BDFS

AtCoder

IDarc170_e

Time8000ms

Memory256MB

Difficulty—

You are given integers $N$ and $P$. There is a graph with $N$ vertices and $N$ edges, where each vertex is labeled $1$ to $N$. The $i$\-th edge connects vertices $i$ and $i+1$ bidirectionally. Here, vertex $N+1$ refers to vertex $1$. Perform the following algorithm to obtain a sequence $D=(D_1,D_2,\ldots,D_N)$ of length $N$: * Set an integer sequence $D$ of length $N$ to $D=(D_1,\ldots,D_N)=(-1,\ldots,-1)$. Also, set a sequence $Q$ of number pairs to $Q=((1,0))$. Repeat the following process while $Q$ is not empty: * Let $(v,d)$ be the first element of $Q$. Remove this element. * If $D_v = -1$, then set $D_v := d$, and for each vertex $x$ adjacent to vertex $v$ such that $D_x=-1$, perform the following process. If there are multiple such $x$ that satisfy the condition, process them in ascending order of vertex number: 1. With probability $\frac{P}{100}$, add $(x,d+1)$ to the **front** of $Q$. 2. If $(x,d+1)$ was not added to the front of $Q$, add it to the **end** of $Q$. Find the expected value of the sum of the elements of the final sequence $D$ obtained, modulo $998244353$. Solve each of the $T$ test cases given. Definition of expected value $\text{mod } 998244353$ It can be proved that the expected value to be found is always a rational number. Furthermore, the constraints of this problem guarantee that if that value is expressed as an irreducible fraction $\frac{P}{Q}$, then $Q$ is not divisible by $998244353$. Here, there is a unique integer $R$ between $0$ and $998244352$, inclusive, such that $R\times Q \equiv P\pmod{998244353}$. Provide this $R$ as the answer. ## Constraints * $1 \leq T \leq 10^4$ * $3 \leq N \leq 10^{18}$ * $1\leq P \leq 99$ * All input numbers are integers. ## Input The input is given from Standard Input in the following format: $T$ $\mathrm{case}_1$ $\vdots$ $\mathrm{case}_T$ Each case is given in the following format: $N$ $P$ [samples]

Samples

Input #1

3
3 50
4 1
1000000000000000000 70

Output #1

499122179
595552585
760296751

In the first test case, the algorithm may operate as follows:

*   Initially, $D=(-1,-1,-1)$ and $Q=((1,0))$. Remove the first element $(1,0)$ from $Q$.
*   $D_1 = -1$, so set $D_1 := 0$. The vertices $x$ adjacent to vertex $1$ such that $D_x= -1$ are $2$ and $3$.
*   Add $(2,1)$ to the front of $Q$. Add $(3,1)$ to the end of $Q$. Now $Q=((2,1),(3,1))$.
*   Remove the first element $(2,1)$ from $Q$.
*   $D_2 = -1$, so set $D_2 := 1$. The vertex $x$ adjacent to vertex $2$ such that $D_x= -1$ is $3$.
*   Add $(3,2)$ to the front of $Q$. Now $Q=((3,2),(3,1))$.
*   Remove the first element $(3,2)$ from $Q$.
*   $D_3 = -1$, so set $D_3 := 2$. There are no vertices $x$ adjacent to vertex $3$ such that $D_x= -1$, so do nothing.
*   Remove the first element $(3,1)$ from $Q$.
*   $D_3 =2$, so do nothing.
*   $Q$ is now empty, so the process ends.

In this case, the final sequence obtained is $D=(0,1,2)$. The probability that the algorithm operates as described above is $\frac{1}{8}$, and the expected sum of the elements of $D$ is $\frac{5}{2}$.

API Response (JSON)

{
  "problem": {
    "name": "BDFS",
    "description": {
      "content": "You are given integers $N$ and $P$. There is a graph with $N$ vertices and $N$ edges, where each vertex is labeled $1$ to $N$. The $i$\\-th edge connects vertices $i$ and $i+1$ bidirectionally. Here, v",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 8000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc170_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given integers $N$ and $P$.\nThere is a graph with $N$ vertices and $N$ edges, where each vertex is labeled $1$ to $N$. The $i$\\-th edge connects vertices $i$ and $i+1$ bidirectionally. Here, v...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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