Randomized and quantum query complexities of finding a king in a tournament

TL;DR

This paper establishes nearly tight bounds on the randomized and quantum query complexities for finding a king in a tournament, improving over decades-old bounds and advancing understanding of quantum and randomized algorithms in graph problems.

Contribution

It provides the first tight bounds (up to logarithmic factors) for randomized and quantum query complexities of finding a king in a tournament.

Findings

Randomized complexity is Theta(n).

Quantum complexity is Theta(√n).

Improves bounds from over 20 years ago.

Abstract

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.