The Forbidden Permutation

2000ms

262144K

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).

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.

Sample Input:

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

Sample Output:

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 $$$p_2$$$.

After these moves, the permutation $$$p$$$ will be $$$[2,5,4,3,1]$$$. The array $$$a$$$ will be good because the condition $$$\mathrm{pos}(a_1) < \mathrm{pos}(a_2)$$$ won't be satisfied. It can be shown that you can't make the array $$$a$$$ good with fewer moves.

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 $$$p_4$$$.

After these moves, the permutation $$$p$$$ will be $$$[3,4,2,1,5]$$$. The array $$$a$$$ will be good because the condition $$$\mathrm{pos}(a_2) < \mathrm{pos}(a_3)$$$ won't be satisfied. It can be shown that you can't make the array $$$a$$$ good with fewer moves.

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 $$$p_6$$$.

Informação

Provedor Codeforces

Código CF1778B