Variations on the Tournament Problem

In 1883, Lewis Carrol wrote a newspaper article to criticize how the second best player was determined in a tennis tournament, and to suggest how such a task could be done correctly. This article has been taken by Donald Knuth as the inspiration for efficiently determining the smallest t elements of a totally ordered set of size n using k-comparisons. In the ensuing research, optimal algorithms for some low values of k and t have been established, by Knuth and Aigner; for k = 2 and t ≤ 3, a few new bounds have been established for special values of n. Surprisingly, very little else is known on this problem, in spite of its illustrious pedigree and its relationship to other classical problems (e.g., selection and sorting with k-sorters). Enticed by the undeniable beauty of the problem, and the obvious promise of fun, we have joined the investigative quest. The purpose of this paper is to share some new results obtained so far. We are glad to report advances in two directions.