An efficient sorting algorithm for a sequence of kings in a tournament

A king u in a tournament is a player who beats any other player v directly or indirectly. That is, either u → v (u beats v) or there exists a third player w such that u→ w and w → v. A sorted sequence of kings in a tournament of n players is a sequence of players, S = (u1, u2, . . . , un), such that ui → ui+1 and ui in a king in sub-tournament Tui = {ui ,ui+1, . . . , un} for i = 1,2, . . . , n−1. The existence of a sorted sequence of kings in any tournament is shown by Lou et al. [Proc. 31st Southeastern Internat. Conf. on Combinatorics, Graph Theory, and Computing, 2000] where a sorting algorithm with a complexity of (n3) is given. In this paper, we present another constructive proof for the existence of a sorted sequence of kings of a tournament and propose an efficient algorithm with a complexity of (n2).  2001 Elsevier Science B.V. All rights reserved.