Problem p01873 - Generation 3

Orig Description

B: Periodic Sequence
Problem
Dr. Period, who serves as a professor at H University, researches the property called the "period" that is said to lurk in everything. As a basic period generally known, a period lurking in a sequence could be considered. That is, if the length N sequence S = S_1, S_2, ... , S_N satisfies the following property, it is a fact that it has a period t (t \≤ N).
For 1 \≤ i \≤ N − t, S_i=S_{i+t}.
Now, Dr. Period is focusing on sequences that can be described more simply using periods. For example, if a length N sequence has a period t (\≤ N), and it can be written as N=kt using an integer k, the sequence can be described as a length t sequence S_1, ... , S_t that is a concatenation of k continuous ones. Dr. Period has decided to call such a sequence a k-part.
Dr. Period is interested in the largest k-part. Therefore, you, as an assistant, have been asked to create a program that takes a sequence as input and outputs the largest k when it is a k-part. Create a program that accurately responds to Dr. Period's request.
Input Format
N
S_1 ... S_N
The first line contains an integer N, which represents the length of the sequence. The second line contains integers S_i (1 \≤ i \≤ N) that represent each element of the length N sequence, separated by spaces. The input satisfies 1 \≤ N \≤ 200,000 and 1 \≤ S_i \≤ 100,000 (1 \≤ i \≤ N).
Output Format
For the given sequence, output the maximum value of k when it is a k-part on one line.
Sample Input 1
6
1 2 3 1 2 3
Sample Output 1
2
Sample Input 2
12
1 2 1 2 1 2 1 2 1 2 1 2
Sample Output 2
6
Sample Input 3
6
1 2 3 4 5 6
Sample Output 3
1

Extracted Specification

An integer N (1 ≤ N ≤ 200,000), representing some quantity or size.
A list of N integers S, where each integer is between 1 and 100,000.

### Example Input:

```
6
1 2 3 1 2 3
```

### Function Signature:
Write a function f(N, S) that takes in the input.
def f(N: int, S: List[int]):
    '''
    N: an integer
    S: a list of integers
    '''

Test Case Generator

Certainly! Here is a test case generator for the given problem statement "Periodic Sequence":

```python
import random

def generate_periodic_sequence(N, t):
    k = N // t
    base_sequence = [random.randint(1, 100000) for _ in range(t)]
    return base_sequence * k

def tcgen():
    N = random.randint(1, 200000)  # Length of the sequence
    t = random.randint(1, N)  # Period length

    # Generate a periodic sequence with period t
    S = generate_periodic_sequence(N, t)
    
    return N, S

# Example of generating a test case
N, S = tcgen()
print(N)
print(" ".join(map(str, S)))
```

**Explanation:**

1. **`generate_periodic_sequence(N, t)`**: This function generates a sequence of length `N` with a period `t`. It first creates a base sequence of length `t` with random values between 1 and 100000. Then, it repeats this base sequence `k` times to form the full sequence of length `N`, where `k` is the integer division of `N` by `t`.

2. **`tcgen`**: This function generates a random test case:
   - It picks a random `N` between 1 and 200000.
   - It picks a random `t` between 1 and `N`.
   - It uses the `generate_periodic_sequence` function to generate the sequence `S` with the chosen `N` and `t`.

3. **Printing the Test Case**: The generated sequence `S` is printed in a format suitable for the problem's input specification.

This generator will create test cases that help in assessing the program's ability to identify the largest k-part in a sequence, ensuring diverse and rigorous testing of the implementation.

Extract Arguments

def extract_arguments(fh):
    N = int(fh.readline().strip())
    S = list(map(int, fh.readline().strip().split()))
    return N, S