103054: [Atcoder]ABC305 E - Art Gallery on Graph

Problem Statement

There is a simple undirected graph with $N$ vertices and $M$ edges, where vertices are numbered from $1$ to $N$, and edges are numbered from $1$ to $M$. Edge $i$ connects vertex $a_i$ and vertex $b_i$.
$K$ security guards numbered from $1$ to $K$ are on some vertices. Guard $i$ is on vertex $p_i$ and has a stamina of $h_i$. All $p_i$ are distinct.

A vertex $v$ is said to be guarded when the following condition is satisfied:

• there is at least one guard $i$ such that the distance between vertex $v$ and vertex $p_i$ is at most $h_i$.

Here, the distance between vertex $u$ and vertex $v$ is the minimum number of edges in the path connecting vertices $u$ and $v$.

List all guarded vertices in ascending order.

Constraints

• $1 \leq N \leq 2 \times 10^5$
• $0 \leq M \leq \min \left(\frac{N(N-1)}{2}, 2 \times 10^5 \right)$
• $1 \leq K \leq N$
• $1 \leq a_i, b_i \leq N$
• The given graph is simple.
• $1 \leq p_i \leq N$
• All $p_i$ are distinct.
• $1 \leq h_i \leq N$
• All input values are integers.

Input

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

$N$ $M$ $K$
$a_1$ $b_1$
$a_2$ $b_2$
$\vdots$
$a_M$ $b_M$
$p_1$ $h_1$
$p_2$ $h_2$
$\vdots$
$p_K$ $h_K$

Output

Print the answer in the following format. Here,

• $G$ is the number of guarded vertices,
• and $v_1, v_2, \dots, v_G$ are the vertex numbers of the guarded vertices in ascending order.

$G$
$v_1$ $v_2$ $\dots$ $v_G$

Sample Input 1

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

Sample Output 1

4
1 2 3 5

The guarded vertices are $1, 2, 3, 5$.
These vertices are guarded because of the following reasons.

• The distance between vertex $1$ and vertex $p_1 = 1$ is $0$, which is not greater than $h_1 = 1$. Thus, vertex $1$ is guarded.
• The distance between vertex $2$ and vertex $p_1 = 1$ is $1$, which is not greater than $h_1 = 1$. Thus, vertex $2$ is guarded.
• The distance between vertex $3$ and vertex $p_2 = 5$ is $1$, which is not greater than $h_2 = 2$. Thus, vertex $3$ is guarded.
• The distance between vertex $5$ and vertex $p_1 = 1$ is $1$, which is not greater than $h_1 = 1$. Thus, vertex $5$ is guarded.

Sample Input 2

3 0 1
2 3

Sample Output 2

1
2

The given graph may have no edges.

Sample Input 3

10 10 2
2 1
5 1
6 1
2 4
2 5
2 10
8 5
8 6
9 6
7 9
3 4
8 2

Sample Output 3

7
1 2 3 5 6 8 9

问题描述

有一个简单的无向图，包含N个顶点和M条边，顶点编号为1到N，边编号为1到M。第i条边连接顶点$a_i$和顶点$b_i$。
有K个编号为1到K的安全警卫位于一些顶点上。警卫i位于顶点$p_i$，具有耐力$h_i$。所有$p_i$都是不同的。

当满足以下条件时，顶点v被认为是“被保护的”：

• 至少存在一个警卫i，使得顶点v和顶点$p_i$之间的距离不超过$h_i$。

这里，顶点u和顶点v之间的距离是连接顶点u和顶点v的路径中边的数量的最小值。

按升序列出所有被保护的顶点。

约束条件

• $1 \leq N \leq 2 \times 10^5$
• $0 \leq M \leq \min \left(\frac{N(N-1)}{2}, 2 \times 10^5 \right)$
• $1 \leq K \leq N$
• $1 \leq a_i, b_i \leq N$
• 给定的图是简单的。
• $1 \leq p_i \leq N$
• 所有$p_i$都是不同的。
• $1 \leq h_i \leq N$
• 所有输入值都是整数。

输入

输入通过标准输入以以下格式给出：

$N$ $M$ $K$
$a_1$ $b_1$
$a_2$ $b_2$
$\vdots$
$a_M$ $b_M$
$p_1$ $h_1$
$p_2$ $h_2$
$\vdots$
$p_K$ $h_K$

输出

以以下格式打印答案。这里，

• $G$是被保护的顶点的数量，
• 顶点$v_1, v_2, \dots, v_G$是按升序排列的被保护的顶点的编号。

$G$
$v_1$ $v_2$ $\dots$ $v_G$

样例输入1

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

样例输出1

4
1 2 3 5

被保护的顶点是1, 2, 3, 5。
这些顶点是由于以下原因被保护的。

• 顶点1和顶点$p_1 = 1$之间的距离为0，不超过$h_1 = 1$。因此，顶点1被保护。
• 顶点2和顶点$p_1 = 1$之间的距离为1，不超过$h_1 = 1$。因此，顶点2被保护。
• 顶点3和顶点$p_2 = 5$之间的距离为1，不超过$h_2 = 2$。因此，顶点3被保护。
• 顶点5和顶点$p_1 = 1$之间的距离为1，不超过$h_1 = 1$。因此，顶点5被保护。

样例输入2

3 0 1
2 3

样例输出2

1
2

给定的图可能没有边。

样例输入3

10 10 2
2 1
5 1
6 1
2 4
2 5
2 10
8 5
8 6
9 6
7 9
3 4
8 2

样例输出3

7
1 2 3 5 6 8 9