ICPC 2009 - F. Deer-Proof Fence
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/2009/F-deer-proof-fence. Edit
competitive_programming/icpc/2009/F-deer-proof-fence/solution.tex to update the written solution and
competitive_programming/icpc/2009/F-deer-proof-fence/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 F
Deer-Proof Fence
Input: fence.in
Uncle Magnus has planted some young saplings on his farm as part of his reforestation project. Unfortunately,
deer like to eat tender sapling shoots and leaves, making it necessary to build protective fences around them.
Since deer and other sapling nibblers can reach partway over the fence, every fence must lie at least a minimum
distance (a margin) from each sapling.
Deer-proof fencing is quite expensive, so Uncle Magnus wants to minimize the total length of fencing used.
Your job is to write a program that computes the minimum length of fencing that is required to enclose and
protect the saplings. Fences may include both straight and curved segments. You may design a single fence that
encloses all saplings or multiple fences that enclose separate groups of saplings.
Figure 6 shows two example configurations, each consisting of three saplings with different margin
requirements. In the top configuration, which corresponds to the first sample input, the minimum-length solution
consists of two separate fences. In the bottom configuration, which corresponds to the second sample input, the
minimum-length solution consists of a single fence.
Figure 6: Deerproof fences.
Input
The input consists of multiple test cases. The first line of each test case contains integers N (0 < N ≤ 9), which is
the number of saplings, and M (0 < M ≤ 200), which is the margin required around each sapling. This line is
followed by N additional lines. Each of these N lines contains two integers x and y that describe the Cartesian
coordinates of a sapling (|x| ≤ 100 and |y| ≤ 100). No two saplings are in the same location. For simplicity the
saplings can all be considered as points and the thickness of deer-proof fences can be considered zero.
The last test case is followed by a line containing two zeros.
Output
For each test case, print the case number (starting with 1) followed by the minimum total length of fencing
required to protect the saplings with the given margin. Print the length with two digits to the right of the decimal
point. Follow the format of the sample output.
Sample Input Output for the Sample Input
3 2 Case 1: length = 29.13
0 0 Case 2: length = 45.13
2 0
10 0
3 4
0 0
2 0
10 0
0 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 2009\\F. Deer-Proof Fence}
\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;
}