Searching for Sorted Sequences of Kings in Tournaments

A tournament Tn is an orientation of a complete graph on n vertices. A king in a tournament is a vertex from which every other vertex is reachable by a path of length at most 2. A sorted sequence of kings in a tournament Tn is an ordered list of its vertices u1, u2, . . . , un such that ui dominates ui+1 (ui → ui+1) and ui is a king in the subtournament induced by {uj : i ≤ j ≤ n} for each i = 1, 2, . . . , n−1. In particular, if Tn is transitive, searching for a sorted sequence of kings in Tn is equivalent to sorting a set of n numbers. In this paper, we try to find a sorted sequence of kings in a general tournament by asking the following type of binary question: “What is the orientation of the edge between two specified vertices u, v?” The cost for finding a sorted sequence of kings is the minimum number of binary questions asked in order to guarantee the finding of a sorted sequence of kings. Using an adversary argument proposed in this paper, we show that the cost for finding a sorted sequence of kings in Tn is Θ(n3/2) in the worst case, thus settling the order of magnitude of this question. We also show that the cost for finding a king in Tn is Ω(n4/3) and O(n3/2) in the worst case. Finally, we show a connection between a sorted sequence of kings and a median order in a tournament.