The Forbidden Permutation
Description
You are given a permutation $p$ of length $n$, an array of $m$ distinct integers $a_1, a_2, \ldots, a_m$ ($1 \le a_i \le n$), and an integer $d$.
Let $\mathrm{pos}(x)$ be the index of $x$ in the permutation $p$. The array $a$ is not good if
- $\mathrm{pos}(a_{i}) < \mathrm{pos}(a_{i + 1}) \le \mathrm{pos}(a_{i}) + d$ for all $1 \le i < m$.
For example, with the permutation $p = [4, 2, 1, 3, 6, 5]$ and $d = 2$:
- $a = [2, 3, 6]$ is a not good array.
- $a = [2, 6, 5]$ is good because $\mathrm{pos}(a_1) = 2$, $\mathrm{pos}(a_2) = 5$, so the condition $\mathrm{pos}(a_2) \le \mathrm{pos}(a_1) + d$ is not satisfied.
- $a = [1, 6, 3]$ is good because $\mathrm{pos}(a_2) = 5$, $\mathrm{pos}(a_3) = 4$, so the condition $\mathrm{pos}(a_2) < \mathrm{pos}(a_3)$ is not satisfied.
In one move, you can swap two adjacent elements of the permutation $p$. What is the minimum number of moves needed such that the array $a$ becomes good? It can be shown that there always exists a sequence of moves so that the array $a$ becomes good.
A permutation is an array consisting of $n$ distinct integers from $1$ to $n$ in arbitrary order. For example, $[2,3,1,5,4]$ is a permutation, but $[1,2,2]$ is not a permutation ($2$ appears twice in the array) and $[1,3,4]$ is also not a permutation ($n=3$, but there is $4$ in the array).
Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^4$). The description of the test cases follows.
The first line of each test case contains three integers $n$, $m$ and $d$ ($2\leq n \leq 10^5$, $2\leq m\leq n$, $1 \le d \le n$), the length of the permutation $p$, the length of the array $a$ and the value of $d$.
The second line contains $n$ integers $p_1, p_2, \ldots, p_n$ ($1\leq p_i \leq n$, $p_i \ne p_j$ for $i \ne j$).
The third line contains $m$ distinct integers $a_1, a_2, \ldots, a_m$ ($1\leq a_i \leq n$, $a_i \ne a_j$ for $i \ne j$).
The sum of $n$ over all test cases doesn't exceed $5 \cdot 10^5$.
For each test case, print the minimum number of moves needed such that the array $a$ becomes good.
Input
Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^4$). The description of the test cases follows.
The first line of each test case contains three integers $n$, $m$ and $d$ ($2\leq n \leq 10^5$, $2\leq m\leq n$, $1 \le d \le n$), the length of the permutation $p$, the length of the array $a$ and the value of $d$.
The second line contains $n$ integers $p_1, p_2, \ldots, p_n$ ($1\leq p_i \leq n$, $p_i \ne p_j$ for $i \ne j$).
The third line contains $m$ distinct integers $a_1, a_2, \ldots, a_m$ ($1\leq a_i \leq n$, $a_i \ne a_j$ for $i \ne j$).
The sum of $n$ over all test cases doesn't exceed $5 \cdot 10^5$.
Output
For each test case, print the minimum number of moves needed such that the array $a$ becomes good.
5
4 2 2
1 2 3 4
1 3
5 2 4
5 4 3 2 1
5 2
5 3 3
3 4 1 5 2
3 1 2
2 2 1
1 2
2 1
6 2 4
1 2 3 4 5 6
2 5
1
3
2
0
2
Note
In the first case, $pos(a_1)=1$, $pos(a_2)=3$. To make the array good, one way is to swap $p_3$ and $p_4$. After that, the array $a$ will be good because the condition $\mathrm{pos}(a_2) \le \mathrm{pos}(a_1) + d$ won't be satisfied.
In the second case, $pos(a_1)=1$, $pos(a_2)=4$. The $3$ moves could be:
- Swap $p_3$ and $p_4$.
- Swap $p_2$ and $p_3$.
- Swap $p_1$ and $p_2$.
In the third case, $pos(a_1)=1$, $pos(a_2)=3$, $pos(a_3)=5$. The $2$ moves can be:
- Swap $p_4$ and $p_5$.
- Swap $p_3$ and $p_4$.
In the fourth case, $pos(a_1)=2$, $pos(a_2)=1$. The array $a$ is already good.
In the fifth case, $pos(a_1)=2$, $pos(a_2)=5$. The $2$ moves are:
- Swap $p_1$ and $p_2$.
- Swap $p_5$ and $p_6$.