I am currently trying to find algorithm to minimize the total time of a tournament.
The game and tournament requirements are as below:
- The game requires $2$ teams of $2$ players in each team (total $4$ players)
- Each game takes about $8$ to $10$ minutes to finish
I found out that the perfect number of tournament players (to have $4$ player in the final game), must be $2^n$
However, it is very hard to always have registered players matching this perfect number. So, I can substitute computer players to complete the perfect number.
For example, if the number of registered players is $55$, the nearest perfect number is $64$, so I will substitute $9$ computer players.
What I reached so far is, the total runs (the number of games completed to reach to the final game in the tournament) can be calculated by:
$[$$log_2$ $n$ $]$ $-$ $1$ , where $n$ is the number of players to the nearest perfect number.
and the tournament finish time (minimum) can be calculated by:
$8$ $*$ $($ $[$$log_2$ $n$ $]$ $-$ $1$ $)$
The above does not seem to be the best solution, having $513$ players will take the same time ($72$ minutes) as of $1024$ players. Note: I cannot exclude any registered player from the tournament.
Is there a better approach to minimize the total time of a tournament?
Each game can be calculated by the below formula:
$[[B + (T * P * U)] * R] + E$ , where:
$T$ is the time allowed for each player to respond and play, otherwise computer player will play
$P$ is the number of players, which is set to $4$ by default
$U$ is the number of turns (how many times each player will play in one round), which is set to $8$ by default
$R$ is the maximum number of rounds that the game will have (here is the key that will determine the total time of the game will take), this number cannot be less than $4$, it is impossible to finish a game in 4 rounds.
$B$ is the time to catch a bluff in the game, which is set to $10$
$E$ is extra time which happens only once, and it is set to $5$ by default
What I suggested for worst case is:
$[[10+(3*4*8)]*9]+5$ $=959$ seconds $=16$ minutes