ICPC 2013 - H. Matrxka
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/2013/H-matr-xka/solution.tex to update the written solution and
competitive_programming/icpc/2013/H-matr-xka/solution.cpp to update the implementation.
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Problem Statement
Copied statement text kept beside the solution archive for direct reference.
ICPC 2013
2013 World Finals
St. Petersburg
HOSTED BY ITMO
Problem H
Matrxka
Time Limit: 5 seconds
Matryoshkas are sets of traditional Russian wooden dolls of decreasing size placed one inside the other.
A matryoshka doll can be opened to reveal a smaller figure of the same sort inside, which has, in turn,
another figure inside, and so on.
The Russian Matryoshka Museum recently exhibited a col-
lection of similarly designed matryoshka sets, differing
only in the number of nested dolls in each set. Unfor-
tunately, some over-zealous (and obviously unsupervised)
children separated these sets, placing all the individual
dolls in a row. There are n dolls in the row, each with an
integer size. You need to reassemble the matryoshka sets,
knowing neither the number of sets nor the number of dolls
in each set. You know only that every complete set consists
of dolls with consecutive sizes from 1 to some number m, Picture from Wikimedia Commons
which may vary between the different sets.
When reassembling the sets, you must follow these rules:
• You can put a doll or a nested group of dolls only inside a larger doll.
• You can combine two groups of dolls only if they are adjacent in the row.
• Once a doll becomes a member of a group, it cannot be transferred to another group or per-
manently separated from the group. It can be temporarily separated only when combining two
groups.
Your time is valuable, and you want to do this reassembly process as quickly as possible. The only
time-consuming part of this task is opening and subsequently closing a doll, so you want to minimize
how often you do this. For example, the minimum number of openings (and subsequent closings) when
combining group [1, 2, 6] with the group [4] is two, since you have to open the dolls with sizes 6 and 4.
When combining group [1, 2, 5] with the group [3, 4], you need to perform three openings.
Write a program to calculate the minimum number of openings required to combine all disassembled
matryoshka sets.
Input
The input consists of a single test case. A test case consists of two lines. The first line contains one
integer n (1 ≤ n ≤ 500) representing the number of individual dolls in the row. The second line
contains n positive integers specifying the sizes of the dolls in the order they appear in the row. Each
size is between 1 and 500 inclusive.
Output
Display the minimum number of openings required when reassembling the matryoshka sets. If reassem-
bling cannot be done (some of the kids might have been excessively zealous and taken some dolls),
display the word impossible.
ICPC 2013
2013 World Finals
St. Petersburg
HOSTED BY ITMO
Sample Input 1 Sample Output 1
7 impossible
1 2 1 2 4 3 3
Sample Input 2 Sample Output 2
7 7
1 2 3 2 4 1 3
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 2013\\H. Matrxka Time Limit: 5 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;
}