Score : $600$ points

问题描述

我们有一个简单的连通无向图，包含 $N$ 个顶点和 $M$ 条边。
顶点按照 Vertex $1$, Vertex $2$, $\dots$, Vertex $N$ 编号。
边按照 Edge $1$, Edge $2$, $\dots$, Edge $M$ 编号。其中边 $i$ 双向连接顶点 $a_i$ 和顶点 $b_i$。不存在直接连接顶点 $1$ 和顶点 $N$ 的边。
每个顶点要么为空，要么被墙占据。最初，所有顶点都是空的。

Aoki 将沿图中的边从 Vertex $1$ 出发前往 Vertex $N$。但是，他不允许移动到被墙占据的顶点上。

Takahashi 决定选择一些顶点建造墙壁，以确保无论 Aoki 选择哪条路径都无法到达 Vertex $N$。
在顶点 $i$ 上建造一面墙需要 Takahashi 支付 $c_i$ 日元（日本货币）。他不能在顶点 $1$ 和顶点 $N$ 上建造墙壁。

请问为了满足条件，Takahashi 需要花费多少日元来建造墙壁？同时，请输出实现最小成本的建墙方式。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

We have a simple connected undirected graph with $N$ vertices and $M$ edges.
The vertices are numbered as Vertex $1$, Vertex $2$, $\dots$, Vertex $N$.
The edges are numbered as Edge $1$, Edge $2$, $\dots$, Edge $M$. Edge $i$ connects Vertex $a_i$ and Vertex $b_i$ bidirectionally. There is no edge that directly connects Vertex $1$ and Vertex $N$.
Each vertex is either empty or occupied by a wall. Initially, every vertex is empty.

Aoki is going to travel from Vertex $1$ to Vertex $N$ along the edges on the graph. However, Aoki is not allowed to move to a vertex occupied by a wall.

Takahashi has decided to choose some of the vertices to build walls on, so that Aoki cannot travel to Vertex $N$ no matter which route he takes.
Building a wall on Vertex $i$ costs Takahashi $c_i$ yen (the currency of Japan). He cannot build a wall on Vertex $1$ and Vertex $N$.

How many yens is required for Takahashi to build walls so that the conditions is satisfied? Also, print the way of building walls to achieve the minimum cost.

Constraints

• $3 \leq N \leq 100$
• $N - 1 \leq M \leq \frac{N(N-1)}{2} - 1$
• $1 \leq a_i \lt b_i \leq N$ $(1 \leq i \leq M)$
• $(a_i, b_i) \neq (1, N)$
• The given graph is simple and connected.
• $1 \leq c_{i} \leq 10^9$ $(2 \leq i \leq N-1)$
• $c_1 = c_N = 0$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$

$a_1$ $b_1$

$a_2$ $b_2$

$\vdots$

$a_M$ $b_M$

$c_1$ $c_2$ $\dots$ $c_N$

Output

Print in the following format. Here, $C,k$, and $p_i$ are defined as follows.

• $C$ is the cost that Takahashi will pay

• $k$ is the number of vertices for Takahashi to build walls on

• $(p_1,p_2,\dots,p_k)$ is a sequence of vertices on which Takahashi will build walls

$C$

$k$

$p_1$ $p_2$ $\dots$ $p_k$

If there are multiple ways to build walls to satisfy the conditions with the minimum cost, print any of them.

Sample Input 1

5 5
1 2
2 3
3 5
2 4
4 5
0 8 3 4 0

Sample Output 1

7
2
3 4

If Takahashi builds walls on Vertex $3$ and Vertex $4$, paying $3 + 4 = 7$ yen, Aoki is unable to travel from Vertex $1$ to Vertex $5$.
There is no way to satisfy the condition with less cost, so $7$ yen is the answer.

Sample Input 2

3 2
1 2
2 3
0 1 0

Sample Output 2

1
1
2

Sample Input 3

5 9
1 2
1 3
1 4
2 3
2 4
2 5
3 4
3 5
4 5
0 1000000000 1000000000 1000000000 0

Sample Output 3

3000000000
3
2 3 4

update @ 2024/3/10 10:19:55