ICPC 2017 - F. Posterize
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/2017/F-posterize/solution.tex to update the written solution and
competitive_programming/icpc/2017/F-posterize/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.
Rapid City
event
sponsor
ICPC 2017
Problem F
Posterize
Time limit: 2 seconds
Pixels in a digital picture can be represented with three integers in the range 0 to 255 that indicate the
intensity of the red, green, and blue colors. To compress an image or to create an artistic effect, many
photo-editing tools include a “posterize” operation which works as follows. Each color channel is examined
separately; this problem focuses only on the red channel. Rather than allow all integers from 0 to 255 for the
red channel, a posterized image allows at most k integers from this range. Each pixel’s original red intensity
is replaced with the nearest of the allowed integers. The photo-editing tool selects a set of k integers that
minimizes the sum of the squared errors introduced across all pixels in the original image. If there are n
pixels that have original red values r1 , . . . , rn , and k allowed integers v1 , . . . , vk , the sum of squared errors
is defined as
n
X
min (ri − vj )2 .
1≤j≤k
i=1
Your task is to compute the minimum achievable sum of squared errors, given parameter k and a description
of the red intensities of an image’s pixels.
Input
The first line of the input contains two integers d (1 ≤ d ≤ 256), the number of distinct red values that occur
in the original image, and k (1 ≤ k ≤ d), the number of distinct red values allowed in the posterized image.
The remaining d lines indicate the number of pixels of the image having various red values. Each such line
contains two integers r (0 ≤ r ≤ 255) and p (1 ≤ p ≤ 226 ), where r is a red intensity value and p is the
number of pixels having red intensity r. Those d lines are given in increasing order of red value.
Output
Display the sum of the squared errors for an optimally chosen set of k allowed integer values.
Rapid City
event
sponsor
ICPC 2017
Sample Input 1 Sample Output 1
2 1 66670000
50 20000
150 10000
Sample Input 2 Sample Output 2
2 2 0
50 20000
150 10000
Sample Input 3 Sample Output 3
4 2 37500000
0 30000
25 30000
50 30000
255 30000
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 2017\\F. Posterize}
\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;
}