ICPC 2009 - C. The Return of Carl
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/2009/C-the-return-of-carl/solution.tex to update the written solution and
competitive_programming/icpc/2009/C-the-return-of-carl/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 C
The Return of Carl
Input file: carl.in
Carl the ant is back! When we last left him (Problem A, 2004 World Finals), Carl was a little mixed-up, always
taking strange, zigzag paths when traveling. But now, Carl has straightened out his life – literally. He now
always takes the straightest, shortest path between any pair of points. This sounds simple, except for one small
thing: Carl now spends most of his time on a paperweight in the shape of a regular octahedron. You may recall
that an octahedron is one of the five Platonic solids and consists of eight equilateral triangles, as shown in
Figure 3.
Figure 3: Regular octahedron
Carl has an innate (some say in-ant) ability to always take the shortest path when going from any starting point
to any destination point on the paperweight. Your job is to verify this by determining the length of the shortest
path when given two such points, not necessarily distinct.
Input
The input contains multiple test cases. Each test case consists of four integers θ1, φ1, θ2 and φ2, where
0 ≤ θi < 360 and 0 ≤ φi ≤ 180. The first two are the spherical coordinates of the start point, and the last two are
the spherical coordinates of the destination point. As shown in Figure 3, θi is the azimuth and φi is the zenith
angle, both of them given in degrees.
The input is terminated by a line containing four negative ones.
The paperweight is fixed for all test cases as follows: the octahedron is centered on the origin and each vertex
lies on one of the axes. Every edge is exactly 10 cm long. You should suppose that Carl’s size is zero and ignore
any supporting mechanisms that may be necessary to hold the paperweight in the correct position.
Output
For each test case, print the case number (starting with 1) and the length in centimeters of the shortest path from
the start point to the destination point, rounded to the nearest thousandth. Follow the format of the sample
output.
Sample Input Output for the Sample Input
0 90 90 90 Case 1: 10.000
0 90 90 45 Case 2: 8.660
0 0 0 180 Case 3: 17.321
-1 -1 -1 -1
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\\C. The Return of Carl}
\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;
}