Crystal Switches

AtCoder

IDabc277_e

Time2000ms

Memory256MB

Difficulty—

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

Samples

Input #1

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.

Input #2

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 $-1$.

API Response (JSON)

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    "name": "Crystal Switches",
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      "content": "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 pass",
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      "statement_type": "Markdown",
      "content": "You are given an undirected graph consisting of $N$ vertices and $M$ edges.  \nFor $i = 1, 2, \\ldots, M$, the $i$\\-th edge is an undirected edge connecting vertex $u_i$ and $v_i$ that is initially pass...",
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