Orig Description
Non-redundant Drive
The people of JAG kingdom hate redundancy. For example, the N cities in JAG kingdom are connected with just $N - 1$ bidirectional roads such that any city is reachable from any city through some roads. Under the condition, the number of paths from a city to another city is exactly one for all pairs of the cities. This is a non-redundant road network :)
One day, you, a citizen of JAG kingdom, decided to travel as many cities in the kingdom as possible with a car. The car that you will use has an infinitely large tank, but initially the tank is empty. The fuel consumption of your car is 1 liter per 1 km, i.e. it consumes 1 liter of gasoline to move 1 km.
Each city has exactly one gas station, and you can supply $g_x$ liters of gasoline to your car at the gas station of the city $x$. Of course, you have a choice not to visit some of the gas stations in your travel. But you will not supply gasoline twice or more at the same gas station, because it is redundant. Each road in the kingdom has a distance between two cities: the distance of $i$-th road is $d_i$ km. You will not pass the same city or the same road twice or more, of course, because it is redundant.
If a quantity of stored gasoline becomes zero, the car cannot move, and hence your travel will end there. But then, you may concern about an initially empty tank. Don't worry. You can start at any gas station of the cities in the kingdom. Furthermore, each road directly connects the gas stations of the its two ends (because the spirit of non-redundancy avoids redundant moves in a city), you therefore can supply gasoline to your car even if your car tank becomes empty just when you arrive the city.
Your task is to write a program computing the maximum number of cities so that you can travel under your non-redundancy policy.
Input
The input consists of a single test case.
$N$
$g_1$ $g_2$ ... $g_N$
$a_1$ $b_1$ $d_1$
$a_2$ $b_2$ $d_2$
...
$a_{N-1}$ $b_{N-1}$ $d_{N-1}$
The first line contains an integer $N$ ($1 \leq N \leq 100,000$), which is the number of cities in JAG kingdom. The second line contains $N$ integers: the $i$-th of them is $g_i$ ($1 \leq g_i \leq 10,000$), the amount of gasoline can be supplied at the gas station of the city $i$. The following $N - 1$ lines give information of roads: the $j$-th line of them contains $a_j$ and $b_j$ , which indicates that the $j$-th road bidirectionally connects the cities $a_j$ and $b_j$ ($1 \leq a_j, b_j \leq N, a_j \ne b_j$) with distance $d_j$ ($1 \leq d_j \leq 10,000$). You can assume that all cities in the kingdom are connected by the roads.
Output
Print the maximum number of cities you can travel from any city under the constraint such that you can supply gasoline at most once per a gas station.
Sample Input 1
5
5 8 1 3 5
1 2 4
2 3 3
2 4 3
1 5 7
Output for the Sample Input 1
4
Sample Input 2
2
10 1
1 2 10
Output for the Sample Input 2
2
Sample Input 3
5
1 3 5 1 1
1 2 5
2 3 3
2 4 3
1 5 5
Output for the Sample Input 3
3
Extracted Specification
An integer N (1 ≤ N ≤ 100,000), representing some quantity or size.
A list of integers G of size N, where each integer is between 1 and 10,000.
A list of tuples representing connections: (a_i, b_i, d_i), where each a_i and b_i are integers (1 ≤ a_i, b_i ≤ N, a_i ≠ b_i) and d_i is an integer (1 ≤ d_i ≤ 10,000).
### Example Input:
```
5
5 8 1 3 5
1 2 4
2 3 3
2 4 3
1 5 7
```
### Function Signature:
Write a function f(N, G, connections) that takes in the input.
```python
def f(N: int, G: List[int], connections: List[Tuple[int, int, int]]):
'''
N: an integer
G: a list of integers
connections: a list of tuples, each containing three integers
'''
```
Test Case Generator
Certainly! Below is a Python code snippet that generates test cases for the "Non-redundant Drive" problem statement. This generator will create random valid inputs for the problem:
```python
import random
def generate_test_case():
# Number of cities (N)
N = random.randint(1, 100000)
# Gasoline amounts for each city (g_1, g_2, ..., g_N)
gasolines = [random.randint(1, 10000) for _ in range(N)]
# Generate the roads
roads = []
available_nodes = list(range(1, N + 1))
random.shuffle(available_nodes)
for i in range(2, N + 1):
a = available_nodes[i - 2]
b = available_nodes[i - 1]
d = random.randint(1, 10000)
roads.append((a, b, d))
# Output the test case
print(N)
print(" ".join(map(str, gasolines)))
for road in roads:
print(f"{road[0]} {road[1]} {road[2]}")
# Example of generating a test case
generate_test_case()
```
### Explanation:
1. **Number of Cities (N)**: We randomly choose a number between 1 and 100,000 inclusive for the number of cities.
2. **Gasoline Amounts**: For each city, we randomly choose a gasoline supply amount between 1 and 10,000 inclusive.
3. **Roads Generation**:
- We shuffle a list of city indices to ensure randomness.
- We connect cities sequentially to form a tree-like structure, ensuring there are no cycles and the graph remains connected.
- For each road, we randomly choose a distance between 1 and 10,000 inclusive.
4. **Output**:
- The number of cities.
- The list of gasoline amounts.
- The list of roads, each with two city indices and a distance.
This code will print a single test case each time it's run. You can run the `generate_test_case()` function multiple times to generate multiple test cases. For large values of `N`, ensure your environment can handle the large output efficiently.
Extract Arguments
def extract_arguments(fh):
N = int(fh.readline().strip())
g = list(map(int, fh.readline().strip().split()))
roads = []
for _ in range(N - 1):
a, b, d = map(int, fh.readline().strip().split())
roads.append((a, b, d))
return N, g, roads