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 of three integers each, where:
- The first integer xi (0 ≤ xi ≤ 1000) represents a value.
- The second integer yi (0 ≤ yi ≤ 1000) represents another value.
- The third integer ri (1 ≤ ri ≤ 25) represents another value.
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
3
0 0 5
8 0 5
8 8 5
3
0 0 5
7 3 6
16 0 5
0
```
### Function Signature:
Write a function f(datasets) that takes in the input.
```python
def f(datasets: List[Tuple[int, List[Tuple[int, int, int]]]]):
'''
datasets: a list of tuples, each containing an integer and a list of tuples of three integers
'''
```
Test Case Generator
Sure, here's a test case generator for the "Chain-Confined Path" problem statement:
```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 tcgen():
datasets = []
while True:
n = random.randint(3, 100)
circles = []
# Generate first circle
x1, y1, r1 = generate_circle()
circles.append((x1, y1, r1))
# Generate remaining n-1 circles ensuring they intersect with the previous one
for _ in range(n - 1):
while True:
x, y, r = generate_circle()
# Ensure this circle intersects with the previous one
prev_x, prev_y, prev_r = circles[-1]
dist = ((x - prev_x)**2 + (y - prev_y)**2)**0.5
if dist < (r + prev_r) and dist > abs(r - prev_r):
circles.append((x, y, r))
break
dataset = [n] + circles
datasets.append(dataset)
# Decide whether to generate more datasets or terminate
if random.random() < 0.2:
break
# Format the final input string
input_str = ""
for dataset in datasets:
input_str += f"{dataset[0]}\n"
for circle in dataset[1:]:
input_str += f"{circle[0]} {circle[1]} {circle[2]}\n"
input_str += "0\n" # End of input
return input_str
# Example usage
print(tcgen())
```
This generator creates random test cases for the "Chain-Confined Path" problem. Each dataset is a list of circles that intersect in a chain-like manner, ensuring the constraints of the problem are met. It also ensures that the circles intersect properly by checking the distances between the centers of consecutive circles.
Feel free to execute the `tcgen` function to generate randomized test cases for your 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:
# with open('input.txt', 'r') as fh:
# datasets = extract_arguments(fh)
# for dataset in datasets:
# print(dataset)