Ascendant Descendant

AtCoder

IDarc183_e

Time7000ms

Memory256MB

Difficulty—

There is a rooted tree with $N$ vertices numbered from $1$ to $N$. The root is vertex $1$, and the parent of vertex $i$ ($2 \leq i \leq N$) is vertex $P_i$ ($P_i<i$). There are also integer sequences of length $M$: $A=(A_1,A_2,\cdots,A_M)$ and $B=(B_1,B_2,\cdots,B_M)$, consisting of integers between $1$ and $N$, inclusive. $A$ is said to be **good** if and only if for each $i$, vertex $A_i$ is an ancestor of vertex $B_i$ or $A_i=B_i$. Initially, $A$ is good. Consider the following operation on $A$. * Choose an integer $i$ ($1 \leq i \leq M-1$) and swap the values of $A_i$ and $A_{i+1}$. Here, $A$ must remain good after the operation. Find the number, modulo $998244353$, of sequences that can result from performing this operation on $A$ zero or more times. ## Constraints * $2 \leq N \leq 250000$ * $2 \leq M \leq 250000$ * $1 \leq P_i <i$ * $1 \leq A_i \leq B_i \leq N$ * Vertex $A_i$ is an ancestor of vertex $B_i$ or $A_i=B_i$. * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $M$ $P_2$ $P_3$ $\cdots$ $P_N$ $A_1$ $A_2$ $\cdots$ $A_M$ $B_1$ $B_2$ $\cdots$ $B_M$ [samples]

Samples

Input #1

Output #1

2

Consider choosing $i=1$. The $A=(2,1,1)$ after the operation is not good, so this operation is invalid.
Consider choosing $i=2$. The $A=(1,1,2)$ after the operation is good, so this operation is valid.
There are two sequences that can result from performing zero or more operations on $A$: $A=(1,2,1)$ and $(1,1,2)$.

Input #2

Output #2

Input #3

8 13
1 2 2 3 4 4 3
5 3 2 5 4 6 2 8 2 6 7 4 7
5 5 8 5 6 6 5 8 3 6 7 4 7

Output #3

Input #4

30 27
1 2 1 1 5 1 7 1 5 10 1 12 12 13 15 16 12 18 19 18 21 21 23 13 18 18 27 27 13
1 18 1 5 11 12 1 1 1 12 1 12 1 15 1 1 21 1 12 10 2 8 3 1 1 30 12
14 27 30 5 11 17 1 18 24 27 29 27 19 15 28 5 21 21 29 11 2 8 3 4 10 30 22

Output #4

API Response (JSON)

{
  "problem": {
    "name": "Ascendant Descendant",
    "description": {
      "content": "There is a rooted tree with $N$ vertices numbered from $1$ to $N$. The root is vertex $1$, and the parent of vertex $i$ ($2 \\leq i \\leq N$) is vertex $P_i$ ($P_i<i$). There are also integer sequences ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 7000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc183_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There is a rooted tree with $N$ vertices numbered from $1$ to $N$. The root is vertex $1$, and the parent of vertex $i$ ($2 \\leq i \\leq N$) is vertex $P_i$ ($P_i<i$).\nThere are also integer sequences ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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