ICPC 2010 - E. Channel
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/2010/E-channel. Edit
competitive_programming/icpc/2010/E-channel/solution.tex to update the written solution and
competitive_programming/icpc/2010/E-channel/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 E
Channel
Problem ID: channel
Joe, a former champion coder, has finally bought the farm. No, no, he’s alive and well; he merely made use of
his vast programming competition winnings to purchase his ancestral farm. He hopes to retire and spend the rest
of his days tending cows (for some reason, he now considers himself an expert on cows).
Sadly, Farmer Joe’s simple bucolic ambitions are not to be. His farm is situated in a cold, northern climate - too
cold for cows! What’s worse, the climate is dry and ill-suited for growing crops. Joe now realizes that he will
have to set up an irrigation scheme for his field. This scheme involves diverting part of a river into a long,
winding channel through his field. Since the crops nearest the channel will thrive, the longer the channel, the
better.
His field is a long rectangular strip of land divided into unit squares. Each square either has dirt, represented by
‘.’, or an immovable rock, represented by ‘#’. The irrigation channel, which is one square wide, enters his land
in the upper left corner and exits it in the lower right. The channel cannot pass through rocks, of course. It is
important that this channel never touches itself, even at a corner (or else the water will seep through and take a
shortcut). Figure 3 and Figure 4 contain examples of the channel touching itself.
C.CCC C...
C.C.C CCCC
CCC.C ..CC
...CC ..C.
...C. ..C.
...CC ..CC
Figure 3 Figure 4
Unfortunately, Joe’s best programming days are long behind him. He has a straightforward solution but it turns
out to be far too time consuming. Can you help him figure out the optimal placement of this channel?
Input
Input consists of several test cases. Each test case starts with a line containing r, the number of rows in his field
(2 r 20), and c, the number of columns (2 c 9). The next r lines each contain a string of length c,
describing his field.
The last test case is followed by a line containing two zeros.
Output
For each test case, display the case number. On the following r lines show how to place the longest possible
channel in the field, subject to the above restrictions. Use the character ‘C’ for the channel. There will always be
a unique solution. Follow the format in the sample output. Print a blank line after the output for each test case.
Sample Input Output for the Sample Input
3 3 Case 1:
.#. C#.
... CCC
.#. .#C
6 7
....... Case 2:
....... CCCCCCC
....... ......C
....#.. CCCCCCC
....... C...#..
.#..... CCC.CCC
0 0 .#CCC.C
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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 2010\\E. Channel}
\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;
}