Problem p02752 - Generation 3

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 tuples (a_i, b_i), where 1 ≤ a_i, b_i ≤ 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, where each tuple contains two integers
    '''

Test Case Generator

Sure, let's write a test case generator for the tree problem described:

### Problem Statement
We need to generate a tree with `N` vertices and `N-1` edges, where `3 ≤ N ≤ 2 * 10^5` and for each edge, it connects two vertices `a_i` and `b_i` where `1 ≤ a_i, b_i ≤ N`.

### Test Case Generator
Here's a Python function to generate such test cases:

```python
import random

def generate_tree(N):
    edges = []
    # A tree with N nodes has exactly N-1 edges
    parents = list(range(1, N + 1))
    random.shuffle(parents)
    for i in range(2, N + 1):
        # Connect each node to a previously connected node to ensure it's a tree
        edges.append((parents[i - 1], parents[random.randint(0, i - 2)]))
    return edges

def tcgen():
    N = random.randint(3, 2 * 10**5)
    edges = generate_tree(N)
    return N, edges

# Example usage:
N, edges = tcgen()
print(N)
for edge in edges:
    print(edge[0], edge[1])
```

### Explanation
1. **generate_tree(N)**:
    - This function generates a random tree with `N` vertices.
    - It creates a list of nodes and shuffles them to ensure randomness.
    - It then iterates from the second node to the N-th node, connecting each node to a previously connected node to form a valid tree.

2. **tcgen()**:
    - Randomly selects `N` in the range `[3, 2 * 10^5]`.
    - Calls `generate_tree(N)` to get a list of edges for a tree with `N` nodes.
    - Returns `N` and the list of edges.

### Example Run
```python
N, edges = tcgen()
print(N)
for edge in edges:
    print(edge[0], edge[1])
```

This will generate a random valid tree and print it in the required format. Each test case output will start with the number of vertices `N`, followed by `N-1` lines of edges with two integers each representing an edge connecting two vertices.

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