Boris and His Amazing Haircut
Contest: Hello 2023
Difficulty rating: 1700
Tags: constructive algorithms, data structures, dp, dsu, greedy, sortings

Boris thinks that chess is a tedious game. So he left his tournament early and went to a barber shop as his hair was a bit messy.

His current hair can be described by an array $$$a_1,a_2,\ldots, a_n$$$, where $$$a_i$$$ is the height of the hair standing at position $$$i$$$. His desired haircut can be described by an array $$$b_1,b_2,\ldots, b_n$$$ in a similar fashion.

The barber has $$$m$$$ razors. Each has its own size and can be used at most once. In one operation, he chooses a razor and cuts a segment of Boris's hair. More formally, an operation is:

- Choose any razor which hasn't been used before, let its size be $$$x$$$;
- Choose a segment $$$[l,r]$$$ ($$$1\leq l \leq r \leq n$$$);
- Set $$$a_i := \min (a_i,x)$$$ for each $$$l\leq i \leq r$$$;

Notice that some razors might have equal sizes — the barber can choose some size $$$x$$$ only as many times as the number of razors with size $$$x$$$.

He may perform as many operations as he wants, as long as any razor is used at most once and $$$a_i = b_i$$$ for each $$$1 \leq i \leq n$$$ at the end. He does not have to use all razors.

Can you determine whether the barber can give Boris his desired haircut?

Input format:
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \leq t \leq 20\,000$$$). The description of the test cases follows.

The first line of each test case contains a positive integer $$$n$$$ ($$$3\leq n\leq 2\cdot 10^5$$$) — the length of arrays $$$a$$$ and $$$b$$$.

The second line of each test case contains $$$n$$$ positive integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \leq a_i \leq 10^9$$$)  — Boris's current hair.

The third line of each test case contains $$$n$$$ positive integers $$$b_1, b_2, \ldots, b_n$$$ ($$$1 \leq b_i \leq 10^9$$$)  — Boris's desired hair.

The fourth line of each test case contains a positive integer $$$m$$$ ($$$1 \leq m \leq 2\cdot 10^5$$$)  — the number of razors.

The fifth line of each test case contains $$$m$$$ positive integers $$$x_1,x_2, \ldots, x_m$$$ ($$$1 \leq x_i \leq 10^9$$$)  — the sizes of the razors.

It is guaranteed that the sum of $$$n$$$ and the sum of $$$m$$$ over all test cases do not exceed $$$2\cdot 10^5$$$.

Output format:
For each test case, print "YES" if the barber can cut Boris's hair as desired. Otherwise, print "NO".

You can output the answer in any case (upper or lower). For example, the strings "yEs", "yes", "Yes", and "YES" will be recognized as positive responses.

Constraints:
Time limit: 2.0 s
Memory limit: 256.0 MB

Note:
In the first test case, Boris's hair is initially $$$[3,3,3]$$$. Let us describe a sequence of $$$2$$$ operations the barber can perform:

- The barber uses the razor with size $$$1$$$ on the segment $$$[2,2]$$$; hence Boris's hair becomes $$$[3,1,3]$$$.
- The barber uses the razor with size $$$2$$$ on the segment $$$[1,3]$$$; hence Boris's hair becomes $$$[2,1,2]$$$ which is the desired haircut.

In the third test case, no operation has to be done since Boris's hair is already as desired.

In the fourth test case, no cuts will be able to increase the third element in $$$[1,1,1]$$$ in a way that the array becomes $$$[1,1,2]$$$.

Example input:
7
3
3 3 3
2 1 2
2
1 2
6
3 4 4 6 3 4
3 1 2 3 2 3
3
3 2 3
10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
10
1 2 3 4 5 6 7 8 9 10
3
1 1 1
1 1 2
12
4 2 4 3 1 5 6 3 5 6 2 1
13
7 9 4 5 3 3 3 6 8 10 3 2 5
5 3 1 5 3 2 2 5 8 5 1 1 5
8
1 5 3 5 4 2 3 1
13
7 9 4 5 3 3 3 6 8 10 3 2 5
5 3 1 5 3 2 2 5 8 5 1 1 5
7
1 5 3 4 2 3 1
3
19747843 2736467 938578397
2039844 2039844 2039844
1
2039844

Example output:
YES
NO
YES
NO
YES
NO
YES

Editorial:
If ai<bi for some i, then an answer does not exist since a cut cannot make a hair taller.
If you choose to perform a cut on some segment [l,r] with a razor of size x, you can "greedily" extend it (decrease l and increase r) as long as x≥ai for each i in that segment and still obtain a correct solution.
There exist some data structures (segment tree, dsu, STL maps and sets, …) ideas, but there is also a simple solution with a STL stack.
Consider an and bn. If bn is greater, the answer is NO since it is an impossible case (see "Stupid Hint" section). If an is greater, then a cut on range [l,n] with a razor of size bn has to be performed. Additionally, l should be as small as possible (see "Hint" section). For each i in the range, if ai becomes exactly equal to bi, we consider the position i satisfied. If an and bn are equal, then we simply pop both arrays' ends (we ignore those values as they are already satisfied) and we continue our algorithm.
Onto the implementation. We keep track (and count) of each razor size we must use. This can simply be done by putting the corresponding sizes into some container (array or vector) and checking at the end whether it is a subset of x1,x2,…xm (the input array of allowed razors). To do this, one can use sorting or maps. This part works in O(nlogn+mlogm) time.
Implementing cuts is more challenging, though. To do this, we keep a monotone stack which represents all cuts which are valid until "now" (more formally, all cuts with their l being ≤i, and the value of l will be determined later). The top of the stack will represent the smallest razor, and the size of razors does not decrease as we pop it. So, we pop the stack as long as the top is smaller than bn (since performing an operation which makes ai less than bi is not valid). After this, if the new top is exactly equal to bn we can conclude that we have satisfied an=bn with some previous cut and we simply continue our algorithm. Otherwise, we add bn to the stack as a cut must be performed. This part works in O(n+m) time.
Total complexity is O(nlogn+mlogm) because of sorting/mapping.