Problem p04049 - Generation 1

Orig Description

Score : 600 points
Problem StatementGiven an undirected tree, let the distance between vertices u and v be the
number of edges on the simple path from u to v.
The diameter of a tree is the maximum among the distances between any two vertices.
We will call a tree good if and only if its diameter is at most K.
You are given an undirected tree with N vertices numbered 1 through N.
For each i (1≦i≦N-1), there is an edge connecting vertices A_i and B_i.
You want to remove zero or more vertices from the tree, so that the resulting tree is good.
When a vertex is removed, all incident edges will also be removed.
The resulting graph must be connected.
Find the minimum number of vertices that you need to remove in order to produce a good tree.
Constraints
2≦N≦2000
1≦K≦N-1
1≦A_i≦N, 1≦B_i≦N
The graph defined by A_i and B_i is a tree.
InputThe input is given from Standard Input in the following format:
N K
A_1 B_1
A_2 B_2
:
A_{N-1} B_{N-1}
OutputPrint the minimum number of vertices that you need to remove in order to produce a good tree.
Sample Input 16 2
1 2
3 2
4 2
1 6
5 6
Sample Output 12
The tree is shown below. Removing vertices 5 and 6 will result in a good tree with the diameter of 2.
Sample Input 26 5
1 2
3 2
4 2
1 6
5 6
Sample Output 20
Since the given tree is already good, you do not need to remove any vertex.

Extracted Specification

An integer N (2 ≤ N ≤ 2000), representing some quantity or size.
An integer K (1 ≤ K ≤ N-1).
A list of tuples, each containing two integers (A_i, B_i) where (1 ≤ A_i ≤ N and 1 ≤ B_i ≤ N), representing pairs of connections.

### Example Input:

```
6 2
1 2
3 2
4 2
1 6
5 6
```

### Function Signature:
Write a function f(N, K, edges) that takes in the input.
```python
def f(N: int, K: int, edges: List[Tuple[int, int]]):
    '''
    N: an integer
    K: an integer
    edges: a list of tuples, each containing two integers
    '''
```

Test Case Generator

Sure, I can help you write a test case generator for the given problem statement about finding the minimum number of vertices to remove in order to produce a good tree. Below is a Python function that generates random test cases for this problem.

```python
import random

def generate_tree(N):
    edges = []
    for i in range(2, N + 1):
        # Connect each new node to an existing random node to ensure a tree structure
        edges.append((i, random.randint(1, i - 1)))
    return edges

def tcgen():
    N = random.randint(2, 2000)
    K = random.randint(1, N - 1)
    
    edges = generate_tree(N)
    
    # Convert edges list to the required format
    edge_list = "\n".join(f"{a} {b}" for a, b in edges)
    
    # Combine everything into the required input format
    test_case = f"{N} {K}\n" + edge_list
    
    return test_case

# Example of generating a test case
print(tcgen())
```

### Explanation

1. **Function `generate_tree(N)`**:
    - This function generates a random tree with `N` nodes.
    - It ensures that the tree is valid by connecting each new node to an existing random node, thus maintaining the tree structure.

2. **Function `tcgen()`**:
    - Randomly selects `N` in the range `[2, 2000]` and `K` in the range `[1, N-1]`.
    - Generates the edges for the tree using `generate_tree(N)`.
    - Formats the edges into the required input string format.
    - Returns the complete test case as a formatted string.

### Example Output

When you run `print(tcgen())`, you might get an output like:

```
299 150
2 1
3 1
4 2
5 4
6 5
...
299 298
```

This output represents a test case where `N=299`, `K=150`, and the subsequent lines represent the edges of the tree.

### Notes

- This generator ensures that the tree structure is valid by always connecting a new node to an existing node.
- The randomness in node connections ensures diversity in test cases.
- The edges are formatted to match the required input format for competitive programming. 

You can use this generator to create many different test cases to test the robustness and correctness of your solution to the problem.

Extract Arguments

def extract_arguments(fh):
    first_line = fh.readline().strip().split()
    N = int(first_line[0])
    K = int(first_line[1])
    edges = []
    for _ in range(N-1):
        edge = tuple(map(int, fh.readline().strip().split()))
        edges.append(edge)
    return N, K, edges

# if __name__ == "__main__":
#     input_path = sys.argv[1]
#     with open(input_path, 'r') as fh: 
#     N, K, edges = extract_arguments(fh)
#     f(N, K, edges)