Problem p04049 - Generation 3

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), representing some other quantity or size.
A list of N-1 pairs of integers (A_i, B_i), where each integer is between 1 and N.

### 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. 
def f(N: int, K: int, edges: List[Tuple[int, int]]): 
    ''' 
    N: an integer 
    K: an integer
    edges: a list of tuples of integers
    '''

Test Case Generator

To write a test case generator for the given problem statement, we need to generate the following:

1. An integer \(N\) (2 ≤ \(N\) ≤ 2000), representing the number of vertices in the tree.
2. An integer \(K\) (1 ≤ \(K\) ≤ \(N-1\)), representing the maximum allowed diameter for the tree to be considered good.
3. \(N-1\) pairs of integers \(A_i\) and \(B_i\) (1 ≤ \(A_i\), \(B_i\) ≤ \(N\)), representing the edges of the tree.

The generated tree must be a valid undirected tree, meaning it should be connected and acyclic.

Here is a Python function to generate such test cases:

```python
import random
import networkx as nx

def generate_tree(N):
    """Generate a random tree with N nodes."""
    G = nx.generators.trees.random_tree(N)
    edges = list(G.edges)
    return edges

def tcgen():
    # Generate random N and K
    N = random.randint(2, 2000)
    K = random.randint(1, N-1)
    
    # Generate a random tree with N nodes
    edges = generate_tree(N)
    
    # Convert edges to the required input format
    A = [edge[0] + 1 for edge in edges]
    B = [edge[1] + 1 for edge in edges]
    
    return N, K, list(zip(A, B))

def print_test_case(N, K, edges):
    print(N, K)
    for A_i, B_i in edges:
        print(A_i, B_i)

# Example usage:
N, K, edges = tcgen()
print_test_case(N, K, edges)
```

This code uses the `networkx` library to generate a random tree and then formats the edges accordingly. The `generate_tree` function ensures that the generated tree is connected and acyclic. The `tcgen` function handles the random selection of \(N\) and \(K\) and the generation of the tree. Finally, the `print_test_case` function formats the output in the required format.

Make sure to install the `networkx` library if you haven't already:

```bash
pip install networkx
```

This will properly generate valid test cases conforming to the constraints given in the problem statement.

Extract Arguments

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