ICPC 2013 - F. Low Power
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/F-low-power/solution.tex to update the written solution and
competitive_programming/icpc/2013/F-low-power/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 F
Low Power
Time Limit: 4 seconds
You are building advanced chips for machines. Making the chips is easy, but the power supply turns out
to be an issue since the available batteries have varied power outputs.
Consider the problem of n machines, each with two chips, where each chip is powered by k batteries.
Surprisingly, it does not matter how much power each chip gets, but a machine works best when its two
chips have power outputs as close as possible. The power output of a chip is simply the smallest power
output of its k batteries.
You have a stockpile of 2nk batteries that you want to assign to the chips. It might not be possible
to allocate the batteries so that in every machine both chips have equal power outputs, but you want
to allocate them so that the differences are as small as possible. To be precise, you want to tell your
customers that in all machines the difference of power outputs of the two chips is at most d, and you
want to make d as small as possible. To do this you must determine an optimal allocation of the batteries
to the machines.
Consider Sample Input 1. There are 2 machines, each requiring 3 batteries per chip, and a supply of
batteries with power outputs 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. You can, for instance, assign the batteries
with power outputs 1, 3, 5 to one chip, those with power 2, 4, 12 to the other chip of the same machine,
those with power 6, 8, 9 to the third chip, and those with power 7, 10, 11 to the fourth. The power
outputs of the chips are 1, 2, 6, and 7, respectively, and the difference between power outputs is 1 in both
machines. Note that there are many other ways to achieve this result.
Input
The input consists of a single test case. A test case consists of two lines. The first line contains two
positive integers: the number of machines n and the number of batteries per chip k (2nk ≤ 106 ). The
second line contains 2nk integers pi specifying the power outputs of the batteries (1 ≤ pi ≤ 109 ).
Output
Display the smallest number d such that you can allocate the batteries so that the difference of power
outputs of the two chips in each machine is at most d.
Sample Input 1 Sample Output 1
2 3 1
1 2 3 4 5 6 7 8 9 10 11 12
Sample Input 2 Sample Output 2
2 2 2
3 1 3 3 3 3 3 3
This page is intentionally left blank.
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\\F. Low Power 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;
}