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```Problem
You are given two non-decreasing sequences A = (A₁, A2,..., AM) and B = (B₁, B2,..., BN).
You can choose any two indices i and j, and then swap A[i]; and B[j].
Note that the you can do the operations as many times as you want, and your goal is to transform A into an
arithmetic sequence. After opeartions, you can rearrange the sequence A.
Now, determine how many distinct arithmetic sequences you can obtain.

Input Format
• The first line contains a single integer T - the number of test cases.
• For each testcase:
• The first line contains two integer M and N-the lengths of the sequences A and B, respectively.
• The second line contains M integers A₁, A2,..., AM.
• The third line contains N integers B₁, B2,..., BN.

Output Format
• For each testcase, output the number of distinct arithmetic sequences you can obtain.

Constraints

• 1 ≤ T ≤ 10³
• 2 ≤ N ≤ M ≤ 105
• -M ≤ A₁, B₁ ≤ M
• The sum of M + N across all test cases does not exceed 2. 10^7.

sample input

33
-10 2
0 1 2
33
0 0 1
0 1 1

Sample Output
4
2
8

out put is : 4```

observation
Because you can swap A[i] and B[j] as many times as you want, your goal is to form distinct arithmetic sequences of length M.
Steps
First, since you are given two sorted sequences, you can merge these two into one in O(M).
Now, you have a sequence C of length M + N, and consider two kinds of arithmetic sequences:
• Common difference is = 0:
That is, the sequence is written as (a, a, ..., a). To find this kind of sequences, count each element in C.
If there are at least M's a in C, then add the desired answer by 1.
• Common difference is != 0:
Also, there may be some arithmetic sequences of the form (a, a + d, a + 2 · d, ..., a + (M − 1) · d) for some d != 0.
In this case, elements in one arithmetic sequence are all different, so please remove duplicate elements in C.
Since the sequence C is non-decreasing, consider the case d > 0 here.
Moreover, since - M ≤ Ai, B; ≤ M and the arithmetic sequence should be of length M, the choices of the common difference d > 0 is finite. In fact, the choices is at
most
2.M/M-1
So, for each possible common difference d, please count the number of arithmetic sequences.
Note that for each common difference d > 0, there would be the corresponding negative common difference -d.

so if the a =[-1,0,1] and b=[0,1,2] and m = 3 I want to know according to give formula 2.m/m-1 (because the merge sequence of this common difference is != 0). which give no of possible choices. how can I obtain the d values. to create the arithmetic sequences.?

What I have tried:

so if a =[-1,0,1] and b=[0,1,2] and m = 3 according to give formula 2.m/m-1 (because the merge sequence of this common difference is != 0). which give number of possible choices. i get the value 3 so there should be 3 possible choices.

here is where I am at so I merged the 2 none decreasing sequences to get the sequence length M+N which C = [-1,0,1,0,1,2] cause the common difference is != 0 I went with the 2nd method("• Common difference is != 0:").

I removed the duplicands and sorted the sequence. so now I got c =[-1,0,1,2] so the sequence values a[1] = -1 , -m <= a[i] if m= 3 I understand -3<=-1 and b[j]<= m which b[1] =1 and 1<= 3. according to the solution and it says that sequence it continue to m. since its none decreasing we consider values d > 0 only.

so it say to get the number of the possible "d" values to create the distinct sequences use the formula 2.M/M-1 and I got answer 3, 3 possible choices form here i only know that the output is [-1,0,1] and [0,1,2] and there negative which counts the total number of 4 distinct arithmetic sequences. I don't know how to get the d values to create the above sequence.

(assumption :- if the possible choice is 3 should i just use 1,2,3 and add each element in to the first of the sequence -1 and continue -1,0,1 or something else? this is where i'm stuck and with out this I can't code)
Posted
Updated 11-May-23 17:21pm
v4
Kenneth Haugland 11-May-23 6:38am
Homework?
ashan prabath 11-May-23 10:57am
no its a part of an assignment.

## Solution 2

What I don't understand is what you are trying to do. An output of 4 from those two sequences makes no sense to me. I assumed you want to determine what the formula for the sequence is so something like what is f(x) to get M[x].

With that in mind, it seems to me you should never merge, remove duplicates, or sort your sequences because that distorts the order and order is extremely important in figuring out the pattern. Here's an example - let's say sequence 1 is [0,2,4] and sequence 2 is [1,3,5]. If you merge and sort those you will have [0,1,2,3,4,5] and that gives an entirely different formula for f(x).

I would first find the differences between successive items. That will tell you a lot about the sequence. The two you showed will have differences of 1. Where this becomes really useful is when the formula for f(x) is M[x] = (2*x)+1 or something similar. That results in a sequence of [1,3,5] and differences of [2,2]. You can think of differences as being like a derivative in calculus. You can integrate a derivative to get the formula so you can approach this similarly : integrate the differences then compare with the original to get the constant offset value. If the differences are not zero then you have potentially a second order formula like f(x) = x^2 + c. In that case you might want to look at the second order differences or second derivative.

I'll stop there. Hopefully this gives you some things to consider.

ashan prabath 11-May-23 22:18pm
@Rick York
I am trying to get the distinct arithmetic sequence according to this.by swapping indices of i and j of A and B.

the distinct arithmetic sequences are (-1,0,1) ,(1,0,-1)
or
(0,1,2),(2,1,0)

i'm trying to find this using the above methord.

## Solution 1

While we are more than willing to help those that are stuck, that doesn't mean that we are here to do it all for you! We can't do all the work, you are either getting paid for this, or it's part of your grades and it wouldn't be at all fair for us to do it all for you.

So we need you to do the work, and we will help you when you get stuck. That doesn't mean we will give you a step by step solution you can hand in!
Start by explaining where you are at the moment, and what the next step in the process is. Then tell us what you have tried to get that next step working, and what happened when you did.

If you are having problems getting started at all, then this may help: How to Write Code to Solve a Problem, A Beginner's Guide[^]

ashan prabath 11-May-23 10:48am
@originalGriff
I accept any help at the moment. And I don't get paid. I need to understand this in Oder to code this. and yes this is part of an assignment.

here is where I am at so I merged the 2 none decreasing sequences to get the sequence length M+N which C = [-1,0,1,0,1,2] cause the common difference is != 0 I went with the 2nd method("• Common difference is != 0:").

I removed the duplicands and sorted the sequence. so now I got c =[-1,0,1,2] so the sequence values a[1] = -1 , -m <= a[i] if m= 3 I understand -3<=-1 and b[j]<= m which b[1] =1 and 1<= 3. according to the solution and it says that sequence it continue to m. since its none decreasing we consider values d > 0 only.

so it say to get the number of the possible "d" values to create the distinct sequences use the formula 2.M/M-1 and I got answer 3, 3 possible choices form here i only know that the output is [-1,0,1] and [0,1,2] and there negative which counts the total number of 4 distinct arithmetic sequences. I don't know how to get the d values to create the above sequence.

(assumption :- if the possible choice is 3 should i just use 1,2,3 and add each element in to the first of the sequence -1 and continue -1,0,1 or something else? this is where i'm stuck and with out this I can't code)

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