A Certain Game

AtCoder

IDabc314_f

Time2000ms

Memory256MB

Difficulty—

$N$ players, player $1$, player $2$, ..., player $N$, participate in a game tournament. Just before the tournament starts, each player forms a one-person team, so there are $N$ teams in total. The tournament has a total of $N-1$ matches. In each match, two different teams are chosen. One team goes first, and the other goes second. Each match will result in exactly one team winning. Specifically, for each $i = 1, 2, \ldots, N-1$, the $i$\-th match proceeds as follows. * The team with player $p_i$ goes first, and the team with player $q_i$ goes second. * Let $a$ and $b$ be the numbers of players in the first and second teams, respectively. The first team wins with probability $\frac{a}{a+b}$, and the second team wins with probability $\frac{b}{a+b}$. * Then, the two teams are combined into a single team. The result of each match is independent of those of the others. For each of the $N$ players, print the expected number of times the team with that player wins throughout the tournament, modulo $998244353$. How to print an expected value modulo $998244353$It can be proved that the sought expected value is always rational. Also, the constraints of this problem guarantee that if the sought expected value is expressed as an irreducible fraction $\frac{y}{x}$, then $x$ is not divisible by $998244353$. Now, there is a unique integer $z$ between $0$ and $998244352$, inclusive, such that $xz \equiv y \pmod{998244353}$. Report this $z$. ## Constraints * $2 \leq N \leq 2 \times 10^5$ * $1 \leq p_i, q_i \leq N$ * Just before the $i$\-th match, player $p_i$ and player $q_i$ belong to different teams. * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $p_1$ $q_1$ $p_2$ $q_2$ $\vdots$ $p_{N-1}$ $q_{N-1}$ [samples]

Samples

Input #1

Output #1

698771048 698771048 964969543 964969543 133099248

We call a team formed by player $x_1$, player $x_2$, $\ldots$, player $x_k$ as team $\lbrace x_1, x_2, \ldots, x_k \rbrace$.

*   The first match is played by team $\lbrace 1 \rbrace$, with player $1$, and team $\lbrace 2 \rbrace$, with player $2$. Team $\lbrace 1 \rbrace$ wins with probability $\frac{1}{2}$, and team $\lbrace 2 \rbrace$ wins with probability $\frac{1}{2}$. Then, the two teams are combined into a single team $\lbrace 1, 2 \rbrace$.
*   The second match is played by team $\lbrace 4 \rbrace$, with player $4$, and team $\lbrace 3 \rbrace$, with player $3$. Team $\lbrace 4 \rbrace$ wins with probability $\frac{1}{2}$, and team $\lbrace 3 \rbrace$ wins with probability $\frac{1}{2}$. Then, the two teams are combined into a single team $\lbrace 3, 4 \rbrace$.
*   The third match is played by team $\lbrace 5 \rbrace$, with player $5$, and team $\lbrace 3, 4 \rbrace$, with player $3$. Team $\lbrace 5 \rbrace$ wins with probability $\frac{1}{3}$, and team $\lbrace 3, 4 \rbrace$ wins with probability $\frac{2}{3}$. Then, the two teams are combined into a single team $\lbrace 3, 4, 5 \rbrace$.
*   The fourth match is played by team $\lbrace 1, 2 \rbrace$, with player $1$, and team $\lbrace 3, 4, 5 \rbrace$, with player $4$. Team $\lbrace 1, 2 \rbrace$ wins with probability $\frac{2}{5}$, and team $\lbrace 3, 4, 5 \rbrace$ wins with probability $\frac{3}{5}$. Then, the two teams are combined into a single team $\lbrace 1, 2, 3, 4, 5 \rbrace$.

The expected numbers of times the teams with players $1, 2, 3, 4, 5$ win throughout the tournament, $E_1, E_2, E_3, E_4, E_5$, are $\frac{9}{10}, \frac{9}{10}, \frac{53}{30}, \frac{53}{30}, \frac{14}{15}$, respectively.

Input #2

Output #2

43970290 310168785 806914186 501498951 950708909 272140427 335124893 168750835 310168785 168750835 280459129 280459129 272140427 476542843 43970290

API Response (JSON)

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  "problem": {
    "name": "A Certain Game",
    "description": {
      "content": "$N$ players, player $1$, player $2$, ..., player $N$, participate in a game tournament. Just before the tournament starts, each player forms a one-person team, so there are $N$ teams in total. The tou",
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    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
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    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc314_f"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "$N$ players, player $1$, player $2$, ..., player $N$, participate in a game tournament. Just before the tournament starts, each player forms a one-person team, so there are $N$ teams in total.\nThe tou...",
      "is_translate": false,
      "language": "English"
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Full JSON Raw Segments