F. Low Power
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...
Problem Statement
Formatted from the contest statement text, with sample tests broken out into copyable blocks.
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 Tests
2 3
1 2 3 4 5 6 7 8 9 10 11 121 2 2
3 1 3 3 3 3 3 32Editorial
The solution write-up is rendered from the LaTeX source, with equations kept live through MathJax.
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
C++ solution used for this page.
#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 and Assets
Raw files are still available here when you want the original TeX, C++, or statement assets.
Show raw files
competitive_programming/icpc/2013/F-low-power/solution.texC++ implementationcompetitive_programming/icpc/2013/F-low-power/solution.cppStatement textcompetitive_programming/icpc/2013/F-low-power/statement.txtStatement PDFcompetitive_programming/icpc/2013/F-low-power/statement.pdfMetadatacompetitive_programming/icpc/2013/F-low-power/meta.jsonYear packetcompetitive_programming/icpc/2013/contest_problems.pdf