ICPC 2009 - A. A Careful Approach
State the problem in your own words. Focus on the mathematical or algorithmic core rather than repeating the full statement.
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This page is built from the copied files in competitive_programming/icpc/2009/A-a-careful-approach. Edit
competitive_programming/icpc/2009/A-a-careful-approach/solution.tex to update the written solution and
competitive_programming/icpc/2009/A-a-careful-approach/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 A
A Careful Approach
Input: approach.in
If you think participating in a programming contest is stressful, imagine being an air traffic controller. With
human lives at stake, an air traffic controller has to focus on tasks while working under constantly changing
conditions as well as dealing with unforeseen events.
Consider the task of scheduling the airplanes that are landing at an airport. Incoming airplanes report their
positions, directions, and speeds, and then the controller has to devise a landing schedule that brings all
airplanes safely to the ground. Generally, the more time there is between successive landings, the “safer” a
landing schedule is. This extra time gives pilots the opportunity to react to changing weather and other surprises.
Luckily, part of this scheduling task can be automated – this is where you come in. You will be given scenarios
of airplane landings. Each airplane has a time window during which it can safely land. You must compute an
order for landing all airplanes that respects these time windows. Furthermore, the airplane landings should be
stretched out as much as possible so that the minimum time gap between successive landings is as large as
possible. For example, if three airplanes land at 10:00am, 10:05am, and 10:15am, then the smallest gap is five
minutes, which occurs between the first two airplanes. Not all gaps have to be the same, but the smallest gap
should be as large as possible.
Input
The input file contains several test cases consisting of descriptions of landing scenarios. Each test case starts
with a line containing a single integer n (2 ≤ n ≤ 8), which is the number of airplanes in the scenario. This is
followed by n lines, each containing two integers ai, bi, which give the beginning and end of the closed interval
[ai, bi] during which the ith plane can land safely. The numbers ai and bi are specified in minutes and satisfy
0 ≤ ai ≤ bi ≤ 1440.
The input is terminated with a line containing the single integer zero.
Output
For each test case in the input, print its case number (starting with 1) followed by the minimum achievable time
gap between successive landings. Print the time split into minutes and seconds, rounded to the closest second.
Follow the format of the sample output.
Sample Input Output for the Sample Input
3 Case 1: 7:30
0 10 Case 2: 20:00
5 15
10 15
2
0 10
10 20
0
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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 2009\\A. A Careful Approach}
\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;
}