ICPC 2011 - K. Trash Removal
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/2011/K-trash-removal. Edit
competitive_programming/icpc/2011/K-trash-removal/solution.tex to update the written solution and
competitive_programming/icpc/2011/K-trash-removal/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 K
Trash Removal
Problem ID: trash
Allied Chute Manufacturers is a company that builds trash chutes. A trash chute is a hollow tube installed in buildings
so that trash dropped in at the top will fall down and be collected in the basement. Designing trash chutes is actually
highly nontrivial. Depending on what kind of trash people are expected to drop into them, the trash chute needs to
have an appropriate size. And since the cost of manufacturing a trash chute is proportional to its size, the company
always would like to build a chute that is as small as possible. Choosing the right size can be tough though.
We will consider a 2-dimensional simplification of the chute design problem. A trash chute points straight down and
has a constant width. Objects that will be dropped into the trash chute are modeled as polygons. Before an object is
dropped into the chute it can be rotated so as to provide an optimal fit. Once dropped, it will travel on a straight path
downwards and will not rotate in flight. The following figure shows how an object is first rotated so it fits into the trash
chute.
Your task is to compute the smallest chute width that will allow a given polygon to pass through.
Input
The input contains several test cases. Each test case starts with a line containing an integer n (3 ≤ n ≤ 100), the
number of points in the polygon that models the trash item.
The next n lines then contain pairs of integers xi and yi (0 ≤ xi , yi ≤ 104 ), giving the coordinates of the polygon
vertices in order. All points in one test case are guaranteed to be mutually distinct and the polygon sides will never
intersect. (Technically, there is one inevitable exception of two neighboring sides sharing their common vertex. Of
course, this is not considered an intersection.)
The last test case is followed by a line containing a single zero.
ICPC 2011 World Finals Problem K: Trash Removal
Output
For each test case, display its case number followed by the width of the smallest trash chute through which it can be
dropped. Display the minimum width with exactly two digits to the right of the decimal point, rounding up to the
nearest multiple of 1/100. Answers within 1/100 of the correct rounded answer will be accepted.
Follow the format of the sample output.
Sample input Output for the Sample Input
3 Case 1: 2.40
0 0 Case 2: 14.15
3 0
0 4
4
0 10
10 0
20 10
10 20
0
ICPC 2011 World Finals Problem K: Trash Removal
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 2011\\K. Trash Removal}
\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;
}