Score : $600$ points

问题描述

我们有一个包含 $N$ 个顶点的有向图。这 $N$ 个顶点分别称为顶点 1、顶点 2、……、顶点 $N$。在时间 $0$，该图没有任何边。

对于每个时刻 $t = 1, 2, \ldots, T$，在时间 $t$，将从顶点 $u_t$ 到顶点 $v_t$ 添加一条有向边。（这条边可以是自环，即可能存在 $u_t = v_t$ 的情况。）

若一个顶点可以从顶点 1 出发，沿着边 恰好 走过 $L$ 次到达，则称这个顶点为 好 顶点。

对于每个 $i = 1, 2, \ldots, N$，请输出顶点 $i$ 首次成为好顶点的时间。如果顶点 $i$ 从未成为好顶点，请输出 $-1$。

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. The $N$ vertices are called Vertex $1$, Vertex $2$, $\ldots$, Vertex $N$. At time $0$, the graph has no edge.

For each $t = 1, 2, \ldots, T$, at time $t$, a directed edge from Vertex $u_t$ to Vertex $v_t$ will be added. (The edge may be a self-loop, that is, $u_t = v_t$ may hold.)

A vertex is called good when it can be reached by starting at Vertex $1$ and traversing an edge exactly $L$ times.

For each $i = 1, 2, \ldots, N$, print the earliest time when Vertex $i$ is good. If there is no time when Vertex $i$ is good, print $-1$ instead.

Constraints

• $2 \leq N \leq 100$
• $1 \leq T \leq N^2$
• $1 \leq L \leq 10^9$
• $1 \leq u_t, v_t \leq N$
• $i \neq j \Rightarrow (u_i, v_i) \neq (u_j, v_j)$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $T$ $L$

$u_1$ $v_1$

$u_2$ $v_2$

$\vdots$

$u_T$ $v_T$

Output

In the following format, for each $i = 1, 2, \ldots, N$, print the earliest time $X_i$ when Vertex $i$ is good. If there is no time when Vertex $i$ is good, $X_i$ should be $-1$.

$X_1$ $X_2$ $\ldots$ $X_N$

Sample Input 1

4 5 3
2 3
3 4
1 2
3 2
2 2

Sample Output 1

-1 4 5 3

At time $0$, the graph has no edge. Afterward, edges are added as follows.

• At time $1$, a directed edge from Vertex $2$ to Vertex $3$ is added.
• At time $2$, a directed edge from Vertex $3$ to Vertex $4$ is added.
• At time $3$, a directed edge from Vertex $1$ to Vertex $2$ is added. Now, Vertex $4$ can be reached from Vertex $1$ in exactly three moves: $1 \rightarrow 2 \rightarrow 3 \rightarrow 4$, making Vertex $4$ good.
• At time $4$, a directed edge from Vertex $3$ to Vertex $2$ is added. Now, Vertex $2$ can be reached from Vertex $1$ in exactly three moves: $1 \rightarrow 2 \rightarrow 3 \rightarrow 2$, making Vertex $2$ good.
• At time $5$, a directed edge (self-loop) from Vertex $2$ to Vertex $2$ is added. Now, Vertex $3$ can be reached from Vertex $1$ in exactly three moves: $1 \rightarrow 2 \rightarrow 2 \rightarrow 3$, making Vertex $3$ good.

Vertex $1$ will never be good.

Sample Input 2

2 1 1000000000
1 2

Sample Output 2

-1 -1

update @ 2024/3/10 10:14:40