From Donkeys to Kings in Tournaments
Amir Abboud, Tomer Grossman, Moni Naor, Tomer Solomon

TL;DR
This paper investigates the complexity of finding k kings in tournament graphs, revealing efficient algorithms for small k and proving high query complexity for larger k, with implications for algorithmic design and computational limits.
Contribution
It establishes tight bounds on the query complexity for finding k kings, providing new algorithms for small k and complexity lower bounds for larger k.
Findings
Randomized query complexity for k ≤ 3 is O(n).
Deterministic query complexity for k ≤ 3 matches that of finding a single king.
Finding k ≥ 4 kings requires Ω(n^2) queries, even randomized.
Abstract
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 kings in a tournament graph. We show that the randomized query complexity of finding kings is , 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 queries). On the other hand, we show that finding kings requires queries, even in the randomized case. We consider the RAM model for . We show an algorithm that finds kings in time , which is optimal for constant values of . Alternatively, one can also find kings in time (the time for matrix multiplication). We provide evidence that this is optimal for large …
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsSports Analytics and Performance
