Random Transition

AtCoder

IDarc133_f

Time5000ms

Memory256MB

Difficulty—

Given is an integer $N$. Snuke is going to do the procedure below. * Let $x$ be a variable and initialize it with a random integer between $0$ and $N$ (inclusive). For each $0 \leq i \leq N$, $x=i$ is chosen with probability $A_i/10^9$. * Repeat the following operation $K$ times. * With probability $x/N$, decrease the value of $x$ by $1$; with probability $1-x/N$, increase it by $1$. (Here, note that $x$ will still be between $0$ and $N$ after the operation.) For each $i=0,1,\cdots,N$, find the probability, modulo $998244353$, that $x=i$ after the procedure. Definition of probability modulo $998244353$It can be proved that the sought probabilities are always rational numbers. Additionally, under the Constraints of this problem, when representing each of them as an irreducible fraction $\frac{P}{Q}$, it can be proved that $Q \neq 0 \pmod{998244353}$. Thus, there uniquely exists an integer $R$ such that $R \times Q \equiv P \pmod{998244353}, 0 \leq R < 998244353$. Answer with this $R$. ## Constraints * $1 \leq N \leq 100000$ * $0 \leq A_i$ * $\sum_{0 \leq i \leq N}A_i = 10^9$ * $1 \leq K \leq 10^9$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $K$ $A_0$ $A_1$ $\cdots$ $A_N$ [samples]

Samples

Input #1

2 1
0 1000000000 0

Output #1

499122177 0 499122177

First, $x$ is always initialized with $x=1$. In the subsequent operation, the value of $x$ is decreased by $1$ with probability $1/2$ and increased by $1$ with probability $1/2$. Eventually, we will have $x=0,1,2$ with probabilities $1/2,0,1/2$, respectively.

Input #2

4 2
200000000 200000000 200000000 200000000 200000000

Output #2

723727156 598946612 349385524 598946612 723727156

Input #3

10 100
21265166 263511538 35931763 26849698 108140810 134702248 36774526 147099145 58335759 4118743 163270604

Output #3

505314898 24510700 872096939 107940764 808162829 831195162 314651262 535843032 665830283 627881537 696038713

API Response (JSON)

{
  "problem": {
    "name": "Random Transition",
    "description": {
      "content": "Given is an integer $N$. Snuke is going to do the procedure below. *   Let $x$ be a variable and initialize it with a random integer between $0$ and $N$ (inclusive). For each $0 \\leq i \\leq N$, $x=i$",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 5000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc133_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Given is an integer $N$.\nSnuke is going to do the procedure below.\n\n*   Let $x$ be a variable and initialize it with a random integer between $0$ and $N$ (inclusive). For each $0 \\leq i \\leq N$, $x=i$...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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