ICPC 2012 - A. Asteroid Rangers
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/2012/A-asteroid-rangers. Edit
competitive_programming/icpc/2012/A-asteroid-rangers/solution.tex to update the written solution and
competitive_programming/icpc/2012/A-asteroid-rangers/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
Asteroid Rangers
Problem ID: asteroids
The year is 2112 and humankind has conquered the solar system. The Space Ranger Corps have set
up bases on any hunk of rock that is even remotely inhabitable. Your job as a member of the Asteroid
Communications Ministry is to make sure that all of the Space Ranger asteroid bases can communicate
with one another as cheaply as possible. You could set up direct communication links from each base
to every other base, but that would be prohibitively expensive. Instead, you want to set up the minimum
number of links so that everyone can send messages to everyone else, potentially relayed by one or more
bases. The cost of any link is directly proportional to the distance between the two bases it connects, so
this doesn’t seem that hard of a problem.
There is one small difficulty, however. Asteroids have a tendency to move about, so two bases that are
currently very close may not be so in the future. Therefore as time goes on, you must be willing to
switch your communication links so that you always have the cheapest relay system in place. Switching
these links takes time and money, so you are interested in knowing how many times you will have to
perform such a switch.
A few assumptions make your task easier. Each asteroid is considered a single point. Asteroids always
move linearly with a fixed velocity. No asteroids ever collide with other asteroids. Also, any relay system
that becomes optimal at a time t ≥ 0 will be uniquely optimal for any time s satisfying t < s < t+10−6 .
The initial optimal relay system will be unique.
Input
Each test case starts with a line containing an integer n (2 ≤ n ≤ 50) indicating the number of asteroid
bases. Following this are n lines, each containing six integers x, y, z, vx , vy , vz . The first three specify
the initial location of an asteroid (−150 ≤ x, y, z ≤ 150), and the last three specify the x, y, and z
components of that asteroid’s velocity in space units per time unit (−100 ≤ vx , vy , vz ≤ 100).
Output
For each test case, display a single line containing the case number and the number of times that the
relay system needs to be set up or modified.
Sample Input Output for Sample Input
3 Case 1: 3
0 0 0 0 0 0 Case 2: 3
5 0 0 0 0 0
10 1 0 -1 0 0
4
0 0 0 1 0 0
0 1 0 0 -1 0
1 1 1 3 1 1
-1 -1 2 1 -1 -1
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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 2012\\A. Asteroid Rangers}
\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;
}