A15577 | The Forbidden Permutation

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.