Problem p04049 - Generation 2

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 another quantity or size.
A list of pairs of integers [(A1, B1), (A2, B2), ..., (AN-1, BN-1)], where each integer in the pairs 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. 
```python
def f(N: int, K: int, edges: List[Tuple[int, int]]):
    '''
    N: an integer
    K: an integer
    edges: 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 generate_tree(n):
    edges = []
    for i in range(2, n + 1):
        # Connect each new node to a random existing node to form a tree
        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)
    
    return N, K, edges

def format_input(N, K, edges):
    input_str = f"{N} {K}\n"
    input_str += "\n".join(f"{a} {b}" for a, b in edges)
    return input_str

# Example of generating a test case and formatting it
N, K, edges = tcgen()
print(format_input(N, K, edges))
```

This test case generator will create valid random trees and format the input appropriately to fit the problem's requirements. Here's a step-by-step breakdown of the code:

1. **generate_tree(n)**:
    - Generates a random tree with `n` nodes.
    - Ensures the created structure is a valid tree by connecting each new node to a previously existing one.

2. **tcgen()**:
    - Randomly selects `N` (number of vertices) between 2 and 2000.
    - Randomly selects `K` (maximum allowed diameter) between 1 and `N-1`.
    - Uses the `generate_tree(N)` function to create the edges of the tree.

3. **format_input(N, K, edges)**:
    - Formats the generated tree and parameters into the required input format.

This generator ensures that you get a variety of test cases, from very small trees to large ones close to the upper constraints, with random connections ensuring different tree shapes.

Extract Arguments

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