ICPC 2015 - D. Cutting Cheese
State the problem in your own words. Focus on the mathematical or algorithmic core rather than repeating the full statement.
Source-first archive entry
This page is built from the copied files in competitive_programming/icpc/2015/D-cutting-cheese. Edit
competitive_programming/icpc/2015/D-cutting-cheese/solution.tex to update the written solution and
competitive_programming/icpc/2015/D-cutting-cheese/solution.cpp to update the implementation.
The website does not replace those files with hand-maintained HTML. It reads the copied source tree during the build and exposes the exact files below.
Problem Statement
Copied statement text kept beside the solution archive for direct reference.
Problem D
Cutting Cheese
Time limit: 3 seconds
Of course you have all heard of the International Cheese
Processing Company. Their machine for cutting a piece of
cheese into slices of exactly the same thickness is a classic.
Recently they produced a machine able to cut a spherical
cheese (such as Edam) into slices – no, not all of the same
thickness, but all of the same weight! But new challenges
lie ahead: cutting Swiss 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.
Picture by Jon Sullivan via Wikimedia Commons
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 × 100 × 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 ≤ n ≤ 10 000 is the number of holes in
the cheese, and 1 ≤ s ≤ 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 ≤ x, y, z ≤ 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 14.611103142
10000 10000 20000 20000 16.269801734
40000 40000 50000 60000 24.092457788
27.002992272
18.023645064
Editorial
Rendered from the copied solution.tex file. The original TeX source remains
available below.
Key Observations
Write the structural observations that make the problem tractable.
State any useful invariant, monotonicity property, graph interpretation, or combinatorial reformulation.
If the constraints matter, explain exactly which part of the solution they enable.
Algorithm
Describe the data structures and the state maintained by the algorithm.
Explain the processing order and why it is sufficient.
Mention corner cases explicitly if they affect the implementation.
Correctness Proof
We prove that the algorithm returns the correct answer.
Lemma 1.
State the first key claim.
Proof.
Provide a concise proof.
Lemma 2.
State the next claim if needed.
Proof.
Provide a concise proof.
Theorem.
The algorithm outputs the correct answer for every valid input.
Proof.
Combine the lemmas and finish the argument.
Complexity Analysis
State the running time and memory usage in terms of the input size.
Implementation Notes
Mention any non-obvious implementation detail that is easy to get wrong.
Mention numeric limits, indexing conventions, or tie-breaking rules if relevant.
Code
Exact copied C++ implementation from solution.cpp.
#include <bits/stdc++.h>
using namespace std;
namespace {
void solve() {
// Fill in the full solution logic for the problem here.
}
} // namespace
int main() {
ios::sync_with_stdio(false);
cin.tie(nullptr);
solve();
return 0;
}
Source Files
Exact copied source-of-truth files. Edit solution.tex for the write-up and solution.cpp for the implementation.
\documentclass[11pt]{article}
\usepackage[margin=1in]{geometry}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{enumitem}
\title{ICPC World Finals 2015\\D. Cutting Cheese}
\author{}
\date{}
\begin{document}
\maketitle
\section*{Problem Summary}
State the problem in your own words. Focus on the mathematical or algorithmic core rather than repeating the full statement.
\section*{Key Observations}
\begin{itemize}[leftmargin=*]
\item Write the structural observations that make the problem tractable.
\item State any useful invariant, monotonicity property, graph interpretation, or combinatorial reformulation.
\item If the constraints matter, explain exactly which part of the solution they enable.
\end{itemize}
\section*{Algorithm}
\begin{enumerate}[leftmargin=*]
\item Describe the data structures and the state maintained by the algorithm.
\item Explain the processing order and why it is sufficient.
\item Mention corner cases explicitly if they affect the implementation.
\end{enumerate}
\section*{Correctness Proof}
We prove that the algorithm returns the correct answer.
\paragraph{Lemma 1.}
State the first key claim.
\paragraph{Proof.}
Provide a concise proof.
\paragraph{Lemma 2.}
State the next claim if needed.
\paragraph{Proof.}
Provide a concise proof.
\paragraph{Theorem.}
The algorithm outputs the correct answer for every valid input.
\paragraph{Proof.}
Combine the lemmas and finish the argument.
\section*{Complexity Analysis}
State the running time and memory usage in terms of the input size.
\section*{Implementation Notes}
\begin{itemize}[leftmargin=*]
\item Mention any non-obvious implementation detail that is easy to get wrong.
\item Mention numeric limits, indexing conventions, or tie-breaking rules if relevant.
\end{itemize}
\end{document}
#include <bits/stdc++.h>
using namespace std;
namespace {
void solve() {
// Fill in the full solution logic for the problem here.
}
} // namespace
int main() {
ios::sync_with_stdio(false);
cin.tie(nullptr);
solve();
return 0;
}