Searching for Maximum Out-Degree Vertices in Tournaments

A vertex x in a tournament T is called a king if for every vertex y of T there is a directed path from x to y of length at most 2. It is not hard to show that every vertex of maximum out-degree in a tournament is a king. However, tournaments may have kings which are not vertices of maximum out-degree. A binary inquiry asks for the orientation of the edge between a pair of vertices and receives the answer. The cost of finding a king in an unknown tournament is the number of binary inquiries required to detect a king. For the cost of finding a king in a tournament, in the worst case, Shen, Sheng and Wu (SIAM J. Comput., 2003) proved a lower and upper bounds of Ω(n) and O(n), respectively. In contrast to their result, we prove that the cost of finding a vertex of maximum out-degree is ( n 2 ) −O(n) in the worst case.