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ICPC 2017
ICPC 2017

G. Replicate Replicate Rfplicbte

event sponsor ICPC 2017 Problem G Replicate Replicate Rfplicbte Time limit: 3 seconds The owner of the Automatic Cellular Manufacturing corporation has just patented a new process for the mass production of identical parts. Her approach uses a two-dimension...

Updated May 21, 2026
Track ICPC
Year 2017
Statement Text + PDF
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Problem statement

Problem Statement

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

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Time limit 3 seconds

The owner of the Automatic Cellular Manufacturing corporation has just patented a new process for the mass production of identical parts. Her approach uses a two-dimensional lattice of two-state cells, each of which is either “empty” or “filled.” The exact details are, of course, proprietary. Initially, a set of cells in the lattice is filled with a copy of the part that is to be reproduced. In a sequence of discrete steps, each cell in the lattice simultaneously updates its state by examining its own state and those of its eight surrounding neighbors. If an odd number of these nine cells are filled, the cell’s state in the next time step will be filled, otherwise it will be empty. Figure G.1 shows several steps in the replication process for a simple pattern consisting of three filled cells.

Figure G.1: The replication process.

However, a bug has crept into the process. After each update step, one cell in the lattice might spontaneously flip its state. For instance, Figure G.2 shows what might happen if a cell flipped its state after the first time step and another flipped its state after the third time step.

Figure G.2: Errors in the replication process. This figure corresponds to Sample Input 1.

Unfortunately, the original patterns were lost, and only the (possibly corrupted) results of the replication remain. Can you write a program to determine a smallest possible nonempty initial pattern that could have resulted in a given final pattern?

Input

The first line of input contains two integers w (1 ≤ w ≤ 300) and h (1 ≤ h ≤ 300), where w and h are the width and height of the bounding box of the final pattern. Following that are h lines, each containing w characters, giving the final pattern. Each character is either ‘.’ (representing an empty cell) or ‘#’ (repre- senting a filled cell). There is at least one filled cell in the first row, in the last row, in the first column, and in the last column.

Rapid City
sponsor
               ICPC 2017

Output

Display a minimum-size nonempty pattern that could have resulted in the given pattern, assuming that at each stage of the replication process at most one cell spontaneously changed state. The size of a pattern is the area of its bounding box. If there is more than one possible minimum-size nonempty starting pattern, any one will be accepted. Use the character ‘.’ for empty cells and ‘#’ for filled cells. Use the minimum number of rows and columns needed to display the pattern.

Sample Tests

Sample 1
Sample Input 
 10 10
 .#...#...#
 ##..##..##
 ##.#.##...
 ##.#.##...
 .#...#####
 ...##..#.#
 ......###.
 ##.#.##...
 #..#..#..#
 ##..##..##
Sample Output 
.#
##
Sample 2
Sample Input 
 8 8
 ##..#.##
 #.####.#
 .#.#.#..
 .##.#.##
 .#.#.#..
 .##.#.##
 #..#.###
 ##.#.##.
Sample Output 
####
#..#
#.##
###.
Sample 3
Sample Input 
 5 4
 #....
 ..###
 ..###
 ..###
Sample Output 
#
Editorial

Editorial

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

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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.

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#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;
}
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