102396: [AtCoder]ABC239 G - Builder Takahashi

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

问题描述

我们有一个简单的连通无向图，有$N$个顶点和$M$条边。

顶点编号为顶点$1$、顶点$2$、$\dots$、顶点$N$。

边编号为边$1$、边$2$、$\dots$、边$M$。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，每个顶点都是空的。

Aoki打算沿着图中的边从顶点$1$前往顶点$N$。然而，Aoki不允许移动到被墙占据的顶点。

Takahashi决定选择一些顶点来建造墙，以便无论Aoki选择哪条路线，都无法到达顶点$N$。

在顶点$i$建造墙需要花费Takahashi$c_i$日元（日本的货币）。**他不能在顶点$1$和顶点$N$建造墙。**

Takahashi需要多少钱来建造墙，以满足条件？同时，打印出实现最低成本的建造墙的方式。

约束

• $3 \leq N \leq 100$
• $N - 1 \leq M \leq \frac{N(N-1)}{2} - 1$
• $1 \leq a_i \lt b_i \leq N$ $(1 \leq i \leq M)$
• $(a_i, b_i) \neq (1, N)$
• 给定的图是简单且连通的。
• $1 \leq c_{i} \leq 10^9$ $(2 \leq i \leq N-1)$
• $c_1 = c_N = 0$
• 输入中的所有值都是整数。

输入

输入以标准输入的以下格式给出：

$N$ $M$
$a_1$ $b_1$
$a_2$ $b_2$
$\vdots$
$a_M$ $b_M$
$c_1$ $c_2$ $\dots$ $c_N$

输出

以以下格式打印。在这里，$C,k$和$p_i$定义如下。

• $C$是Takahashi将支付的费用
• $k$是Takahashi需要在上面建造墙的顶点数
• $(p_1,p_2,\dots,p_k)$是Takahashi将在上面建造墙的顶点序列

$C$
$k$
$p_1$ $p_2$ $\dots$ $p_k$

如果有多种满足条件的方法来建造墙，以最低成本打印其中的任何一种。

样例输入1

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

样例输出1

7
2
3 4

如果Takahashi在顶点$3$和顶点$4$上建造墙，支付$3 + 4 = 7$日元，Aoki将无法从顶点$1$前往顶点$5$。

没有以更低成本满足条件的方法，因此$7$日元是答案。

样例输入2

3 2
1 2
2 3
0 1 0

样例输出2

1
1
2

样例输入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

样例输出3

3000000000
3
2 3 4