ICPC 2008 - J. The Sky is the Limit
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/2008/J-the-sky-is-the-limit/solution.tex to update the written solution and
competitive_programming/icpc/2008/J-the-sky-is-the-limit/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 J
The Sky is the Limit
Input file: skyline.in
The city of Banff hired an advertising agency to promote the city’s attractions to potential visitors. One of the
planned slogans stated that the mountain ranges around the city form the most beautiful skyline in Canada. But the
Institute for Consumer Protection in Canada (ICPC) decided that “the most beautiful skyline” was a subjective and
unverifiable claim, and could therefore be considered misleading.
The advertising agency then came up with the slogan “Banff – the longest skyline in Canada.” Although not as
catchy, it is hopefully verifiable, and therefore admissible under Canada’s tricky advertising laws.
This is where you come in. What the advertising agency needs is a program that determines the length of a skyline.
Consider each mountain as a two-dimensional triangle having two upper sides the same length. A skyline is the
outline of one or more mountains. The skyline’s length is the total length of the outline. The left illustration below
shows three mountains. The right illustration shows (with bold lines) the skyline and (with dashed lines) the portion
of the mountains’ upper edges that are not part of the skyline. Note that parts of the horizon line that lie between
mountains are not considered part of the skyline.
Input
Each input file contains one or more test cases, which are descriptions of mountain ranges. Each description starts
with a line containing a positive integer N, which specifies the number of mountains in the range. Each of the next N
lines describes a mountain with three integers X, H, and B, which specify the horizontal position of the mountain’s
peak relative to some fixed point, the height of the peak, and the width of the base of the mountain, respectively. The
base of each mountain coincides with a horizontal line. The values satisfy the conditions N ≤ 100, H > 0, and B > 0.
The last test case is followed by a line containing a zero.
Output
For each test case, print the case number (beginning with 1) and the length of the skyline. Print the length rounded to
the nearest integer, with 0.5 rounded up. Print a blank line after the output of each test case. Use the format shown in
the sample output below.
Sample Input Output for the Sample Input
1 Case 1: 141
100 50 100
3 Case 2: 138
20 30 35
37 24 29
60 20 13
0
This page is intentionally blank.
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 2008\\J. The Sky is the Limit}
\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;
}