Score : $500$ points

问题描述

你被给予一个由 $N$ 个顶点和 $M$ 条边构成的无向图。
对于 $i = 1, 2, \ldots, M$，第 $i$ 条边是一个无向边，连接顶点 $u_i$ 和 $v_i$，如果 $a_i = 1$ 则初始时该边是可通行的，如果 $a_i = 0$ 则初始时不可通行。另外，在 $K$ 个顶点上存在开关：顶点 $s_1$、顶点 $s_2$、$\ldots$、顶点 $s_K$。

高桥初始时位于顶点 $1$，他将重复执行以下两种动作之一——移动或击打开关，每次可以选择任一动作，并可以重复任意次数。

• 移动：选择当前所在顶点通过一条边相邻的一个顶点，并移动到那个顶点。
• 击打开关：如果当前所在顶点上有开关，则击打它。这将使图中每条边的通行性翻转。也就是说，原本可通行的边会变为不可通行，反之亦然。

确定高桥是否能够到达顶点 $N$，如果可以，请输出他在到达顶点 $N$ 前至少需要执行 移动 动作的次数。

以上为通义千问 qwen-max 翻译，仅供参考。

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 $-1$.

update @ 2024/3/10 11:40:01