D. Gem Island
Gem Island is a tiny island in the middle of the Pacific Ocean. Until recently, it was known as one of the poorest, but also most peaceful, places on Earth. Today, it is neither poor nor peaceful. What happened? One sunny morning, not too long ago, all inha...
Problem Statement
Formatted from the contest statement text, with sample tests broken out into copyable blocks.
Gem Island is a tiny island in the middle of the Pacific Ocean. Until recently, it was known as one of the poorest, but also most peaceful, places on Earth. Today, it is neither poor nor peaceful. What happened? One sunny morning, not too long ago, all inhabitants of Gem Island woke up to a surprise. That morn- ing, each of them suddenly held one sparkling gem in their hand. The gems had magically appeared overnight. This was cause for much rejoicing – everybody was suddenly rich, they could finally afford all the things they had ever dreamed of, and the name of their island made so much more sense now. The next morning, one of the inhabitants woke up to another surprise – her gem had magically split into two gems! The same thing happened on each of the following nights, when exactly one of the gems (apparently uniformly at random among all the gems on the island) would split into two. After a while, the inhabitants of Gem Island possessed a widely varying number of gems. Some had a lot and many had only a few. How come some inhabitants had more gems than others? Did they cheat, were they just lucky, or was something else at work? The island elders have asked for your help. They want you to determine if the uneven distribution of gems is explained by pure chance. If so, that would greatly reduce tensions on the island. The island has n inhabitants. You are to determine the gem distribution after d nights of gem splitting. In particular, you are interested in the expected number of gems collectively held by the r people with the largest numbers of gems. More formally, suppose that after d nights the numbers of gems held by the n inhabitants are listed in non-increasing order as a1 ≥ a2 ≥ . . . ≥ an . What is the expected value of a1 + · · · + ar ?
Input
The input consists of a single line containing the three integers n, d, and r (1 ≤ n, d ≤ 500, 1 ≤ r ≤ n), as described in the problem statement above.
Output
Display the expected number of gems that the top r inhabitants hold after d nights, with an absolute or relative error of at most 10−6 .
Sample Tests
2 3 13.5 3 3 24.9 5 10 312.2567433Editorial
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/2018/D-gem-island/solution.texC++ implementationcompetitive_programming/icpc/2018/D-gem-island/solution.cppStatement textcompetitive_programming/icpc/2018/D-gem-island/statement.txtStatement PDFcompetitive_programming/icpc/2018/D-gem-island/statement.pdfMetadatacompetitive_programming/icpc/2018/D-gem-island/meta.jsonYear packetcompetitive_programming/icpc/2018/contest_problems.pdf