Spanning Tree

AtCoder

IDjsc2021_g

Time2000ms

Memory256MB

Difficulty—

We have a graph with $N$ vertices numbered $1, 2, \dots, N$. Initially, it has no edges. Now, let us add some number of undirected edges to $G$ so that the following condition holds for any $i, j\,(i ≠ j)$ after addition. * If $A_{i, j} = 1$, there is an edge directly connecting Vertex $i$ and Vertex $j$; * if $A_{i, j} = 0$, there is no edge directly connecting Vertex $i$ and Vertex $j$; * if $A_{i, j} = -1$, either is fine. Among the graphs that can be $G$ after addition, how many are trees? Since the count can be enormous, find it modulo $(10^9 + 7)$. ## Constraints * All values in input are integers. * $2 ≤ N ≤ 300$ * $-1 ≤ A_{i, j} = A_{j, i} ≤ 1$ * $A_{i, i} = 0$ ## Input Input is given from Standard Input in the following format: $N$ $A_{1, 1}$ $\cdots$ $A_{1, N}$ $\vdots$ $A_{N, 1}$ $\cdots$ $A_{N, N}$ [samples]

Samples

Input #1

Output #1

2

We need an edge between Vertex $1$ and Vertex $2$, and we must not add an edge between Vertex $1$ and Vertex $4$ or between Vertex $3$ and Vertex $4$.
Thus, we have the following two valid graphs:
![image](https://img.atcoder.jp/ghi/0f55bf9e7a13aef4bddde12f87a23d5d.png)

Input #2

Output #2

Input #3

Output #3

Input #4

11
0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
-1 0 -1 -1 -1 -1 -1 -1 -1 -1 -1
-1 -1 0 -1 -1 -1 -1 -1 -1 -1 -1
-1 -1 -1 0 -1 -1 -1 -1 -1 -1 -1
-1 -1 -1 -1 0 -1 -1 -1 -1 -1 -1
-1 -1 -1 -1 -1 0 -1 -1 -1 -1 -1
-1 -1 -1 -1 -1 -1 0 -1 -1 -1 -1
-1 -1 -1 -1 -1 -1 -1 0 -1 -1 -1
-1 -1 -1 -1 -1 -1 -1 -1 0 -1 -1
-1 -1 -1 -1 -1 -1 -1 -1 -1 0 -1
-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 0

Output #4

357947677

When we distinguish the vertices, there are $11^9$ trees with $11$ vertices.

API Response (JSON)

{
  "problem": {
    "name": "Spanning Tree",
    "description": {
      "content": "We have a graph with $N$ vertices numbered $1, 2, \\dots, N$. Initially, it has no edges.   Now, let us add some number of undirected edges to $G$ so that the following condition holds for any $i, j\\,(",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "jsc2021_g"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "We have a graph with $N$ vertices numbered $1, 2, \\dots, N$. Initially, it has no edges.  \nNow, let us add some number of undirected edges to $G$ so that the following condition holds for any $i, j\\,(...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments