Problem p01541 - Generation 1

Orig Description

Long and Narrow
 Takahashi and Ringo-chan decided to go sightseeing in New York. Takahashi had no particular place he wanted to visit, but due to Ringo-chan's enthusiastic invitation, they decided to visit the "long and narrow" place shown on the map below. 
Google Map -(C)2012 Google
The "long and narrow" place is very long and can be considered as a straight line. There are several tourist carriages running from the starting point to the goal point on this "long and narrow" place. These carriages are operated according to the following rules. 
There are n carriages, and all carriages depart from the starting point and move towards the goal point.
For all cases below, carriage size can be ignored.
Since it is difficult to change speed along the way, all carriages move at a constant speed of Si (an integer of 1 or more) minutes per 1 km.
Carriages have a predetermined departure order, which must be followed. However, there is no particular rule regarding the order in which carriages arrive at the goal point.
Multiple carriages cannot depart from the starting point at the same time, and there must be at least 1 minute between the departure of one carriage and the next carriage.
The goal point is large enough that it doesn't matter how many carriages arrive at the same time. Also, carriages that arrive at the goal point

Extracted Specification

An integer n (1 ≤ n ≤ 10^3), representing some quantity or size.
A list of integers S of size n, where each integer is between 1 and 10^5.

### Example Input:

```
5
10 20 30 40 50
```

### Function Signature:

Write a function f(n, S) that takes in the input.
```python
def f(n: int, S: List[int]):
    ''' 
    n: an integer 
    S: a list of integers 
    '''
```

Test Case Generator

Sure, I can help you write a test case generator for the given problem statement "Long and Narrow." Let's first summarize the problem statement and constraints:

### Problem Statement: Long and Narrow

**Description:**

1. There are \(n\) carriages, each moving from the starting point to the goal point on a straight line.
2. Each carriage moves at a constant speed of \(S_i\) minutes per 1 km.
3. Carriages have a predetermined departure order.
4. Carriages cannot depart at the same time; there must be at least 1 minute between the departure of one carriage and the next.
5. The goal point can accommodate all carriages arriving at the same time.

**Input:**

1. An integer \(n\) (1 ≤ \(n\) ≤ 10^5), representing the number of carriages.
2. An array of integers \(S\) of size \(n\), where each integer \(S[i]\) (1 ≤ \(S[i]\) ≤ 10^9) represents the speed of the \(i\)-th carriage in minutes per 1 km.

**Output:**

The departure and arrival times of the carriages are not directly needed for the test case generation but may be useful for validating the correctness of a solution.

### Test Case Generator

Here is a Python function that generates random test cases for the problem:

```python
import random

def tcgen():
    # Number of carriages
    n = random.randint(1, 10**5)
    
    # Speeds of the carriages
    S = [random.randint(1, 10**9) for _ in range(n)]
    
    return n, S

# Example usage
if __name__ == "__main__":
    n, S = tcgen()
    print(n)
    print(' '.join(map(str, S)))
```

### Explanation:

1. `n` is chosen randomly between 1 and 100,000.
2. `S` is a list of `n` random integers, each between 1 and 1,000,000,000, representing the speed of each carriage.

This function will generate a valid random test case for the "Long and Narrow" problem, ensuring that the input constraints are respected.

Extract Arguments

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