Farthest City

AtCoder

IDabc281_g

Time4000ms

Memory256MB

Difficulty—

You are given positive integers $N$ and $M$. Find the number, modulo $M$, of simple connected undirected graphs with $N$ vertices numbered $1, \dots, N$ that satisfy the following condition. * For every $u = 2, \dots, N-1$, the shortest distance from vertex $1$ to vertex $u$ is strictly smaller than the shortest distance from vertex $1$ to vertex $N$. Here, the shortest distance from vertex $u$ to vertex $v$ is the minimum number of edges in a simple path connecting vertices $u$ and $v$. Two graphs are considered different if and only if there are two vertices $u$ and $v$ that are connected by an edge in exactly one of those graphs. ## Constraints * $3 \leq N \leq 500$ * $10^8 \leq M \leq 10^9$ * $N$ and $M$ are integers. ## Input The input is given from Standard Input in the following format: $N$ $M$ [samples]

Samples

Input #1

4 1000000000

Output #1

8

The following eight graphs satisfy the condition.
![image](https://img.atcoder.jp/abc281/5c77dfe15dfa3c03666e654bf8cfdc01.png)

Input #2

3 100000000

Output #2

Input #3

500 987654321

Output #3

610860515

Be sure to find the number modulo $M$.

API Response (JSON)

{
  "problem": {
    "name": "Farthest City",
    "description": {
      "content": "You are given positive integers $N$ and $M$.   Find the number, modulo $M$, of simple connected undirected graphs with $N$ vertices numbered $1, \\dots, N$ that satisfy the following condition. *   Fo",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 4000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc281_g"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given positive integers $N$ and $M$.  \nFind the number, modulo $M$, of simple connected undirected graphs with $N$ vertices numbered $1, \\dots, N$ that satisfy the following condition.\n\n*   Fo...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments