ICPC 2014 - E. Maze Reduction
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/2014/E-maze-reduction. Edit
competitive_programming/icpc/2014/E-maze-reduction/solution.tex to update the written solution and
competitive_programming/icpc/2014/E-maze-reduction/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 E
Maze Reduction
Time Limit: 2 seconds
Jay runs a small carnival that has various rides and attractions. Unfortunately, times are tough. A recent
roller coaster accident, flooding in the restrooms, and an unfortunate clown incident have given Jay’s
carnival a bad reputation with the public. With fewer paying customers and reduced revenue, he will
need to cut some costs to stay in business.
One of the biggest carnival attractions is a large, confusing maze. It consists of a variety of circular
rooms connected by narrow, twisting corridors. Visitors love getting lost in it and trying to map it out. It
has come to Jay’s attention that some of the rooms might be effectively identical to each other. If that’s
the case, he will be able to reduce its size without anyone noticing.
Two rooms A and B are effectively identical if, when you are dropped into either room A or B (and you
know the map of the maze), you cannot tell whether you began in A or B just by exploring the maze.
The corridor exits are evenly spaced around each room, and you cannot mark or leave anything in a
room (in particular, you cannot tell whether you have previously visited it). The only identifying feature
that rooms have is their number of exits. Corridors are also twisty enough to be indistinguishable from
each other, but when you enter a room you know which corridor you came from, so you can navigate a
little by using the order they appear around the room.
Jay has appealed to the Association for Carnival Mazery for help. That’s you! Write a program to
determine all the sets of effectively identical rooms in the maze.
Input
The input consists of a single test case. The first line contains an integer n, the number of rooms in the
maze (1 ≤ n ≤ 100). Rooms are numbered from 1 to n. Following this are n lines, describing each room
in order. Each line consists of an integer k, indicating that this room has k corridors (0 ≤ k < 100), and
then k distinct integers listing the rooms each corridor connects to (in clockwise order, from an arbitrary
starting point). Rooms do not connect to themselves.
Output
Display one line for each maximal set of effectively identical rooms (ignoring sets of size 1) containing
the room numbers in the set in increasing order. Order the sets by their smallest room numbers. If there
are no such sets, display none instead.
Sample Input 1 Sample Output 1
13 2 4
2 2 4 5 6
3 1 3 5 7 8 9 10 11 12 13
2 2 4
3 1 3 6
2 2 6
2 4 5
2 8 9
2 7 9
2 7 8
2 11 13
2 10 12
2 11 13
2 10 12
Sample Input 2 Sample Output 2
6 none
3 3 4 5
0
1 1
1 1
2 1 6
1 5
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 2014\\E. Maze Reduction Time Limit: 2 seconds}
\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;
}