PermutationForces II
Description
You are given a permutation $a$ of length $n$. Recall that permutation is an array consisting of $n$ distinct integers from $1$ to $n$ in arbitrary order.
You have a strength of $s$ and perform $n$ moves on the permutation $a$. The $i$-th move consists of the following:
- Pick two integers $x$ and $y$ such that $i \leq x \leq y \leq \min(i+s,n)$, and swap the positions of the integers $x$ and $y$ in the permutation $a$. Note that you can select $x=y$ in the operation, in which case no swap will occur.
You want to turn $a$ into another permutation $b$ after $n$ moves. However, some elements of $b$ are missing and are replaced with $-1$ instead. Count the number of ways to replace each $-1$ in $b$ with some integer from $1$ to $n$ so that $b$ is a permutation and it is possible to turn $a$ into $b$ with a strength of $s$.
Since the answer can be large, output it modulo $998\,244\,353$.
The input consists of multiple test cases. The first line contains an integer $t$ ($1 \leq t \leq 1000$) — the number of test cases. The description of the test cases follows.
The first line of each test case contains two integers $n$ and $s$ ($1 \leq n \leq 2 \cdot 10^5$; $1 \leq s \leq n$) — the size of the permutation and your strength, respectively.
The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le n$) — the elements of $a$. All elements of $a$ are distinct.
The third line of each test case contains $n$ integers $b_1, b_2, \ldots, b_n$ ($1 \le b_i \le n$ or $b_i = -1$) — the elements of $b$. All elements of $b$ that are not equal to $-1$ are distinct.
It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$.
For each test case, output a single integer — the number of ways to fill up the permutation $b$ so that it is possible to turn $a$ into $b$ using a strength of $s$, modulo $998\,244\,353$.
Input
The input consists of multiple test cases. The first line contains an integer $t$ ($1 \leq t \leq 1000$) — the number of test cases. The description of the test cases follows.
The first line of each test case contains two integers $n$ and $s$ ($1 \leq n \leq 2 \cdot 10^5$; $1 \leq s \leq n$) — the size of the permutation and your strength, respectively.
The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le n$) — the elements of $a$. All elements of $a$ are distinct.
The third line of each test case contains $n$ integers $b_1, b_2, \ldots, b_n$ ($1 \le b_i \le n$ or $b_i = -1$) — the elements of $b$. All elements of $b$ that are not equal to $-1$ are distinct.
It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$.
Output
For each test case, output a single integer — the number of ways to fill up the permutation $b$ so that it is possible to turn $a$ into $b$ using a strength of $s$, modulo $998\,244\,353$.
Samples
<div class="test-example-line test-example-line-even test-example-line-0">6</div><div class="test-example-line test-example-line-odd test-example-line-1">3 1</div><div class="test-example-line test-example-line-odd test-example-line-1">2 1 3</div><div class="test-example-line test-example-line-odd test-example-line-1">3 -1 -1</div><div class="test-example-line test-example-line-even test-example-line-2">3 2</div><div class="test-example-line test-example-line-even test-example-line-2">2 1 3</div><div class="test-example-line test-example-line-even test-example-line-2">3 -1 -1</div><div class="test-example-line test-example-line-odd test-example-line-3">4 1</div><div class="test-example-line test-example-line-odd test-example-line-3">1 4 3 2</div><div class="test-example-line test-example-line-odd test-example-line-3">4 3 1 2</div><div class="test-example-line test-example-line-even test-example-line-4">6 4</div><div class="test-example-line test-example-line-even test-example-line-4">4 2 6 3 1 5</div><div class="test-example-line test-example-line-even test-example-line-4">6 1 5 -1 3 -1</div><div class="test-example-line test-example-line-odd test-example-line-5">7 4</div><div class="test-example-line test-example-line-odd test-example-line-5">1 3 6 2 7 4 5</div><div class="test-example-line test-example-line-odd test-example-line-5">2 5 -1 -1 -1 4 -1</div><div class="test-example-line test-example-line-even test-example-line-6">14 14</div><div class="test-example-line test-example-line-even test-example-line-6">1 2 3 4 5 6 7 8 9 10 11 12 13 14</div><div class="test-example-line test-example-line-even test-example-line-6">-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1</div><div class="test-example-line test-example-line-even test-example-line-6"></div>
1
2
0
2
12
331032489
Note
In the first test case, $a=[2,1,3]$. There are two possible ways to fill out the $-1$s in $b$ to make it a permutation: $[3,1,2]$ or $[3,2,1]$. We can make $a$ into $[3,1,2]$ with a strength of $1$ as follows: $$[2,1,3] \xrightarrow[x=1,\,y=1]{} [2,1,3] \xrightarrow[x=2,\,y=3]{} [3,1,2] \xrightarrow[x=3,\,y=3]{} [3,1,2].$$ It can be proven that it is impossible to make $[2,1,3]$ into $[3,2,1]$ with a strength of $1$. Thus only one permutation $b$ satisfies the constraints, so the answer is $1$.
In the second test case, $a$ and $b$ the same as the previous test case, but we now have a strength of $2$. We can make $a$ into $[3,2,1]$ with a strength of $2$ as follows: $$[2,1,3] \xrightarrow[x=1,\,y=3]{} [2,3,1] \xrightarrow[x=2,\,y=3]{} [3,2,1] \xrightarrow[x=3,\,y=3]{} [3,2,1].$$ We can still make $a$ into $[3,1,2]$ using a strength of $1$ as shown in the previous test case, so the answer is $2$.
In the third test case, there is only one permutation $b$. It can be shown that it is impossible to turn $a$ into $b$, so the answer is $0$.