Art Gallery on Graph

AtCoder

IDabc305_e

Time2000ms

Memory256MB

Difficulty—

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

Samples

Input #1

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.

Input #2

3 0 1
2 3

Output #2

1
2

The given graph may have no edges.

Input #3

Output #3

7
1 2 3 5 6 8 9

API Response (JSON)

{
  "problem": {
    "name": "Art Gallery on Graph",
    "description": {
      "content": "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 $b_i$",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc305_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "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 $b_i$...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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