-nd test case,

• You inevitably repeat visiting vertices $1 \rightarrow 2 \rightarrow 3 \rightarrow 1 \rightarrow \dots$. Thus, you will end up at vertex $2$ after making the move $10^{10^{100}}$ times, so the answer is No.

update @ 2024/3/10 08:39:51