ICPC 2008 - G. Net Loss
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/2008/G-net-loss. Edit
competitive_programming/icpc/2008/G-net-loss/solution.tex to update the written solution and
competitive_programming/icpc/2008/G-net-loss/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 G
Net Loss
Input file: netloss.in
Rose N. Blatt is designing an embedded neural network to place inside a cell phone. When trained by the phone’s
owner, the neural network will enable the user to dictate text messages in a hands-free way. The key idea in Rose’s
design is the use of complicated polynomial response functions in each of the nodes of the network (rather than the
more traditional thresholding functions used in many other neural nets). Figure 1 shows a portion of such a neural
network (the polynomials are not accurately graphed).
-0.3x4 + 0.2x3 + 0.7
-x9 + 0.2x8 +0.13x3
P P P P
P P P P P P
x7 + 0.1x2 - 0.5x
P P P P
-x2 + x -0.3
P P
Figure 1
When Rose was ready to select her polynomials, she discovered a problem. Due to the limited amount of memory
available, she did not have enough space to store all of the coefficients of the polynomials in her network. She has
decided to use an approximation to each polynomial in the form of a continuous polygonal curve with two segments,
y = a1x + a0 and y = b1x + b0. The segments meet at a point whose x-coordinate, c, is between -1 and +1. Rose wants
B B B B B B B B
the approximation to be the best in the sense that the distance between p and the approximation function g is
minimal. We define the distance between p and g as the integral of the square of their difference:
1
d(p,g) =
∫
-1
(p(x)-g(x))2 dx P P
For instance, if the polynomial is x2 – 0.2, then the best polygonal approximation, with lines meeting at c = 0, is
P P
shown in Figure 2 (the dotted line shows the graph of the polygonal approximation).
⎧- x - 0.367, - 1 ≤ x ≤ 0
g(x) = ⎨
⎩ x - 0.367, 0 ≤ x ≤ 1
Figure 2
In the few bytes that are available for each node, Rose can store the values of a1, a0, b1, b0, and c as signed numbers. B B B B B B B B
Fortunately Rose has a program that supplies her with a good guess for the value of c. Given this value, you are
asked to help Rose find the optimal values for a1, a0, b1, and b0 in the approximations to her polynomials.
B B B B B B B B
Input
The input file contains multiple test cases. Each test case consists of three lines. The first line contains a positive
integer n, 1 ≤ n ≤ 10, representing the degree of the polynomial p(x). This is followed by a line containing n +1
numbers between -1 and 1 inclusive, which are the coefficients of p(x) from highest order term down to the constant
term, expressed with at most three places after the decimal point. The last line for each test case contains the value
for c, -1 < c < 1, expressed with at most three places after the decimal point.
A line containing the integer zero follows the last test case.
Output
For each test case, print the case number (beginning with 1) and the four optimal values, displaying each with exactly
three places after the decimal point. The first and second values are the parameters a1 and a0 of the line segment
B B B B
y = a1x + a0 defining g in the range -1 ≤ x ≤ c. The third and fourth values are the parameters b1 and b0 of the line
B B B B B B B B
segment y = b1x + b0 defining g in the range c ≤ x ≤ 1. The distance d(p,g) between p and g (as defined earlier)
B B B B
should be the minimum over all such choices for a1, a0, b1, and b0.
B B B B B B B B
Sample Input Output for the Sample Input
2 Case 1: -1.000 -0.367 1.000 -0.367
1.0 0.0 -0.2 Case 2: -0.499 -0.036 1.198 -1.236
0.0
3
1 0 -1 0
0.707
0
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 2008\\G. Net Loss}
\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;
}