102774: [AtCoder]ABC277 E - Crystal Switches

Problem Statement

You are given an undirected graph consisting of $N$ vertices and $M$ edges.
For $i = 1, 2, \ldots, M$, the $i$-th edge is an undirected edge connecting vertex $u_i$ and $v_i$ that is initially passable if $a_i = 1$ and initially impassable if $a_i = 0$. Additionally, there are switches on $K$ of the vertices: vertex $s_1$, vertex $s_2$, $\ldots$, vertex $s_K$.

Takahashi is initially on vertex $1$, and will repeat performing one of the two actions below, Move or Hit Switch, which he may choose each time, as many times as he wants.

• Move: Choose a vertex adjacent to the vertex he is currently on via an edge, and move to that vertex.
• Hit Switch: If there is a switch on the vertex he is currently on, hit it. This will invert the passability of every edge in the graph. That is, a passable edge will become impassable, and vice versa.

Determine whether Takahashi can reach vertex $N$, and if he can, print the minimum possible number of times he performs Move before reaching vertex $N$.

Constraints

• $2 \leq N \leq 2 \times 10^5$
• $1 \leq M \leq 2 \times 10^5$
• $0 \leq K \leq N$
• $1 \leq u_i, v_i \leq N$
• $u_i \neq v_i$
• $a_i \in \lbrace 0, 1\rbrace$
• $1 \leq s_1 \lt s_2 \lt \cdots \lt s_K \leq N$
• All values in the input are integers.

Input

The input is given from Standard Input in the following format:

$N$ $M$ $K$
$u_1$ $v_1$ $a_1$
$u_2$ $v_2$ $a_2$
$\vdots$
$u_M$ $v_M$ $a_M$
$s_1$ $s_2$ $\ldots$ $s_K$

Output

If Takahashi cannot reach vertex $N$, print $-1$; if he can, print the minimum possible number of times he performs Move before reaching vertex $N$.

Sample Input 1

5 5 2
1 3 0
2 3 1
5 4 1
2 1 1
1 4 0
3 4

Sample Output 1

5

Takahashi can reach vertex $N$ as follows.

• Move from vertex $1$ to vertex $2$.
• Move from vertex $2$ to vertex $3$.
• Hit the switch on vertex $3$. This inverts the passability of every edge in the graph.
• Move from vertex $3$ to vertex $1$.
• Move from vertex $1$ to vertex $4$.
• Hit the switch on vertex $4$. This again inverts the passability of every edge in the graph.
• Move from vertex $4$ to vertex $5$.

Here, Move is performed five times, which is the minimum possible number.

Sample Input 2

4 4 2
4 3 0
1 2 1
1 2 0
2 1 1
2 4

Sample Output 2

-1

The given graph may be disconnected or contain multi-edges. In this sample input, there is no way for Takahashi to reach vertex $N$, so you should print $-1$.

问题描述

你有一个包含$N$个顶点和$M$条边的无向图。
对于$i = 1, 2, \ldots, M$，第$i$条边是连接顶点$u_i$和$v_i$的边，如果$a_i = 1$则初始可通行，如果$a_i = 0$则初始不可通行。 此外，图中有$K$个顶点有开关：顶点$s_1$，顶点$s_2$，$\ldots$，顶点$s_K$。

最初，Takahashi在顶点$1$，他会重复执行以下两个动作之一，移动或按开关，他可以在每次选择，多次执行。

• 移动：选择一个与当前所在顶点相邻的顶点，移动到那个顶点。
• 按开关：如果当前所在顶点有一个开关，按它。这将反转图中每条边的可通行性。即，可通行的边将变为不可通行，反之亦然。

确定Takahashi是否能够到达顶点$N$，如果可以，输出他到达顶点$N$前执行次数最少的次数。

约束

• $2 \leq N \leq 2 \times 10^5$
• $1 \leq M \leq 2 \times 10^5$
• $0 \leq K \leq N$
• $1 \leq u_i, v_i \leq N$
• $u_i \neq v_i$
• $a_i \in \lbrace 0, 1\rbrace$
• $1 \leq s_1 \lt s_2 \lt \cdots \lt s_K \leq N$
• 输入中的所有值都是整数。

输入

输入以标准输入给出以下格式：

$N$ $M$ $K$
$u_1$ $v_1$ $a_1$
$u_2$ $v_2$ $a_2$
$\vdots$
$u_M$ $v_M$ $a_M$
$s_1$ $s_2$ $\ldots$ $s_K$

输出

如果Takahashi不能到达顶点$N$，输出$-1$； 如果他可以到达，输出他到达顶点$N$前执行次数最少的次数。

样例输入 1

5 5 2
1 3 0
2 3 1
5 4 1
2 1 1
1 4 0
3 4

样例输出 1

5

Takahashi可以按照以下方式到达顶点$N$。

• 从顶点$1$移动到顶点$2$。
• 从顶点$2$移动到顶点$3$。
• 在顶点$3$上按下开关。这将反转图中每条边的可通行性。
• 从顶点$3$移动到顶点$1$。
• 从顶点$1$移动到顶点$4$。
• 在顶点$4$上按下开关。这再次反转图中每条边的可通行性。
• 从顶点$4$移动到顶点$5$。

在这里，移动执行了五次，这是最少的次数。

样例输入 2

4 4 2
4 3 0
1 2 1
1 2 0
2 1 1
2 4

样例输出 2

-1

给定的图可能是不连通的或包含多条边。 在这个样例输入中，Takahashi没有到达顶点$N$的方法，所以你应该输出$-1$。