ICPC 2011 - H. Mining Your Own Business
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/2011/H-mining-your-own-business/solution.tex to update the written solution and
competitive_programming/icpc/2011/H-mining-your-own-business/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 H
Mining Your Own Business
Problem ID: mining
John Digger is the owner of a large illudium phosdex mine. The mine is made up of a series of tunnels that meet at
various large junctions. Unlike some owners, Digger actually cares about the welfare of his workers and has a concern
about the layout of the mine. Specifically, he worries that there may a junction which, in case of collapse, will cut off
workers in one section of the mine from other workers (illudium phosdex, as you know, is highly unstable). To counter
this, he wants to install special escape shafts from the junctions to the surface. He could install one escape shaft at
each junction, but Digger doesn’t care about his workers that much. Instead, he wants to install the minimum number
of escape shafts so that if any of the junctions collapses, all the workers who survive the junction collapse will have a
path to the surface.
Write a program to calculate the minimum number of escape shafts and the total number of ways in which this
minimum number of escape shafts can be installed.
Input
The input consists of several test cases. The first line of each case contains a positive integer N (N ≤ 5·104 ) indicating
the number of mine tunnels. Following this are N lines each containing two distinct integers s and t, where s and t
are junction numbers. Junctions are numbered consecutively starting at 1. Each pair of junctions is joined by at most
a single tunnel. Each set of mine tunnels forms one connected unit (that is, you can get from any one junction to any
other).
The last test case is followed by a line containing a single zero.
Output
For each test case, display its case number followed by the minimum number of escape shafts needed for the system
of mine tunnels and the total number of ways these escape shafts can be installed. You may assume that the result fits
in a signed 64-bit integer.
Follow the format of the sample output.
ICPC 2011 World Finals Problem H: Mining Your Own Business
Sample input Output for the Sample Input
9 Case 1: 2 4
1 3 Case 2: 4 1
4 1
3 5
1 2
2 6
1 5
6 3
1 6
3 2
6
1 2
1 3
2 4
2 5
3 6
3 7
0
ICPC 2011 World Finals Problem H: Mining Your Own Business
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 2011\\H. Mining Your Own Business}
\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;
}