Problem p03615 - Generation 3

Orig Description

Score : 700 points
Problem StatementYou are given N points (x_i,y_i) located on a two-dimensional plane.
Consider a subset S of the N points that forms a convex polygon.
Here, we say a set of points S forms a convex polygon when there exists a convex polygon with a positive area that has the same set of vertices as S. All the interior angles of the polygon must be strictly less than 180°.
For example, in the figure above, {A,C,E} and {B,D,E} form convex polygons; {A,C,D,E}, {A,B,C,E}, {A,B,C}, {D,E} and {} do not.
For a given set S, let n be the number of the points among the N points that are inside the convex hull of S (including the boundary and vertices). Then, we will define the score of S as 2^{n-|S|}.
Compute the scores of all possible sets S that form convex polygons, and find the sum of all those scores.
However, since the sum can be extremely large, print the sum modulo 998244353.
Constraints
1≤N≤200
0≤x_i,y_i<10^4 (1≤i≤N)
If i≠j, x_i≠x_j or y_i≠y_j.
x_i and y_i are integers.
InputThe input is given from Standard Input in the following format:
N
x_1 y_1
x_2 y_2
:
x_N y_N
OutputPrint the sum of all the scores modulo 998244353.
Sample Input 14
0 0
0 1
1 0
1 1
Sample Output 15
We have five possible sets as S, four sets that form triangles and one set that forms a square. Each of them has a score of 2^0=1, so the answer is 5.
Sample Input 25
0 0
0 1
0 2
0 3
1 1
Sample Output 211
We have three "triangles" with a score of 1 each, two "triangles" with a score of 2 each, and one "triangle" with a score of 4. Thus, the answer is 11.
Sample Input 31
3141 2718
Sample Output 30
There are no possible set as S, so the answer is 0.

Extracted Specification

An integer N (1 ≤ N ≤ 200), representing some quantity or size.
N pairs of integers (x_i, y_i) where each x_i and y_i is between 0 and 10000. All pairs are distinct.

### Example Input:

```
4
0 0
0 1
1 0
1 1
```

### 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 tuples, where each tuple contains two integers
    '''

Test Case Generator

Certainly! Below is a test case generator for the given problem statement.

```python
import random

def generate_test_case():
    # Constraints
    N = random.randint(1, 200)
    points = set()
    
    while len(points) < N:
        x = random.randint(0, 9999)
        y = random.randint(0, 9999)
        # Ensure all points are unique
        if (x, y) not in points:
            points.add((x, y))
    
    points_list = list(points)
    
    return N, points_list

def format_test_case(N, points_list):
    formatted_input = f"{N}\n"
    for x, y in points_list:
        formatted_input += f"{x} {y}\n"
        
    return formatted_input

# Example Usage
N, points_list = generate_test_case()
test_case_input = format_test_case(N, points_list)
print(test_case_input)
```

This generator creates a test case for the given problem by generating a random number \(N\) of unique points within the specified range [0, 9999] for both \(x\) and \(y\) coordinates.

### Steps to Generate a Test Case:

1. **Generate a Random Number of Points**: `N` is chosen randomly between 1 and 200.
2. **Generate Unique Points**: Use a set to ensure the generated points are unique.
3. **Format the Input**: Convert the list of points into the required input format for the problem.

### Example Usage

Running the generator will produce a test case in the format expected by the problem. The `format_test_case` function converts the generated points into a string format that mimics standard input.

### Example Output

Here's an example of what the output might look like:

```
5
1234 5678
2345 6789
3456 7890
4567 8901
5678 9012
```

This output can be directly used as input for your problem, ensuring it adheres to the constraints and requirements given in the problem statement.

Extract Arguments

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

# Example usage:
# if __name__ == "__main__":
#     input_path = sys.argv[1]
#     with open(input_path, 'r') as fh: 
#     N, points = extract_arguments(fh)
#     print(N, points)