EP Ethan Pham Project Euler, competitive programming, and notes on the things I keep studying.
ICPC 2010
ICPC 2010

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 ten...

Updated May 21, 2026
Track ICPC
Year 2010
Statement Text + PDF
TeXC++Statement textStatement PDF
Problem statement

Problem Statement

Formatted from the contest statement text, with sample tests broken out into copyable blocks.

Statement text Statement PDF

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 Tests

Sample
Sample Input 
3 3
.#.
...
.#.
6 7
.......
.......
.......
....#..
.......
.#.....
0 0

This page intentionally left blank.
Sample Output 
Case 1:
C#.
CCC
.#C

Case 2:
CCCCCCC
......C
CCCCCCC
C...#..
CCC.CCC
.#CCC.C
Editorial

Editorial

The solution write-up is rendered from the LaTeX source, with equations kept live through MathJax.

Raw TeX

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

  1. Describe the data structures and the state maintained by the algorithm.

  2. Explain the processing order and why it is sufficient.

  3. 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.

Implementation

Code

C++ solution used for this page.

C++

Clean code view with a raw-file link when you want the original source.

Raw file 
#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;
}
Resources

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TeX write-upcompetitive_programming/icpc/2010/E-channel/solution.texC++ implementationcompetitive_programming/icpc/2010/E-channel/solution.cppStatement textcompetitive_programming/icpc/2010/E-channel/statement.txtStatement PDFcompetitive_programming/icpc/2010/E-channel/statement.pdfMetadatacompetitive_programming/icpc/2010/E-channel/meta.jsonYear packetcompetitive_programming/icpc/2010/contest_problems.pdf