ICPC 2012 - K. Stacking Plates
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/K-stacking-plates. Edit
competitive_programming/icpc/2012/K-stacking-plates/solution.tex to update the written solution and
competitive_programming/icpc/2012/K-stacking-plates/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
Stacking Plates
Problem ID: stacking
The Plate Shipping Company is an Internet retailer that, as their name suggests, exclusively sells plates.
They pride themselves in offering the widest selection of dinner plates in the universe from a large
number of manufacturers.
In a recent cost analysis the company has discovered that they spend a large amount of money on packing
the plates for shipment. Part of the reason is that plates have to be stacked before being put into shipping
containers. And apparently, this is taking more time than expected. Maybe you can help.
A shipment of plates consists of plates from several manufacturers. The plates from each manufacturer
come stacked, that is, each arranged in a single stack with plates ordered by size (the smallest at the top,
the largest at the bottom). We will call such a stack properly ordered. To ship all these plates, you must
combine them into a single stack, again properly ordered. To join the manufacturers’ stacks into a single
stack, two kinds of operations are allowed:
• Split: a single stack can be split into two stacks by lifting any top portion of the stack and putting
it aside to form a new stack.
• Join: two stacks can be joined by putting one on top of the other. This is allowed only if the
bottom plate of the top stack is no larger than the top plate of the bottom stack, that is, the joined
stack has to be properly ordered.
Note that a portion of any stack may never be put directly on top of another stack. It must first be split
and then the split portion must be joined with the other stack. Given a collection of stacks, you have to
find the minimum number of operations that transforms them into a single stack. The following example
corresponds to the sample input, and shows how two stacks can be transformed to a single stack in five
operations:
Input
Each test case starts with a line containing a single integer n (1 ≤ n ≤ 50), the number of stacks that
have to be combined for a shipment. This is followed by n lines, each describing a stack. These lines
start with an integer h (1 ≤ h ≤ 50), the height of the stack. This number is followed by h positive
integers that give the diameters of the plates, from top to bottom. All diameters are at most 10 000.
These numbers will be in non-decreasing order.
Output
For each test case, display the case number and the minimum number of operations (splits and joins)
that have to be performed to combine the given stacks into a single stack.
Sample Input Output for Sample Input
2 Case 1: 5
3 1 2 4 Case 2: 2
2 3 5
3
4 1 1 1 1
4 1 1 1 1
4 1 1 1 1
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\\K. Stacking Plates}
\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;
}