Takahashi And Pass-The-Ball Game

AtCoder

IDabc310_g

Time2000ms

Memory256MB

Difficulty—

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: $N$ $K$ $A _ 1$ $A _ 2$ $\cdots$ $A _ N$ $B _ 1$ $B _ 2$ $\cdots$ $B _ N$ [samples]

Samples

Input #1

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

Output #1

3 0 499122179 499122178 5

During two operations, the five Takahashi have the following number of balls.
![image](https://img.atcoder.jp/abc310/eeca44e66744660173a72967840e158a.png)
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.

Input #2

3 1000
1 1 1
1 10 100

Output #2

111 0 0

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

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

Output #3

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

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

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

API Response (JSON)

{
  "problem": {
    "name": "Takahashi And Pass-The-Ball Game",
    "description": {
      "content": "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 followi",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc310_g"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There are $N$ Takahashi.\nThe $i$\\-th Takahashi has an integer $A_i$ and $B_i$ balls.\nAn integer $x$ between $1$ and $K$, inclusive, will be chosen uniformly at random, and they will repeat the followi...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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