Ancestor Relation

AtCoder

IDarc197_d

Time2000ms

Memory256MB

Difficulty—

You are given an $N\times N$ matrix $A = (A_{i,j})$ ($1\le i,j\le N$) whose entries are $0$ or $1$. Find, modulo $998244353$, the number of trees $G$ on $N$ vertices numbered $1$ to $N$ that satisfy the following condition. * $A_{i,j}=1$ if and only if at least one of the following holds: * When $G$ is rooted at vertex $1$, Vertex $j$ is an ancestor of vertex $i$. That is, vertex $j$ lies on the unique path in $G$ between vertices $1$ and $i$. * When $G$ is rooted at vertex $1$, Vertex $i$ is an ancestor of vertex $j$. That is, vertex $i$ lies on the unique path in $G$ between vertices $1$ and $j$. Here, the endpoints of a path are considered to be on that path. Note that $G$ being a tree guarantees uniqueness of the path between any two vertices. There are $T$ test cases; solve each one. ## Constraints * $1\leq T\leq 10^5$ * $2\leq N\leq 400$ * $A_{i,j}\in \lbrace 0,1\rbrace$ $(1\leq i,j\leq N)$ * $A_{i,i}=1$ $(1\leq i\leq N)$ * $A_{i,j}=A_{j,i}$ $(1\leq i,j\leq N)$ * The sum of $N^2$ over all test cases is at most $400^2$. ## Input The input is given from Standard Input in the following format: $T$ $\text{case}_1$ $\vdots$ $\text{case}_T$ Each case is given in the following format: $N$ $A_{1,1}$ $\cdots$ $A_{1,N}$ $\vdots$ $A_{N,1}$ $\cdots$ $A_{N,N}$ [samples]

Samples

Input #1

5
3
1 1 1
1 1 0
1 0 1
3
1 1 1
1 1 1
1 1 1
3
1 0 0
0 1 0
0 0 1
3
1 0 1
0 1 1
1 1 1
7
1 1 1 1 1 1 1
1 1 0 1 0 1 1
1 0 1 1 1 1 0
1 1 1 1 1 1 1
1 0 1 1 1 1 0
1 1 1 1 1 1 1
1 1 0 1 0 1 1

Output #1

1
2
0
0
8

In the first test case, the following one tree $G$ satisfies the condition: ![image](https://img.atcoder.jp/arc197/047f79d01c371fe5f47850c631892671.png)
In the second test case, the following two trees $G$ satisfy the condition: ![image](https://img.atcoder.jp/arc197/ff998867883faff858791a57f8497f6d.png)

API Response (JSON)

{
  "problem": {
    "name": "Ancestor Relation",
    "description": {
      "content": "You are given an $N\\times N$ matrix $A = (A_{i,j})$ ($1\\le i,j\\le N$) whose entries are $0$ or $1$. Find, modulo $998244353$, the number of trees $G$ on $N$ vertices numbered $1$ to $N$ that satisfy t",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc197_d"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given an $N\\times N$ matrix $A = (A_{i,j})$ ($1\\le i,j\\le N$) whose entries are $0$ or $1$.\nFind, modulo $998244353$, the number of trees $G$ on $N$ vertices numbered $1$ to $N$ that satisfy t...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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