ICPC 2011 - D. Chips Challenge
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/2011/D-chips-challenge. Edit
competitive_programming/icpc/2011/D-chips-challenge/solution.tex to update the written solution and
competitive_programming/icpc/2011/D-chips-challenge/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 D
Chips Challenge
Problem ID: chips
A prominent microprocessor company has enlisted your help to lay out some interchangeable components (widgets)
on some of their computer chips. Each chip’s design is an N ×N square of slots. One slot can hold a single component,
and you are to try to fit in as many widgets as possible.
Modern processor designs are complex, of course. You unfortunately have several restrictions:
• Some of the slots are disabled.
• Some of the slots are already occupied by other components and cannot be used for widgets.
• There are sibling memory buses connected to the horizontal and vertical edges of the chip and their bandwidth
loads need to match. As such, there must be exactly as many components in the first row as in the first column,
exactly as many in the second row as in the second column, and so on. Component counts include both the
components already specified on the chip and the added widgets.
• Similarly, the power supply is connected at the end of each row and column. To avoid hot spots, any given row
or column must have no more than A/B of the total components on the chip for a given A and B.
A specification for a chip is N lines of N characters, where ‘.’ indicates an open slot, ‘/’ indicates a disabled slot,
and ‘C’ indicates a slot already occupied by a component. For example:
CC/..
././/
..C.C
/.C..
/./C/
If no more than 3/10 of the components may be in any one row or column, the maximum number of widgets that can
be added to this 5 × 5 chip is 7. A possible arrangement is below, where ‘W’ indicates a widget added in an open slot.
CC/W.
W/W//
W.C.C
/.CWW
/W/C/
Input
The input consists of several test cases. Each case starts with a line containing three integers: The size of the chip
N (1 ≤ N ≤ 40), and A and B (1 ≤ B ≤ 1000, 0 ≤ A ≤ B) as described above. Each of the following N lines
contains N characters describing the slots, one of ‘.’, ‘/’ or ‘C’, as described above.
The last test case is followed by a line containing three zeros.
ICPC 2011 World Finals Problem D: Chips Challenge
Output
For each test case, display a single line beginning with the case number. If there is a solution, display the maximum
number of widgets that can be added to the chip. Display “impossible” if there is no solution.
Follow the format of the sample output.
Sample input Output for the Sample Input
2 1 1 Case 1: 0
/. Case 2: 1
// Case 3: impossible
2 50 100 Case 4: 7
/. Case 5: impossible
C/
2 100 100
./
C.
5 3 10
CC/..
././/
..C.C
/.C..
/./C/
5 2 10
CC/..
././/
..C.C
/.C..
/./C/
0 0 0
ICPC 2011 World Finals Problem D: Chips Challenge
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 2011\\D. Chips Challenge}
\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;
}