This is the hard version of the problem. In this version, you are given a generic tree and the constraints on $$$n$$$ and $$$q$$$ are higher. You can make hacks only if both versions of the problem are solved.

You are given a rooted tree consisting of $$$n$$$ vertices. The vertices are numbered from $$$1$$$ to $$$n$$$, and the root is the vertex $$$1$$$. You are also given a permutation $$$p_1, p_2, \ldots, p_n$$$ of $$$[1,2,\ldots,n]$$$.

You need to answer $$$q$$$ queries. For each query, you are given two integers $$$x$$$, $$$y$$$; you need to swap $$$p_x$$$ and $$$p_y$$$ and determine if $$$p_1, p_2, \ldots, p_n$$$ is a valid DFS order$$$^\dagger$$$ of the given tree.

Please note that the swaps are persistent through queries.

$$$^\dagger$$$ A DFS order is found by calling the following $$$\texttt{dfs}$$$ function on the given tree.

Note that the DFS order is not unique.

Input format:
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1\le t\le10^4$$$). The description of the test cases follows.

The first line of each test case contains two integers $$$n$$$, $$$q$$$ ($$$2\le n\le 3\cdot 10^5$$$, $$$2\le q\le 10^5$$$) — the number of vertices in the tree and the number of queries.

The next line contains $$$n-1$$$ integers $$$a_2,a_3,\ldots,a_n$$$ ($$$1\le a_i<i$$$) — the parent of each vertex in the given tree.

The next line contains $$$n$$$ integers $$$p_1,p_2,\ldots,p_n$$$ ($$$1\le p_i\le n$$$, all $$$p_i$$$ are distinct) — the initial permutation $$$p$$$.

The next $$$q$$$ lines each contain two integers $$$x$$$, $$$y$$$ ($$$1\le x,y\le n,x\neq y$$$) — the positions of the elements to swap in the permutation.

It is guaranteed that the sum of all $$$n$$$ does not exceed $$$3\cdot 10^5$$$, and the sum of all $$$q$$$ does not exceed $$$10^5$$$.
Output format:
For each test case, print $$$q$$$ lines corresponding to the $$$q$$$ queries. For each query, output $$$\texttt{YES}$$$ if there is a DFS order that exactly equals the current permutation, and output $$$\texttt{NO}$$$ otherwise.

You can output $$$\texttt{Yes}$$$ and $$$\texttt{No}$$$ in any case (for example, strings $$$\texttt{yEs}$$$, $$$\texttt{yes}$$$, $$$\texttt{Yes}$$$, and $$$\texttt{YES}$$$ will be recognized as a positive response).
Example Input 0:
3
3 3
1 1
1 2 3
2 3
3 2
1 3
7 4
1 1 2 2 3 3
1 2 3 4 5 6 7
3 5
2 5
3 7
4 6
5 4
1 1 3 4
2 3 4 5 1
5 1
4 5
3 4
2 3

Example Output 0:
YES
YES
NO
YES
NO
NO
YES
YES
NO
NO
YES

Notes:
In the first test case, the permutation $$$p_1, p_2, \ldots, p_n$$$ after each modification is $$$[1,3,2],[1,2,3],[3,2,1]$$$, respectively. The first two permutations are valid DFS orders; the third is not a DFS order.

In the second test case, the permutation $$$p_1, p_2, \ldots, p_n$$$ after each modification is $$$[1,2,5,4,3,6,7],[1,3,5,4,2,6,7],[1,3,7,4,2,6,5],[1,3,7,6,2,4,5]$$$, respectively.

Example input:
3
3 3
1 1
1 2 3
2 3
3 2
1 3
7 4
1 1 2 2 3 3
1 2 3 4 5 6 7
3 5
2 5
3 7
4 6
5 4
1 1 3 4
2 3 4 5 1
5 1
4 5
3 4
2 3


Example output:
YES
YES
NO
YES
NO
NO
YES
YES
NO
NO
YES