Stochastic Dominance

AtCoder

IDagc076_d

Time8000ms

Memory256MB

Difficulty—

Let $L=10^8$. You are given an integer $N$ and a non-negative integer sequence $A=(A_1,A_2,\cdots,A_M)$ of length $M$ ($0 \leq A_i < L$). For each $k=1,2,\cdots,N$, solve the following problem: There are $k$ red balls numbered from $1$ to $k$ and $k$ blue balls numbered from $1$ to $k$. Now, you will write a real number on each ball in the following way. Here, all random choices are independent. * For each $1 \leq i \leq k$, the number written on red ball $i$ is $L \times (i-1)+A_{j_i}$. Here, $j_i$ is an integer chosen uniformly at random from the range between $1$ and $M$ (inclusive). * For each $1 \leq i \leq k$, the number written on blue ball $i$ is a **real number** chosen uniformly at random from the range $[0,k \times L)$. Find the probability, modulo $998244353$, that the following condition is satisfied after writing numbers on all balls. * For any real number $x$ ($0 \leq x \leq k \times L$), "the number of red balls with a number written that is at most $x$" $\geq$ "the number of blue balls with a number written that is at most $x$". Definition of probability modulo $998244353$It can be proved that the desired probability is always a rational number. Also, under the constraints of this problem, it can be proved that if the desired rational number is represented as an irreducible fraction $\frac{P}{Q}$, then $Q \neq 0 \pmod{998244353}$. Therefore, there uniquely exists an integer $R$ satisfying $R \times Q \equiv P \pmod{998244353}, 0 \leq R \lt 998244353$. Output this $R$. ## Constraints * $1 \leq N \leq 250000$ * $1 \leq M \leq 250000$ * $0 \leq A_i < L=10^8$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $M$ $A_1$ $A_2$ $\cdots$ $A_M$ [samples]

Samples

Input #1

2 1
50000000

Output #1

499122177
686292993

*   $k=1$: The probability of satisfying the condition is $1/2$.
*   $k=2$: The probability of satisfying the condition is $5/16$.

Input #2

3 5
78682095 25048034 71067122 94666780 1927105

Output #2

741897823
406774904
435399949

Input #3

10 7
40490214 25131781 2271243 52064315 34467030 27419560 56103210

Output #3

API Response (JSON)

{
  "problem": {
    "name": "Stochastic Dominance",
    "description": {
      "content": "Let $L=10^8$. You are given an integer $N$ and a non-negative integer sequence $A=(A_1,A_2,\\cdots,A_M)$ of length $M$ ($0 \\leq A_i < L$). For each $k=1,2,\\cdots,N$, solve the following problem: There ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 8000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "agc076_d"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Let $L=10^8$.\nYou are given an integer $N$ and a non-negative integer sequence $A=(A_1,A_2,\\cdots,A_M)$ of length $M$ ($0 \\leq A_i < L$). For each $k=1,2,\\cdots,N$, solve the following problem:\nThere ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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