Algorithm Voting: Determining the Outcome

This problem analyzes a voting scenario where 'n' programmers choose their favorite algorithm from 'm' options. The voting process eliminates algorithms with fewer votes until only one remains. We need to determine whether the process will always result in a single winner within a finite number of rounds or if it can continue indefinitely.

Understanding the Voting Dynamics

For each test case, the key observation is that if the number of programmers (n) is less than or equal to the number of algorithm options (m), a single winner will always be selected in a finite number of rounds. This is because, in the worst-case scenario, each programmer could initially vote for a different option. However, as the voting progresses, the options with fewer votes will be eliminated, ultimately leading to a single winner.

Analyzing the Case When n > m

When the number of programmers exceeds the number of algorithm choices, the outcome becomes more complex. Let's consider a scenario where initially, each algorithm has an equal number of votes. After the first round, some algorithms will be eliminated, and some will remain with the same number of votes. Let's say 'k' algorithms remain with the same number of votes. In the second round, at least one of these 'k' algorithms will be eliminated, leading to a reduction in the number of algorithms with the same number of votes. This process continues, and eventually, after a certain number of rounds, only one algorithm will remain. The number of rounds required in this case can be calculated as log2(n) - log2(m). This is a finite value.

Conclusion

Therefore, the voting process will always result in a single winning algorithm in a finite number of rounds if the number of programmers is less than or equal to the number of algorithm options (n ≤ m). When the number of programmers exceeds the number of algorithm choices (n > m), the voting process will also end in a finite number of rounds. This is because the number of algorithms with the same number of votes decreases with each round, ultimately leading to a single winner.