ICPC 2008 - A. Air Conditioning Machinery
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/2008/A-air-conditioning-machinery. Edit
competitive_programming/icpc/2008/A-air-conditioning-machinery/solution.tex to update the written solution and
competitive_programming/icpc/2008/A-air-conditioning-machinery/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
T
Air Conditioning Machinery
Input file: ducts.in
You are a technician for the Air Conditioning Machinery company (ACM). Unfortunately, when you arrive at a
customer site to install some air conditioning ducts, you discover that you are running low on supplies. You have
only six duct segments, and they are all of the same kind, called an “elbow.”
You must install a duct in a confined space: a rectangular prism whose sides are multiples of a unit length. Think of
the confined space as consisting of an array of unit cubes. Each elbow occupies exactly four unit cubes, as shown in
Figure 1 below. A unit cube can be occupied by at most one elbow. Each elbow has exactly two openings, as
indicated by the gray squares in the elbow shown in Figure 1. You may assemble the elbows into longer ducts, but
your duct must be completely contained inside the given space. One way to connect two elbows is shown in Figure 2.
Your task is to connect an inflow to an outflow. The inflow and the outflow are located on the exterior surface of the
confined space, aligned with the unit cubes, as shown in Figure 3. To keep expenses down, you must accomplish this
task while using the minimum number of elbows.
out
in
Figure 1 Figure 2 Figure 3
Input
The input consists of several test cases, each of which consists of a single line containing eleven input values
separated by blanks. The input values for each test case are as follows.
The first three input values are integers (xmax, ymax, and zmax) that indicate the size of the confined space in the x, y,
B B B B B B
and z dimensions, respectively. Each unit cube in the confined space can be identified by coordinates (x, y, z) where
1 ≤ x ≤ xmax, 1 ≤ y ≤ ymax, and 1 ≤ z ≤ zmax. xmax, ymax, and zmax are all positive and not greater than 20.
B B B B B B B B B B B B
The next three input values are integers that indicate the location of the inflow by identifying the x, y, and z
coordinates of the unit cube that connects to the inflow.
The next input value is a two-character string that indicates the direction of the inward flow, using one of the
following codes: +x, -x, +y, -y, +z, -z. The inflow connection is on the face of the unit cube that receives this inward
flow. For example, if the data specifies an inflow direction of +y, the inflow connection is on the face of the unit cube
that faces in the negative y direction.
The next three input values are integers that indicate the location of the outflow by identifying the x, y, and z
coordinates of the unit cube that connects to the outflow.
The last input value is a two-character string that indicates the direction of the outward flow, using the same codes
described above. The outflow connection is on the face of the unit cube that generates this outward flow. For
example, if the data specifies an outflow direction of +y, the outflow connection is on the face of the unit cube that
faces in the positive y direction.
The last line of the input file consists of a single zero to indicate end of input.
Output
For each test case, print the case number (starting with 1) followed by the minimum number of elbows that are
required to connect the inflow to the outflow without going outside the confined space. If the task cannot be
accomplished with your supply of six elbow segments, print the word Impossible instead. Use the format in the
T T
sample data.
Sample Input Output for the Sample Input
5 4 3 3 1 1 +z 5 4 3 +x Case 1: 2
5 4 3 3 1 1 +z 1 2 3 -x Case 2: Impossible
0
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\\A. Air Conditioning Machinery}
\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;
}