ICPC 2014 - C. Crane Balancing
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/2014/C-crane-balancing/solution.tex to update the written solution and
competitive_programming/icpc/2014/C-crane-balancing/solution.cpp to update the implementation.
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Problem Statement
Copied statement text kept beside the solution archive for direct reference.
Problem C
Crane Balancing
Time Limit: 1 second
Wherever there is large-scale construction, you will find cranes that do the lifting. One hardly ever
thinks about what marvelous examples of engineering cranes are: a structure of (relatively) little weight
that can lift much heavier loads. But even the best-built cranes may have a limit on how much weight
they can lift.
The Association of Crane Manufacturers (ACM) needs a program to compute the range of weights that a
crane can lift. Since cranes are symmetric, ACM engineers have decided to consider only a cross section
of each crane, which can be viewed as a polygon resting on the x-axis.
Figure C.1: Crane cross section
Figure C.1 shows a cross section of the crane in the first sample input. Assume that every 1 × 1 unit
of crane cross section weighs 1 kilogram and that the weight to be lifted will be attached at one of the
polygon vertices (indicated by the arrow in Figure C.1). Write a program that determines the weight
range for which the crane will not topple to the left or to the right.
Input
The input consists of a single test case. The test case starts with a single integer n (3 ≤ n ≤ 100), the
number of points of the polygon used to describe the crane’s shape. The following n pairs of integers
xi , yi (−2 000 ≤ xi ≤ 2 000, 0 ≤ yi ≤ 2 000) are the coordinates of the polygon points in order. The
weight is attached at the first polygon point and at least two polygon points are lying on the x-axis.
Output
Display the weight range (in kilograms) that can be attached to the crane without the crane toppling
over. If the range is [a, b], display bac .. dbe. For example, if the range is [1.5, 13.3], display 1 .. 14.
If the range is [a, ∞), display bac .. inf. If the crane cannot carry any weight, display unstable
instead.
Sample Input 1 Sample Output 1
7 0 .. 1017
50 50
0 50
0 0
30 0
30 30
40 40
50 40
Sample Input 2 Sample Output 2
7 unstable
50 50
0 50
0 0
10 0
10 30
20 40
50 40
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 2014\\C. Crane Balancing Time Limit: 1 second}
\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;
}