ICPC 2015 - K. Tours
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/2015/K-tours. Edit
competitive_programming/icpc/2015/K-tours/solution.tex to update the written solution and
competitive_programming/icpc/2015/K-tours/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 K
Tours
Time limit: 3 seconds
The Arca Carania Mountain national park is opening up for tourist traffic. The national park has a
number of sites worth seeing and roads that connect pairs of sites. The park commissioners have put
together a set of round tours in the park in which visitors can ride buses to view various sites. Each
round tour starts at some site (potentially different sites for different tours), visits a number of other sites
without repeating any, and then returns to where it started. At least 3 different sites are visited in each
round tour. At least one round tour is possible in the national park.
The park commissioners have decided that, for any given road, all buses will be operated by a single
company. The commissioners do not want to be accused of favoritism, so they want to be sure that each
possible round tour in the park has exactly the same number of roads assigned to each bus company.
They realize this may be difficult to achieve. Thus, they want to learn what numbers of bus companies
allow for a valid assignment of companies to roads.
Consider Sample Input 1, which is illustrated in Figure K.1. There are a total of three round tours for
these sites. Some company is assigned road 1-3. It must also be assigned some road on the round tour
1-2-3-4-1, say 2-3. But then it is assigned to two of the three roads on the round tour 1-2-3-1, and no
other company can match this – so there can be no other companies. In Sample Input 2 there is only one
round tour, so it is enough to assign the roads of this tour equally between companies.
i i
Site 2 Site 3
i i
Site 1 Site 4
Figure K.1: Sample Input 1.
Input
The first line of input contains two integers n (1 ≤ n ≤ 2 000), which is the number of sites in the park,
and m (1 ≤ m ≤ 2 000), which is the number of roads between the sites. Following that are m lines,
each containing two integers ai and bi (1 ≤ ai < bi ≤ n), meaning the sites ai and bi are connected by
a bidirectional road. No pair of sites is listed twice.
Output
Display all integers k such that it is possible to assign the roads to k companies in the desired way. These
integers should be in ascending order.
Sample Input 1 Sample Output 1
4 5 1
1 2
2 3
3 4
1 4
1 3
Sample Input 2 Sample Output 2
6 6 1 3
1 2
2 3
1 3
1 4
2 5
3 6
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 2015\\K. Tours}
\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;
}