Edge Ordering

AtCoder

IDdiverta2019_f

Time2000ms

Memory256MB

Difficulty—

You are given a simple connected undirected graph $G$ consisting of $N$ vertices and $M$ edges. The vertices are numbered $1$ to $N$, and the edges are numbered $1$ to $M$. Edge $i$ connects Vertex $a_i$ and $b_i$ bidirectionally. It is guaranteed that the subgraph consisting of Vertex $1,2,\ldots,N$ and Edge $1,2,\ldots,N-1$ is a spanning tree of $G$. An allocation of weights to the edges is called a _good allocation_ when the tree consisting of Vertex $1,2,\ldots,N$ and Edge $1,2,\ldots,N-1$ is a minimum spanning tree of $G$. There are $M!$ ways to allocate the edges distinct integer weights between $1$ and $M$. For each good allocation among those, find the total weight of the edges in the minimum spanning tree, and print the sum of those total weights modulo $10^{9}+7$. ## Constraints * All values in input are integers. * $2 \leq N \leq 20$ * $N-1 \leq M \leq N(N-1)/2$ * $1 \leq a_i, b_i \leq N$ * $G$ does not have self-loops or multiple edges. * The subgraph consisting of Vertex $1,2,\ldots,N$ and Edge $1,2,\ldots,N-1$ is a spanning tree of $G$. ## Input Input is given from Standard Input in the following format: $N$ $M$ $a_1$ $b_1$ $\vdots$ $a_M$ $b_M$ [samples]

Samples

Input #1

Output #1

6

An allocation is good only if Edge $3$ has the weight $3$. For these good allocations, the total weight of the edges in the minimum spanning tree is $3$, and there are two good allocations, so the answer is $6$.

Input #2

Output #2

Input #3

Output #3

657573092

Print the sum of those total weights modulo $10^{9}+7$.

API Response (JSON)

{
  "problem": {
    "name": "Edge Ordering",
    "description": {
      "content": "You are given a simple connected undirected graph $G$ consisting of $N$ vertices and $M$ edges. The vertices are numbered $1$ to $N$, and the edges are numbered $1$ to $M$. Edge $i$ connects Vertex $a",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "diverta2019_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given a simple connected undirected graph $G$ consisting of $N$ vertices and $M$ edges. The vertices are numbered $1$ to $N$, and the edges are numbered $1$ to $M$.\nEdge $i$ connects Vertex $a...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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