Problem p02752 - Generation 1

Orig Description

Score : 1100 points
Problem Statement
We have a tree G with N vertices numbered 1 to N.
The i-th edge of G connects Vertex a_i and Vertex b_i.
Consider adding zero or more edges in G, and let H be the graph resulted.
Find the number of graphs H that satisfy the following conditions, modulo 998244353.
H does not contain self-loops or multiple edges.
The diameters of G and H are equal.
For every pair of vertices in H that is not directly connected by an edge, the addition of an edge directly connecting them would reduce the diameter of the graph.
Constraints
 3 \le N \le 2 \times 10^5 
 1 \le a_i, b_i \le N 
The given graph is a tree.
Input
Input is given from Standard Input in the following format:
N
a_1 b_1
\vdots
a_{N-1} b_{N-1}
Output
Print the answer.
Sample Input 16
1 6
2 1
5 2
3 4
2 3
Sample Output 13
For example, adding the edges (1, 5), (3, 5) in G satisfies the conditions.
Sample Input 23
1 2
2 3
Sample Output 21
The only graph H that satisfies the conditions is G.
Sample Input 39
1 2
2 3
4 2
1 7
6 1
2 5
5 9
6 8
Sample Output 327
Sample Input 419
2 4
15 8
1 16
1 3
12 19
1 18
7 11
11 15
12 9
1 6
7 14
18 2
13 12
13 5
16 13
7 1
11 10
7 17
Sample Output 478732

Extracted Specification

An integer N (3 ≤ N ≤ 2 × 10^5), representing some 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
1 6
2 1
5 2
3 4
2 3
```

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

Test Case Generator

Certainly! Below is a test case generator for the given problem statement involving a tree with \( N \) vertices and \( N-1 \) edges.

```python
import random

def tcgen():
    N = random.randint(3, 2 * 10**5)
    
    edges = []
    available_vertices = list(range(2, N + 1))
    random.shuffle(available_vertices)
    
    # Start with a tree where the first node is connected to every other node
    for i in range(2, N + 1):
        edges.append((1, available_vertices.pop()))

    # Randomly shuffle the edges to avoid any specific pattern
    random.shuffle(edges)

    return N, edges

def format_test_case(N, edges):
    test_case = f"{N}\n"
    for a, b in edges:
        test_case += f"{a} {b}\n"
    return test_case

# Generate a test case
N, edges = tcgen()
test_case = format_test_case(N, edges)

print(test_case)
```

### Explanation

- **N**: Generates a random integer between 3 and \( 2 \times 10^5 \) to represent the number of vertices in the tree.
- **edges**: Constructs the edges of the tree. The tree is initially constructed with a root node (1) connected to all other nodes, ensuring it stays a tree. The edges are then shuffled to add randomness.
- **format_test_case**: Formats the generated edges neatly as required for input.

### Sample Output
When you run the generator, it might produce an output similar to this (the values will vary due to randomness):

```
7
1 2
1 3
1 4
1 5
1 6
1 7
```

This represents a tree with 7 vertices and 6 edges, where 1 is the root connected to all other nodes. The output is formatted according to the input constraints of the problem.

Extract Arguments

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