Problem p00764 - Generation 2

Orig Description

Chain-Confined Path
There is a chain consisting of multiple circles on a plane.
The first (last) circle of the chain only intersects with the next (previous) circle,
and each intermediate circle intersects only with the two neighboring circles.
Your task is to find the shortest path that satisfies the following conditions.
The path connects the centers of the first circle and the last circle.
The path is confined in the chain, that is, all the points on the path are located within or on at least one of the circles.
Figure E-1 shows an example of such a chain and the corresponding shortest path.
Figure E-1: An  example chain and the corresponding shortest path
Input
The input consists of multiple datasets.
Each dataset represents the shape of a chain in the following format.
n
x1 y1 r1 
x2 y2 r2 
...
xn yn rn 
The first line of a dataset contains an integer n (3 ≤ n ≤ 100) 
representing the number of the circles.
Each of the following n lines contains three integers separated by a single space.
(xi, yi) 
and ri represent 
the center position and the radius of the i-th circle Ci.
You can assume that 0 ≤ xi ≤ 1000, 
0 ≤ yi ≤ 1000, and
1 ≤ ri ≤ 25.
You can assume that Ci  and 
Ci+1  (1 ≤ i ≤ n−1) intersect at two separate points.
When j ≥ i+2, Ci  and Cj  are apart and either of them does not contain the other.
In addition, you can assume that any circle does not contain the center of any other circle.
The end of the input is indicated by a line containing a zero. 
Figure E-1 corresponds to the first dataset of Sample Input below.
Figure E-2 shows the shortest paths for the subsequent datasets of Sample Input. 
Figure E-2: Example chains and the corresponding shortest paths
Output
For each dataset, output a single line containing the length of the shortest chain-confined path between the centers of the first circle and the last circle.
The value should not have an error greater than 0.001.
No extra characters should appear in the output. 
Sample Input
10
802 0 10
814 0 4
820 1 4
826 1 4
832 3 5
838 5 5
845 7 3
849 10 3
853 14 4
857 18 3
3
0 0 5
8 0 5
8 8 5
3
0 0 5
7 3 6
16 0 5
9
0 3 5
8 0 8
19 2 8
23 14 6
23 21 6
23 28 6
19 40 8
8 42 8
0 39 5
11
0 0 5
8 0 5
18 8 10
8 16 5
0 16 5
0 24 5
3 32 5
10 32 5
17 28 8
27 25 3
30 18 5
0
Output for the Sample Input
58.953437
11.414214
16.0
61.874812
63.195179

Extracted Specification

An integer n (3 ≤ n ≤ 100), representing some quantity or size.
n lines follow, each containing three integers separated by a single space.
The first two integers (0 ≤ xi ≤ 1000, 0 ≤ yi ≤ 1000) represent coordinates.
The third integer (1 ≤ ri ≤ 25) represents a size or radius.
The end of the input is indicated by a line containing a zero.

### Example Input:

```
10
802 0 10
814 0 4
820 1 4
826 1 4
832 3 5
838 5 5
845 7 3
849 10 3
853 14 4
857 18 3
0
```

### Function Signature:
Write a function f(inputs) that takes in the input. 
def f(inputs: List[Tuple[int, List[Tuple[int, int, int]]]]): 
    ''' 
    inputs: a list of tuples, where each tuple contains an integer and a list of tuples
    '''

Test Case Generator

Sure, let's create a test case generator for the given problem statement "Chain-Confined Path". Here is the Python code for generating test cases:

```python
import random

def generate_circle():
    x = random.randint(0, 1000)
    y = random.randint(0, 1000)
    r = random.randint(1, 25)
    return (x, y, r)

def circles_intersect(c1, c2):
    x1, y1, r1 = c1
    x2, y2, r2 = c2
    dist_sq = (x2 - x1) ** 2 + (y2 - y1) ** 2
    return (r1 - r2) ** 2 < dist_sq < (r1 + r2) ** 2

def tcgen():
    n = random.randint(3, 100)
    
    circles = [generate_circle()]
    
    for _ in range(1, n):
        while True:
            new_circle = generate_circle()
            if circles_intersect(circles[-1], new_circle):
                circles.append(new_circle)
                break
    
    dataset = f"{n}\n" + "\n".join(f"{x} {y} {r}" for x, y, r in circles)
    return dataset

def generate_multiple_datasets(count):
    datasets = [tcgen() for _ in range(count)]
    return "\n0\n".join(datasets) + "\n0"

# Example usage
print(generate_multiple_datasets(5))
```

This code will generate multiple test cases for the "Chain-Confined Path" problem. Here’s a breakdown of what each part does:

1. **generate_circle**: Generates a random circle with coordinates `x`, `y`, and radius `r` within the provided constraints.
2. **circles_intersect**: Checks if two circles intersect at two separate points as required by the problem statement.
3. **tcgen**: Generates a single dataset with `n` circles that meet the problem constraints. It ensures that each circle intersects with its predecessor.
4. **generate_multiple_datasets**: Generates the specified number of datasets and concatenates them with the termination line (`0`).

You can call `generate_multiple_datasets` with the desired number of datasets to generate the complete input for the problem.

Extract Arguments

def extract_arguments(fh):
    datasets = []
    while True:
        n = int(fh.readline().strip())
        if n == 0:
            break
        circles = []
        for _ in range(n):
            xi, yi, ri = map(int, fh.readline().strip().split())
            circles.append((xi, yi, ri))
        datasets.append((n, circles))
    return datasets

# Example usage:
# if __name__ == "__main__":
#     import sys
#     input_path = sys.argv[1]
#     with open(input_path, 'r') as fh:
#         datasets = extract_arguments(fh)
#         for dataset in datasets:
#             f(dataset)