Background

Original link: https://oier.team/problems/X3G。

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You are right, but this is a Dream Bear weekly contest problem, not a place for the setter to "generate electricity", so you need to solve a graph theory problem.

Problem Description

By convention, $\operatorname{lca}_G(u,v)$ denotes the lowest common ancestor of nodes with labels $u,v$ in a labeled rooted tree $G$. You are given a rooted tree $T$ with root label $1$, size $n$, and node labels $1\sim n$, as well as a node set $S$ of size $m$. You need to find how many distinct labeled rooted trees $T'$ of size $n$ with node labels $1\sim n$ satisfy that for any $u,v\in S$, we have $\operatorname{lca}_T(u,v)=\operatorname{lca}_{T'}(u,v)$.

Output the answer modulo $998\,244\,353$.

We say two labeled rooted trees of size $n$ are different if and only if at least one of the following holds:

• Their root nodes are different.
• There exists an edge that appears in one tree but not in the other.

Input Format

The first line contains two positive integers $n,m$.

The second line contains $n-1$ positive integers $p_2,p_3,\cdots,p_n$, where $p_i$ is the parent label of node $i$ for $i=2\sim n$.

The last line contains $m$ positive integers, indicating the labels of the selected nodes in the set $S$.

Output Format

One line with one integer, the answer modulo $998\,244\,353$.

5 3
1 1 2 2
3 4 5

1

5 3
1 1 2 2
2 3 4

8

20 10
20 8 7 16 3 15 1 10 17 3 13 15 1 17 1 14 14 8 4
3 7 10 19 15 13 4 6 18 5

553508927

Hint

Sample Explanation #1

Only the tree that is exactly the same as $T$ satisfies the requirement.

Sample Explanation #2

For sample 2, all $8$ valid $p$ arrays are as follows (the root has $p_i=0$):

$\{0,1,1,2,1\},\{0,1,1,2,2\},\{0,1,1,2,3\},\{0,1,1,2,4\},$
$\{0,1,1,5,2\},\{0,5,1,2,1\},\{0,1,5,2,1\},\{5,1,1,2,0\}$。

Constraints

This problem uses bundled testdata.

For $100\%$ of the data, $2\le m\le n\le 10^6$. It is guaranteed that the input $p$ describes a labeled rooted tree, and $S$ describes a set of nodes.

Translated by ChatGPT 5