ICPC 2015 - L. Weather Report
State the problem in your own words. Focus on the mathematical or algorithmic core rather than repeating the full statement.
Source-first archive entry
This page is built from the copied files in competitive_programming/icpc/2015/L-weather-report. Edit
competitive_programming/icpc/2015/L-weather-report/solution.tex to update the written solution and
competitive_programming/icpc/2015/L-weather-report/solution.cpp to update the implementation.
The website does not replace those files with hand-maintained HTML. It reads the copied source tree during the build and exposes the exact files below.
Problem Statement
Copied statement text kept beside the solution archive for direct reference.
Problem L
Weather Report
Time limit: 2 seconds
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 Input 1 Sample Output 1
2 1.457510
0.9 0.049999 0.05 0.000001
Sample Input 2 Sample Output 2
20 40.000000
0.25 0.25 0.25 0.25
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 2015\\L. Weather Report}
\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;
}