Searching for Maximum Out-Degree Vertices in Tournaments

01/15/2018

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by   Gregory Gutin, et al.

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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^4/3) and O(n^3/2), 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.