E. Infiltration
Problem ID: infiltration Good morning, agent W-12. Your mission, should you choose to accept it, is as follows. We are infiltrating the ever so insidious Association of Chaos and Mischief (ACM) in order to take down their command structure. Unfortunately, t...
Problem Statement
Formatted from the contest statement text, with sample tests broken out into copyable blocks.
Good morning, agent W-12. Your mission, should you choose to accept it, is as follows. We are infiltrating the ever so insidious Association of Chaos and Mischief (ACM) in order to take down their command structure. Unfortunately, they appear to be prepared for such an eventuality, and have given their command structure an annoyingly complex design which makes our infiltration quite difficult. The ACM command structure is divided into several cells. For each pair of cells A and B, either A controls B or B controls A. But this “control” relation can be cyclic, so it could happen that A controls B and B controls C and C controls A. We can send in agents to infiltrate any particular cell, which gives us control over that cell and the cells that it controls, but not any other cells. So in the example above, infiltrating A would give us control over A and B, but not C. For a successful infiltration of the ACM, we must obtain control over all of its cells, otherwise the cells that are out of our control will discover us and start causing some of their trademark chaos and mischief. As you know, we’re on a tight spending leash from higher authority these days, so we need to execute this mission as efficiently as possible. Your mission is to figure out the minimum number of cells we need to infiltrate in order to succeed. This mission briefing will self-destruct in five hours. Good luck!
Input
The first line of a test case contains the number n of cells the ACM has (1 ≤ n ≤ 75). Each of the next n lines contains a binary string of length n where the ith character of the j th line is 1 if cell j controls cell i, and 0 otherwise (1 ≤ i, j ≤ n). The ith character of the ith line is 0 and for i 6= j, either the ith character of the j th line is 1 or the j th character of the ith line is 1, but not both.
Output
For each test case, display its case number followed by the minimum number m of cells that must be infiltrated to obtain complete control of the ACM. Then display m numbers c1 , . . . , cm in any order, indicating the list of cells to infiltrate (cells are numbered from 1 to n). If more than one set of m cells gives complete control, any one will be accepted.
Sample Input Output for Sample Input 2 Case 1: 1 2 00 Case 2: 2 1 2 10 Case 3: 2 2 3 3 010 001 100 5 01000 00011 11001 10100 10010
Editorial
The solution write-up is rendered from the LaTeX source, with equations kept live through MathJax.
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
C++ solution used for this page.
#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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