Finding kings in tournaments

Abstract

A tournament is an orientation of a complete graph. It is well-known that any tournament has a vertex from which every other vertex can be reached by a path of length at most 2. Such a vertex is called a king or a 2-king. It is also known that to find such a vertex, Ωn4/3 queries (to the adjacency matrix) are necessary and On3/2 probes are sufficient. It is a long standing open problem to narrow this gap between the upper and lower bound. We first show that – the adversary Ajtai et al. (2016) and Shen et al. (2003) used to prove the known Ωn4/3 lower bound cannot be used to prove a better lower bound, by giving an algorithm that achieves the bound against the same adversary; Clearly in any tournament there is a vertex from which every other vertex is reachable by a path of length at most d for any d≥2 and such a vertex is called a d-king. The bounds for finding a 2-king have been generalized (Ajtai et al., 2016) to obtain generalized upper and lower bounds to find a d-king. We show that – our algorithm against the weak adversary works against such an adversary for finding d-kings too. More generally, if we can find a 2-king in On4/3 time, then we can find a d-king in asymptotically optimal time for any d≥2. This was conjectured in an earlier paper. Then we address the complexity of finding a set of d-kings, i.e. a small subset of vertices such that every vertex is reachable from one of them by a path of length at most d. Such a set is called a d-cover. – We generalize the lower bound for finding a d-king to give a lower bound for finding k sized d-covers. We complement it with an algorithm matching this bound for k∈Ω(lgn). For d=1 for example, our results imply that we can find a (lgn−lglgn+k)-sized dominating set in On2/k time and that this bound is optimal. Finally we develop a dynamic data structure so that whenever a new vertex is added to the tournament, we can find a king of the new tournament in O(n) time.