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.
N-1 tuples (a_i, b_i, c_i) where each tuple consists of:
- Two integers a_i and b_i (1 ≤ a_i, b_i ≤ N)
- One integer c_i (1 ≤ c_i ≤ 10^9).
Two integers Q (1 ≤ Q ≤ 10^5) and K (1 ≤ K ≤ N).
Q tuples (x_j, y_j) where each tuple consists of:
- Two integers x_j and y_j (1 ≤ x_j, y_j ≤ N)
- x_j ≠ y_j and 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.
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 generate test cases for the given tree problem, we need to:
1. Randomly generate a tree with \(N\) vertices.
2. Ensure that the tree is connected and has \(N-1\) edges.
3. Generate \(Q\) queries that ask for the shortest path between two vertices via a specific vertex \(K\).
Here is a Python function that generates such test cases:
```python
import random
def generate_tree(N):
edges = []
for i in range(2, N + 1):
u = random.randint(1, i - 1)
v = i
c = random.randint(1, 10**9)
edges.append((u, v, c))
return edges
def generate_queries(Q, N, 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(N)
Q = random.randint(1, 10**5)
K = random.randint(1, N)
queries = generate_queries(Q, N, K)
return N, edges, Q, K, queries
def print_test_case():
N, edges, Q, K, queries = tcgen()
print(N)
for u, v, c in edges:
print(u, v, c)
print(Q, K)
for x, y in queries:
print(x, y)
# Example usage:
print_test_case()
```
### Explanation:
1. **generate_tree(N)**: This function generates a tree with \(N\) vertices. It ensures the tree is connected by connecting each new vertex \(v\) to an existing vertex \(u\) (always ensuring \(u < v\) to avoid cycles). Each edge has a randomly assigned weight between 1 and \(10^9\).
2. **generate_queries(Q, N, K)**: This function generates \(Q\) queries. Each query is a pair of vertices \(x\) and \(y\) such that \(x \neq y\), \(x \neq K\), and \(y \neq K\).
3. **tcgen()**: This function integrates the above functions to generate a complete test case with \(N\) vertices, a list of edges, \(Q\) queries, and the specific vertex \(K\).
4. **print_test_case()**: This function prints the generated test case in the specified format.
This approach ensures the generated test cases adhere to the problem constraints and are suitable for thorough testing of the solution.
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