ICPC 2012 - F. Keys
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/2012/F-keys. Edit
competitive_programming/icpc/2012/F-keys/solution.tex to update the written solution and
competitive_programming/icpc/2012/F-keys/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
Keys
Problem ID: keys
Adam carries a bunch of keys attached to key rings, some of which may be connected to each other. The
rings are common key rings, so a key can be attached to or detached from a ring by sliding along the
spiral. In the same way, two rings can be connected or disconnected. Adam wants to give some of the
keys to Brenda. Since manipulating the keys and rings is often an annoying task (and also dangerous to
one’s fingernails), Adam is looking for a way to minimize the number of key and ring operations.
Every key attachment, key detachment, ring connection, or ring disconnection is considered one opera-
tion. Since manipulating two rings is significantly easier than sliding a key, we first want to minimize the
number of keys being detached and attached. Among solutions with the same minimal number of key
operations, you need to find the one with the minimal number of ring connections and disconnections.
When all the operations are complete, Adam and Brenda must each carry one connected group of rings
and keys. The only exception is when either of them would have no keys at all—in such a case, no ring
is needed. Each key must be attached to exactly one ring. Some rings (but not keys) may be considered
leftovers and may remain disconnected from the two groups.
The left side of the following figure shows an initial configuration consisting of four keys on three rings.
Adam wishes to give Brenda the two keys labeled N and R. This can be accomplished by two key
operations and one ring operation, resulting in the configuration shown on the right side of the figure.
Input
Each test case contains one or more lines, each containing a two letter string. Lowercase letters (a - z)
represent key rings and uppercase letters (A - Z) represent keys. The two letters on a line specify either
a key attached to a ring or two rings connected together. The end of each test case is denoted by a line
containing the digit zero.
Keys denoted by letters A through M remain with Adam, and keys denoted by letters N through Z are
given to Brenda.
No line contains two uppercase letters. No pair of letters are specified more than once in the same
test case. Each key is connected to exactly one ring. There are no “circles” in the ring configurations
(disconnecting any two rings will increase the number of connected groups). All existing keys and rings
are mentioned at least once.
Output
For each test case, display the case number followed by the minimal number of key attach/detach oper-
ations and the minimal number of ring connect/disconnect operations.
If there is no way to split the keys as requested, display the case number and the word impossible
instead of the two integers.
Sample Input Output for Sample Input
ab Case 1: 2 1
bc Case 2: 0 2
aA Case 3: impossible
aN Case 4: 0 7
Rb
cB
0
aA
bB
Cc
0
aA
aZ
0
aA
bB
cC
xX
yY
ax
xb
by
yc
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 2012\\F. Keys}
\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;
}