Problem D
Cutting Cheese
Swiss cheese such as Emmentaler has holes in it, and the holes may have different sizes. A slice with holes contains less cheese and has a lower weight than a slice without holes. So here is the challenge: cut a cheese with holes in it into slices of equal weight.
By smart sonar techniques (the same techniques used to scan unborn babies and oil fields), it is possible to locate the holes in the cheese up to micrometer precision. For the present problem you may assume that the holes are perfect spheres.
Each uncut block has size $100 \times 100 \times 100$ where each dimension is measured in millimeters. Your task is to cut it into $s$ slices of equal weight. The slices will be $100$ mm wide and $100$ mm high, and your job is to determine the thickness of each slice.
Input
The first line of the input contains two integers $n$ and $s$, where $0 \leq n \leq 10\, 000$ is the number of holes in the cheese, and $1 \le s \le 100$ is the number of slices to cut. The next $n$ lines each contain four positive integers $r$, $x$, $y$, and $z$ that describe a hole, where $r$ is the radius and $x$, $y$, and $z$ are the coordinates of the center, all in micrometers.
The cheese block occupies the points $(x,y,z)$ where $0 \le x,y,z \le 100\, 000$, except for the points that are part of some hole. The cuts are made perpendicular to the $z$ axis.
You may assume that holes do not overlap but may touch, and that the holes are fully contained in the cheese but may touch its boundary.
Output
Display the $s$ slice thicknesses in millimeters, starting from the end of the cheese with $z=0$. Your output should have an absolute or relative error of at most $10^{-6}$.
| Sample Input 1 | Sample Output 1 |
|---|---|
0 4 |
25.000000000 25.000000000 25.000000000 25.000000000 |
| Sample Input 2 | Sample Output 2 |
|---|---|
2 5 10000 10000 20000 20000 40000 40000 50000 60000 |
14.611103142 16.269801734 24.092457788 27.002992272 18.023645064 |