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