ICPC 2018 - H. Single Cut of Failure
State the problem in your own words. Focus on the mathematical or algorithmic core rather than repeating the full statement.
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competitive_programming/icpc/2018/H-single-cut-of-failure/solution.tex to update the written solution and
competitive_programming/icpc/2018/H-single-cut-of-failure/solution.cpp to update the implementation.
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Problem Statement
Copied statement text kept beside the solution archive for direct reference.
Problem H
Single Cut of Failure
Time limit: 6 seconds
The Intrusion and Crime Prevention Company (ICPC) builds intrusion detection systems for homes and
businesses. The International Collegiate Programming Contest (in a strange coincidence also known as
ICPC) is considering hiring the company to secure the room that contains the problem set for next year’s
World Finals.
The contest staff wants to prevent the intrusion attempts that were made in past years, such as rappelling
down the outside of the building to enter through a window, crawling through air ducts, impersonating
Bill Poucher, and the creative use of an attack submarine. For that reason, the problems will be stored in
a room that has a single door and no other exits.
ICPC (the company) proposes to install sensors on the four sides of the door, where pairs of sensors are
connected by wires. If somebody opens the door, any connected sensor pair will detect this and cause
an alarm to sound.
The system has one design flaw, however. An intruder might cut the wires before opening the door. To
assess the security of the system, you need to determine the minimum number of line segments that cut
all wires. Figure H.1 shows two configurations of wires on the door (corresponding to the two sample
inputs), and minimum-size cuts that intersect all wires.
(a) Four wires (blue) that can be (b) Five wires that need two cuts.
intersected with a single cut (red).
Figure H.1: Illustrations of Sample Inputs 1 and 2.
Input
The input starts with a line containing three integers n, w, and h, which represent the number of wires
installed (1 ≤ n ≤ 106 ) and the dimensions of the door (1 ≤ w, h ≤ 108 ). This is followed by n
lines, each describing a wire placement. Each of these lines contains four integers x1 , y1 , x2 , and y2
(0 ≤ x1 , x2 ≤ w, 0 ≤ y1 , y2 ≤ h), meaning that a wire goes from (x1 , y1 ) to (x2 , y2 ). Each wire
connects different sides of the door. No wire is anchored to any of the four corners of the door. All
locations in the input are distinct.
Output
Display a minimum-size set of straight line cuts that intersect all wires. First, display the number of cuts
needed. Then display the cuts, one per line in the format x1 y1 x2 y2 for the cut between (x1 , y1 ) and
(x2 , y2 ). Each cut has to start and end on different sides of the door. Cuts cannot start or end closer than
10−6 to any wire anchor location or any corner of the door.
Cuts may be displayed in any order. The start and end locations of each cut may be displayed in either
order. If there are multiple sets of cuts with the same minimum size, display any of them.
Sample Input 1 Sample Output 1
4 4 6 1
0 1 4 4 0 4 4 3
0 5 2 0
0 3 3 6
2 6 4 2
Sample Input 2 Sample Output 2
5 4 6 2
0 2 2 0 0 4 4 4.5
0 3 2 6 0 1 4 1
1 6 3 0
1 0 4 4
3 6 4 2
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 2018\\H. Single Cut of Failure}
\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;
}