Problem p03006 - Generation 1

Orig Description

Score : 300 points
Problem StatementThere are N balls in a two-dimensional plane. The i-th ball is at coordinates (x_i, y_i).
We will collect all of these balls, by choosing two integers p and q such that p \neq 0 or q \neq 0 and then repeating the following operation:
Choose a ball remaining in the plane and collect it. Let (a, b) be the coordinates of this ball. If we collected a ball at coordinates (a - p, b - q) in the previous operation, the cost of this operation is 0. Otherwise, including when this is the first time to do this operation, the cost of this operation is 1.
Find the minimum total cost required to collect all the balls when we optimally choose p and q.
Constraints
1 \leq N \leq 50
|x_i|, |y_i| \leq 10^9
If i \neq j, x_i \neq x_j or y_i \neq y_j.
All values in input are integers.
InputInput is given from Standard Input in the following format:
N
x_1 y_1
:
x_N y_N
OutputPrint the minimum total cost required to collect all the balls.
Sample Input 12
1 1
2 2
Sample Output 11
If we choose p = 1, q = 1, we can collect all the balls at a cost of 1 by collecting them in the order (1, 1), (2, 2).
Sample Input 23
1 4
4 6
7 8
Sample Output 21
If we choose p = -3, q = -2, we can collect all the balls at a cost of 1 by collecting them in the order (7, 8), (4, 6), (1, 4).
Sample Input 34
1 1
1 2
2 1
2 2
Sample Output 32

Extracted Specification

### Canonicalized Input Description:

An integer N (1 ≤ N ≤ 50), representing some quantity or size.
N pairs of integers (x, y), where each integer x and y is between -10^9 and 10^9. No two pairs (x, y) have the same x or the same y.

### Example Input:

```
2
1 1
2 2
```

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

Test Case Generator

Sure! Here is a test case generator for the given problem statement:

```python
import random

def tcgen():
    # Generate a random number of balls N (1 ≤ N ≤ 50)
    N = random.randint(1, 50)
    
    # Generate N unique coordinates (x_i, y_i) where |x_i|, |y_i| ≤ 10^9 and no two balls have the same coordinates
    coordinates = set()
    while len(coordinates) < N:
        x = random.randint(-10**9, 10**9)
        y = random.randint(-10**9, 10**9)
        coordinates.add((x, y))
    
    # Convert the set of coordinates to a list
    coordinates = list(coordinates)
    
    # Format the input in the required structure
    input_data = f"{N}\n" + "\n".join(f"{x} {y}" for x, y in coordinates)
    
    return input_data

# Example usage
if __name__ == "__main__":
    test_case = tcgen()
    print(test_case)
```

### Explanation:

1. **Generating N**: The number of balls `N` is randomly selected between 1 and 50.
2. **Generating Coordinates**: Using a set to store coordinates ensures that all coordinates are unique. Each coordinate (x, y) is randomly selected such that \(|x|, |y| \leq 10^9\).
3. **Formatting Input**: The generated coordinates are formatted into the required input format for the problem: first the number `N`, followed by each pair of coordinates on a new line.
   
This code ensures that the generated test cases adhere to the problem constraints and are formatted correctly for use in competitive programming.

Extract Arguments

def extract_arguments(fh):
    N = int(fh.readline().strip())
    coordinates = []
    for _ in range(N):
        x, y = map(int, fh.readline().strip().split())
        coordinates.append((x, y))
    return N, coordinates