ICPC 2014 - G. Metal Processing Plant
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/2014/G-metal-processing-plant/solution.tex to update the written solution and
competitive_programming/icpc/2014/G-metal-processing-plant/solution.cpp to update the implementation.
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Problem Statement
Copied statement text kept beside the solution archive for direct reference.
Problem G
Metal Processing Plant
Time Limit: 4 seconds
Yulia works for a metal processing plant in Eka-
terinburg. This plant processes ores mined in the
Ural mountains, extracting precious metals such
as chalcopyrite, platinum and gold from the ores.
Every month the plant receives n shipments of un-
processed ore. Yulia needs to partition these ship-
ments into two groups based on their similarity.
Then, each group is sent to one of two ore pro-
cessing buildings of the plant.
To perform this partitioning, Yulia first calculates
a numeric distance d(i, j) for each pair of ship-
ments 1 ≤ i ≤ n and 1 ≤ j ≤ n, where the Picture from Wikimedia Commons
smaller the distance, the more similar the ship-
ments i and j are. For a subset S ⊆ {1, . . . , n} of shipments, she then defines the disparity D of S as
the maximum distance between a pair of shipments in the subset, that is,
D(S) = max d(i, j).
i,j∈S
Yulia then partitions the shipments into two subsets A and B in such a way that the sum of their dispar-
ities D(A) + D(B) is minimized. Your task is to help her find this partitioning.
Input
The input consists of a single test case. The first line contains an integer n (1 ≤ n ≤ 200) indicating
the number of shipments. The following n − 1 lines contain the distances d(i, j). The ith of these lines
contains n − i integers and the j th integer of that line gives the value of d(i, i + j). The distances are
symmetric, so d(j, i) = d(i, j), and the distance of a shipment to itself is 0. All distances are integers
between 0 and 109 (inclusive).
Output
Display the minimum possible sum of disparities for partitioning the shipments into two groups.
Sample Input 1 Sample Output 1
5 4
4 5 0 2
1 3 7
2 0
4
Sample Input 2 Sample Output 2
7 15
1 10 5 5 5 5
5 10 5 5 5
100 100 5 5
10 5 5
98 99
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 2014\\G. Metal Processing Plant Time Limit: 4 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;
}