L. Weather Report
You have been hired by the Association for Climatological Measurement, a scientific organization interested in tracking global weather trends over a long period of time. Of course, this is no easy task. They have deployed many small devices around the world...
Problem Statement
Formatted from the contest statement text, with sample tests broken out into copyable blocks.
You have been hired by the Association for Climatological Measurement, a scientific organization interested in tracking global weather trends over a long period of time. Of course, this is no easy task. They have deployed many small devices around the world, designed to take periodic measurements of the local weather conditions. These are cheap devices with somewhat restricted capabilities. Every day they ob- serve which of the four standard kinds of weather occurred: Sunny, Cloudy, Rainy, or Frogs. After every n of these ob- servations have been made, the results are reported to the main server for analysis. However, the massive number of devices has caused the available communication bandwidth to be overloaded. The Association needs your help to come up with a method of compressing these reports into fewer bits. For a particular device’s location, you may assume that the weather each day is an independent random event, and you are given the predicted probabilities of the four possible weather types. Each of the 4n possible weather reports for a device must be encoded as a unique sequence of bits, such that no sequence is a prefix of any other sequence (an important property, or else the server would not know when each sequence ends). The goal is to use an encoding that minimizes the expected number of transmitted bits.
Input
The first line of input contains an integer 1 ≤ n ≤ 20, the number of observations that go into each report. The second line contains four positive floating-point numbers, psunny , pcloudy , prainy , and pfrogs , representing the respective weather probabilities. These probabilities have at most 6 digits after the decimal point and sum to 1.
Output
Display the minimum expected number of bits in the encoding of a report, with an absolute or relative error of at most 10−4 .
Sample Tests
2
0.9 0.049999 0.05 0.0000011.457510 20
0.25 0.25 0.25 0.2540.000000Editorial
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;
}
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