From Donkeys to Kings in Tournaments

A tournament is an orientation of a complete graph. A vertex that can reach every other vertex within two steps is called a \emph{king}. We study the complexity of finding $k$ kings in a tournament graph. We show that the randomized query complexity of finding $k \le 3$ kings is $O(n)$, and for the deterministic case it takes the same amount of queries (up to a constant) as finding a single king (the best known deterministic algorithm makes $O(n^{3/2})$ queries). On the other hand, we show that finding $k \ge 4$ kings requires $\Omega(n^2)$ queries, even in the randomized case. We consider the RAM model for $k \geq 4$. We show an algorithm that finds $k$ kings in time $O(kn^2)$, which is optimal for constant values of $k$. Alternatively, one can also find $k \ge 4$ kings in time $n^{\omega}$ (the time for matrix multiplication). We provide evidence that this is optimal for large $k$ by suggesting a fine-grained reduction from a variant of the triangle detection problem.