ICPC 2008 - C. Conveyor Belt
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/C-conveyor-belt. Edit
competitive_programming/icpc/2008/C-conveyor-belt/solution.tex to update the written solution and
competitive_programming/icpc/2008/C-conveyor-belt/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
Conveyor Belt
Input file: belt.in
Many mechanical systems work with rotating shafts connected with conveyor belts. The shafts have a variety of sizes
and rotate in either a clockwise or a counterclockwise manner. The exact way in which a belt will connect two shafts
depends on their rotations, as shown in Figures 1 and 2.
Figure 1 Figure 2
Figure 3 Figure 4
One task in setting up such mechanical systems is to link together two given shafts, subject to these constraints:
• If the two shafts being connected are too far apart, the belt may start to vibrate chaotically when perturbed
slightly. To prevent this, you can connect shafts only when the distance between the points where the belt
leaves one shaft and touches the other is less than some distance d (the exact value of d varies depending on
U U
the type of belt).
• No belt can cross over itself, as shown in Figure 3.
• No belt can pass through another shaft (or touch one
rotating the wrong way), as shown in Figure 4.
• The belt is not a loop; it goes in only one direction,
from the starting shaft to the ending shaft. The starting
shaft “pushes” the belt and the ending shaft “pulls” it.
As an example, consider the problem of connecting shaft A to
shaft D in Figure 5. Suppose that the distance needed to
connect A to C (shown in blue dashed line) or to connect B to
D is greater than the limit allowed. Then the shortest distance
to connect A to D is shown in solid line, going from shaft Figure 5
T T
A to B to C and then D. Notice that the connection cannot go
from B to E and then D as the belt would cross itself.
You must write a program that calculates the minimum length of the belt to connect the two given shafts.
Input
The input consists of multiple test cases. Each test case starts with a line containing an integer N (1 ≤ N ≤ 20)
indicating the number of shafts, numbered from 0 to N -1. Starting on the next line are N four-tuples of the form
x y r s, where x and y are the integer coordinates of a shaft center, r is the integer radius of the shaft, and s is either
“C” for clockwise or “CC” for counterclockwise (0 ≤ x, y ≤ 10000 and 0 < r ≤ 1000). Positive x is right and positive y
is up. The first four-tuple specifies shaft 0, the second one shaft 1 and so on. These four-tuples may extend over
multiple lines, though no four-tuple will be split across two lines. No two shafts touch or overlap. The last line of
each test case contains two integers i and j indicating the starting and ending shafts, followed by a floating point
value d specifying the maximum distance constraint.
The last test case is followed by a line containing a single zero.
Output
For each test case, print the case number (starting with 1) followed by the length of the path of minimum distance.
Print Cannot reach destination shaft if the destination shaft cannot be reached. Use the format shown
T T
in the sample output.
When measuring the distance, start where the belt first leaves the starting shaft and end where the belt first touches
the ending shaft. Your distance calculation must include both the length of belt between shafts as well as the distance
the belt travels around intermediate shafts. Your answer should be rounded to the nearest hundredth, though you need
not print trailing zeroes after the decimal point.
Sample Input Output for the Sample Input
5 Case 1: length = 271
24 50 14 C 93 78 20 C 118 8 15 CC Case 2: length = 228.23
167 32 13 C 159 88 15 CC Case 3: Cannot reach destination shaft
0 3 82.5
5
24 50 14 C 93 78 20 C 118 8 15 CC
167 32 13 C 159 88 15 C
0 3 82.5
5
24 50 14 C 93 78 20 C 118 8 15 CC
167 32 13 C 159 88 15 C
0 3 8.5
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\\C. Conveyor Belt}
\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;
}