ICPC 2010 - G. The Islands
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/2010/G-the-islands. Edit
competitive_programming/icpc/2010/G-the-islands/solution.tex to update the written solution and
competitive_programming/icpc/2010/G-the-islands/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 G
The Islands
Problem ID: islands
Wen Chen is the captain of a rescue boat. One of his important tasks is to visit a group of islands once a day to
check if everything is all right. Captain Wen starts
from the west-most island, makes a pass to the
east-most island visiting some of the islands, then
makes a second pass from the east-most island
back to the first one visiting the remaining
islands. In each pass Captain Wen moves steadily
east (in the first pass) or west (in the second pass),
but moves as far north or south as he needs to
reach the islands. The only complication is that
there are two special islands where Wen gets fuel
for his boat, so he must visit them in separate
passes. Figure 7 shows the two special islands in
pink (1 and 3) and one possible path Captain Wen
could take.
Calculate the length of the shortest path to visit all
Figure 7
the islands in two passes when each island’s
location and the identification of the two special islands are given.
Input
The input consists of multiple test cases. The data for each case begins with a line containing 3 integers n
(4 ≤ n ≤ 100), b1, and b2 (0 < b1, b2 < n-1 and b1 ≠ b2), where n is the number of islands (numbered 0 to n-1) and
b1 and b2 are the two special islands. Following this, there are n lines containing the integer x- and y-coordinates
of each island (0 ≤ x, y ≤ 2000), starting with island 0. No two islands have the same x-coordinate and they are
in order from west-most to east-most (that is, minimum x-coordinate to maximum x-coordinate).
Input for the last case is followed by a line containing 3 zeroes.
Output
For each case, display two lines. The first line contains the case number and the length of the shortest tour
Captain Wen can take to visit all the islands, rounded and displayed to the nearest hundredth. The second line
contains a space-separated list of the islands in the order that they should be visited, starting with island 0 and 1,
and ending with island 0. Each test case will have a unique solution. Follow the format in the sample output.
Sample Input Output for the Sample Input
5 1 3 Case 1: 18.18
1 3 0 1 4 3 2 0
3 4 Case 2: 24.30
4 1 0 1 3 4 2 0
7 5
8 3
5 3 2
0 10
3 14
4 7
7 10
8 12
0 0 0
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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 2010\\G. The Islands}
\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;
}