Problem p03634 - Generation 1

Orig Description

Score : 400 points
Problem StatementYou are given a tree with N vertices.
Here, a tree is a kind of graph, and more specifically, a connected undirected graph with N-1 edges, where N is the number of its vertices.
The i-th edge (1≤i≤N-1) connects Vertices a_i and b_i, and has a length of c_i.
You are also given Q queries and an integer K. In the j-th query (1≤j≤Q):
find the length of the shortest path from Vertex x_j and Vertex y_j via Vertex K.
Constraints
3≤N≤10^5 
1≤a_i,b_i≤N (1≤i≤N-1) 
1≤c_i≤10^9 (1≤i≤N-1) 
The given graph is a tree.
1≤Q≤10^5 
1≤K≤N 
1≤x_j,y_j≤N (1≤j≤Q) 
x_j≠y_j (1≤j≤Q) 
x_j≠K,y_j≠K (1≤j≤Q) 
InputInput is given from Standard Input in the following format:  
N  
a_1 b_1 c_1  
:  
a_{N-1} b_{N-1} c_{N-1}
Q K
x_1 y_1
:  
x_{Q} y_{Q}
OutputPrint the responses to the queries in Q lines.
In the j-th line j(1≤j≤Q), print the response to the j-th query.
Sample Input 15
1 2 1
1 3 1
2 4 1
3 5 1
3 1
2 4
2 3
4 5
Sample Output 13
2
4
The shortest paths for the three queries are as follows:
Query 1: Vertex 2 → Vertex 1 → Vertex 2 → Vertex 4 : Length 1+1+1=3 
Query 2: Vertex 2 → Vertex 1 → Vertex 3 : Length 1+1=2 
Query 3: Vertex 4 → Vertex 2 → Vertex 1 → Vertex 3 → Vertex 5 : Length 1+1+1+1=4 
Sample Input 27
1 2 1
1 3 3
1 4 5
1 5 7
1 6 9
1 7 11
3 2
1 3
4 5
6 7
Sample Output 25
14
22
The path for each query must pass Vertex K=2.
Sample Input 310
1 2 1000000000
2 3 1000000000
3 4 1000000000
4 5 1000000000
5 6 1000000000
6 7 1000000000
7 8 1000000000
8 9 1000000000
9 10 1000000000
1 1
9 10
Sample Output 317000000000

Extracted Specification

An integer N (3 ≤ N ≤ 10^5), representing some quantity or size.
A list of N-1 tuples, each containing three integers (a_i, b_i, c_i), where 1 ≤ a_i, b_i ≤ N and 1 ≤ c_i ≤ 10^9.
An integer Q (1 ≤ Q ≤ 10^5).
An integer K (1 ≤ K ≤ N).
A list of Q tuples, each containing two integers (x_j, y_j), where 1 ≤ x_j, y_j ≤ N and x_j ≠ y_j, x_j ≠ K, y_j ≠ K.

### Example Input:

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

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

Test Case Generator

To create a test case generator for the given problem, we'll generate a tree structure with \(N\) vertices, random edges with lengths, and multiple queries. Here's a detailed step-by-step solution:

1. **Generate Vertices and Edges:**
    - We need to generate \(N-1\) edges to form a tree.
    - Ensure the tree remains connected by using a union-find (disjoint-set) structure or simply connecting a new node to any of the existing nodes.

2. **Generate Edge Lengths:**
    - Each edge length \(c_i\) should be a random integer between 1 and \(10^9\).

3. **Generate Queries:**
    - Each query should consist of two vertices \(x_j\) and \(y_j\) (both different and neither equal to \(K\)).
    - Generate \(Q\) such queries.

4. **Constraints:**
    - \(3 \leq N \leq 10^5\)
    - \(1 \leq c_i \leq 10^9\)
    - \(1 \leq Q \leq 10^5\)
    - Vertices in queries should not be the same and should not be equal to \(K\).

Here's the Python code for the test case generator:

```python
import random

def generate_tree_edges(N):
    edges = []
    for i in range(2, N+1):
        u = i
        v = random.randint(1, i-1)  # Connect to any vertex that is already part of the tree
        c = random.randint(1, 10**9)
        edges.append((u, v, c))
    return edges

def generate_queries(N, Q, K):
    queries = []
    for _ in range(Q):
        while True:
            x = random.randint(1, N)
            y = random.randint(1, N)
            if x != y and x != K and y != K:
                break
        queries.append((x, y))
    return queries

def tcgen():
    N = random.randint(3, 10**5)
    edges = generate_tree_edges(N)
    
    Q = random.randint(1, 10**5)
    K = random.randint(1, N)
    
    queries = generate_queries(N, Q, K)
    
    input_data = f"{N}\n"
    input_data += "\n".join(f"{u} {v} {c}" for u, v, c in edges)
    input_data += f"\n{Q} {K}\n"
    input_data += "\n".join(f"{x} {y}" for x, y in queries)
    
    return input_data

# Example usage
print(tcgen())
```

**Explanation:**

1. **generate_tree_edges(N):** This function generates \(N-1\) edges to form a connected tree. Each edge connects a new vertex \(i\) to any previous vertex \(v\) ensuring connectivity.

2. **generate_queries(N, Q, K):** This function generates \(Q\) queries. It ensures that the vertices \(x\) and \(y\) in each query are different and neither is equal to \(K\).

3. **tcgen():** This function calls the above helper functions to generate the tree and queries, formats them as a single input string, and returns it.

You can place this generator script in your testing environment to generate and verify multiple test cases for the problem.

Extract Arguments

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