Product of Max of Sum of Subtree

AtCoder

IDagc073_c

Time2000ms

Memory256MB

Difficulty—

There is a tree with $N$ vertices numbered from $1$ to $N$, where the $i$\-th edge connects vertices $A_i$ and $B_i$. We will now randomly write real numbers between $-(N-1)$ and $1$ (inclusive) on each vertex of this tree. The random number distributions are all independent uniform distributions. Then, the **score** of the tree is defined as follows: * For each vertex $i$, define $x_i$ as follows: * The maximum value of the sum of values written on vertices of connected subgraphs that contain vertex $i$ * If there exists $i$ that does not satisfy $0 \leq x_i \leq 1$, the tree's score is $0$. * Otherwise, the tree's score is $\prod_{1 \leq i \leq N} x_i$. Find the expected value $\text{mod }{998244353}$ of the tree's score. Definition of expected value $\text{mod }{998244353}$It can be proved that the sought expected value is always a rational number. Also, under the constraints of this problem, it can be proved that when the value is expressed as an irreducible fraction $\frac{P}{Q}$, we have $Q \neq 0 \pmod{998244353}$. Therefore, an integer $R$ satisfying $R \times Q \equiv P \pmod{998244353}, 0 \leq R < 998244353$ is uniquely determined. Answer this $R$. ## Constraints * $1 \leq N \leq 5000$ * $1 \leq A_i, B_i \leq N$ * The input graph is a tree. ## Input The input is given from Standard Input in the following format: $N$ $A_1$ $B_1$ $A_2$ $B_2$ $\vdots$ $A_{N-1}$ $B_{N-1}$ [samples]

Samples

Input #1

Output #1

499122177

The expected value of the score is $1/2$.

Input #2

2
1 2

Output #2

873463809

The expected value of the score is $1/8$.

Input #3

3
1 2
2 3

Output #3

842653798

The expected value of the score is $13/648$.

Input #4

Output #4

132050425

The expected value of the score is $41/187500$.

Input #5

Output #5

243711918

The expected value of the score is $2477/30064771072$.

API Response (JSON)

{
  "problem": {
    "name": "Product of Max of Sum of Subtree",
    "description": {
      "content": "There is a tree with $N$ vertices numbered from $1$ to $N$, where the $i$\\-th edge connects vertices $A_i$ and $B_i$. We will now randomly write real numbers between $-(N-1)$ and $1$ (inclusive) on ea",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "agc073_c"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There is a tree with $N$ vertices numbered from $1$ to $N$, where the $i$\\-th edge connects vertices $A_i$ and $B_i$.\nWe will now randomly write real numbers between $-(N-1)$ and $1$ (inclusive) on ea...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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