竞赛图中寻找国王的随机化与量子查询复杂度
Randomized and quantum query complexities of finding a king in a tournament
AI总结：
本文研究了竞赛图中寻找国王的查询复杂度，在随机化和量子查询模型中分别给出了紧确界 Theta(n) 和 Theta(sqrt{n})，并探讨了寻找最大出度顶点的复杂度。
摘要：
竞赛图是一个完全有向图。众所周知，每个竞赛图都至少包含一个顶点 v，使得所有其他顶点都可以通过长度至多为 2 的路径从 v 到达。所有这样的顶点 v 都被称为底层竞赛图的 *国王*。尽管该领域近期研究活跃，但在具有 n 个顶点的竞赛图中寻找国王的确定性查询复杂度（具有对边方向的查询访问权限）的最著名上下界均来自 20 多年前，且这两个界并不匹配：最著名的下界是 Omega(n^{4/3})，最著名的上界是 O(n^{3/2}) [Shen, Sheng, Wu, SICOMP'03]。我们的贡献是分别在 *随机化* 和 *量子* 查询模型中展示了本质上 *紧* 的界（至多相差对数因子），即 Theta(n) 和 Theta(sqrt{n})。我们还研究了在竞赛图中寻找最大出度顶点的随机化和量子查询复杂度。
英文摘要：
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(sqrt{n}) 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
