Suppose that there are n=2^k teams in an elimination tournament, where there are n/2 games in the first round, with the n/2=2^{k-1} winners playing in the second round, and so on. Develop a recurrence relation for the number of rounds in the tournament.

Jadon Melendez 2022-07-18 Answered
Suppose that there are n = 2 k teams in an elimination tournament, where there are n/2 games in the first round, with the n / 2 = 2 k 1 winners playing in the second round, and so on. Develop a recurrence relation for the number of rounds in the tournament.
Would f ( n ) = f ( n / 2 ) + 1 for n = 2 k be a recurrence relation for the question above? would I also have to say f ( 1 ) = 1 ??
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Answers (2)

grocbyntza
Answered 2022-07-19 Author has 25 answers
Step 1
For all recurrence relations, you will have to define the first term.
I'm not so sure about your notation with f(n)... perhaps you mean T n where n represents the nth power of 2 of the number of teams and T n is the number of rounds?
In which case, I think yes.
Step 2
So your recurrence relation would be:
T 1 = 0
T n + 1 = T n + 1
where:
T n is the number of rounds in the tournament
2 n is the number of teams in the tournament.
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Leila Jennings
Answered 2022-07-20 Author has 5 answers
Step 1
This is an old question, but I got a different answer so I thought I'd share it here for anyone in the future:
Each "level" of the tournament bracket has n = 2 k teams. That means, for each level of the bracket, n 2 games are played (every two teams plays one game). So, the number of games played at a tournament is equal to the number of games played at the highest level ( n 2 ), where there are n teams), plus the number of games played at a tournament one level smaller.
Step 2
Hence:
T 1 = 0
T n = T n 1 + n 2
Where T n is the number of games played in a tournament with n = 2 k teams.
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