Score : $550$ points

问题陈述

设有 $N$ 名高桥。

第 $i$ 名高桥拥有整数 $A_i$ 和 $B_i$ 个球。

将从包括 1 到 $K$ 的整数中均匀随机选择一个整数 $x$，他们将重复以下操作 $x$ 次：

• 对于每一个 $i$，第 $i$ 名高桥将其所有的球都给到第 $A_i$ 名高桥。

注意所有 $N$ 名高桥同时执行此操作。

对于每一名 $i=1,2,\ldots,N$，求在操作结束后第 $i$ 名高桥拥有的球数的期望值，结果对 $998244353$ 取模。

如何求解模 $998244353$ 下的期望值？ 可以证明所求概率始终是一个有理数。此外，本问题的约束条件保证了如果该概率表示为不可约分数 $\frac{y}{x}$，则 $x$ 不会被 $998244353$ 整除。此时存在一个唯一的满足 $0\leq z<998244353$ 的整数 $z$，使得 $y\equiv xz\pmod{998244353}$，请报告这个 $z$ 值。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

There are $N$ Takahashi.

The $i$-th Takahashi has an integer $A_i$ and $B_i$ balls.

An integer $x$ between $1$ and $K$, inclusive, will be chosen uniformly at random, and they will repeat the following operation $x$ times.

• For every $i$, the $i$-th Takahashi gives all his balls to the $A_i$-th Takahashi.

Beware that all $N$ Takahashi simultaneously perform this operation.

For each $i=1,2,\ldots,N$, find the expected value, modulo $998244353$, of the number of balls the $i$-th Takahashi has at the end of the operations.

How to find a expected value modulo $998244353$ It can be proved that the sought probability is always a rational number. Additionally, the constraints of this problem guarantee that if the sought probability is represented as an irreducible fraction $\frac{y}{x}$, then $x$ is not divisible by $998244353$. Here, there is a unique $0\leq z\lt998244353$ such that $y\equiv xz\pmod{998244353}$, so report this $z$.

Constraints

• $1\leq N\leq 2\times10^5$
• $1\leq K\leq 10^{18}$
• $K$ is not a multiple of $998244353$.
• $1\leq A _ i\leq N\ (1\leq i\leq N)$
• $0\leq B _ i\lt998244353\ (1\leq i\leq N)$
• All input values are integers.

Input

The input is given from Standard Input in the following format:

$N$ $K$

$A _ 1$ $A _ 2$ $\cdots$ $A _ N$

$B _ 1$ $B _ 2$ $\cdots$ $B _ N$

Output

Print the expected value of the number of balls the $i$-th Takahashi has at the end of the operations for $i=1,2,\ldots,N$, separated by spaces, in a single line.

Sample Input 1

5 2
3 1 4 1 5
1 1 2 3 5

Sample Output 1

3 0 499122179 499122178 5

During two operations, the five Takahashi have the following number of balls.

If $x=1$ is chosen, the five Takahashi have $4,0,1,2,5$ balls.
If $x=2$ is chosen, the five Takahashi have $2,0,4,1,5$ balls.

Thus, the sought expected values are $3,0,\dfrac52,\dfrac32,5$. Print these values modulo $998244353$, that is, $3,0,499122179,499122178,5$, separated by spaces.

Sample Input 2

3 1000
1 1 1
1 10 100

Sample Output 2

111 0 0

After one or more operations, the first Takahashi gets all balls.

Sample Input 3

16 1000007
16 12 6 12 1 8 14 14 5 7 6 5 9 6 10 9
719092922 77021920 539975779 254719514 967592487 476893866 368936979 465399362 342544824 540338192 42663741 165480608 616996494 16552706 590788849 221462860

Sample Output 3

817852305 0 0 0 711863206 253280203 896552049 935714838 409506220 592088114 0 413190742 0 363914270 0 14254803

Sample Input 4

24 100000000007
19 10 19 15 1 20 13 15 8 23 22 16 19 22 2 20 12 19 17 20 16 8 23 6
944071276 364842194 5376942 671161415 477159272 339665353 176192797 2729865 676292280 249875565 259803120 103398285 466932147 775082441 720192643 535473742 263795756 898670859 476980306 12045411 620291602 593937486 761132791 746546443

Sample Output 4

918566373 436241503 0 0 0 455245534 0 356196743 0 906000633 0 268983266 21918337 0 733763572 173816039 754920403 0 273067118 205350062 0 566217111 80141532 0

update @ 2024/3/10 08:49:29