On partial sorting in restricted rounds

Let n and k be integers such that n ≥ 2 and 1 ≤ k ≤n. In this paper, we consider the problem of finding an ordered list of the k best players out of n participants by organizing a tournament of rounds of pairwise matches (comparisons). Assuming that (i) in each match there is a winner (no ties) (ii) the relative strength of the players is constant throughout the tournament and (iii) the players’ strengths are transitive, the problem is equivalent to partially sorting n different, comparable objects, allowing parallelization in rounds. The rounds are restricted as one player can only play one match in each round. We propose concrete pairing algorithms and make conjectures about their performance in terms of the worst case number of rounds and matches required. The research article was started by professor Antal Iványi who sadly passed away during the work and was completed in his honor by the co-author. He hopes, in this modest way, to reflect his deep admiration for professor Iványi’s many contributions to the theory, practice and appreciation of algorithm design and analysis.