Randomized and quantum query complexities of finding a king in a tournament
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A tournament is a complete directed graph. It is well known that every tournament contains at least one vertex v such that every other vertex is reachable from v by a path of length at most 2. All such vertices v are called *kings* of the underlying tournament. Despite active recent research in the area, the best-known upper and lower bounds on the deterministic query complexity (with query access to directions of edges) of finding a king in a tournament on n vertices are from over 20 years ago, and the bounds do not match: the best-known lower bound is Omega(n^4/3) and the best-known upper bound is O(n^3/2) [Shen, Sheng, Wu, SICOMP'03]. Our contribution is to show essentially *tight* bounds (up to logarithmic factors) of Theta(n) and Theta(sqrtn) in the *randomized* and *quantum* query models, respectively. We also study the randomized and quantum query complexities of finding a maximum out-degree vertex in a tournament.
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