From Donkeys to Kings in Tournaments

Authors Amir Abboud, Tomer Grossman, Moni Naor , Tomer Solomon



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Author Details

Amir Abboud
  • Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel
Tomer Grossman
  • Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel
Moni Naor
  • Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel
Tomer Solomon
  • Tel Aviv University, Israel

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Amir Abboud, Tomer Grossman, Moni Naor, and Tomer Solomon. From Donkeys to Kings in Tournaments. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 3:1-3:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ESA.2024.3

Abstract

A tournament is an orientation of a complete graph. A vertex that can reach every other vertex within two steps is called a king. We study the complexity of finding k kings in a tournament graph. We show that the randomized query complexity of finding k ≤ 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 ≥ 4 kings requires Ω(n²) queries, even in the randomized case. We consider the RAM model for k ≥ 4. We show an algorithm that finds k kings in time O(kn²), which is optimal for constant values of k. Alternatively, one can also find k ≥ 4 kings in time n^{ω} (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.

Subject Classification

ACM Subject Classification
  • Theory of computation
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Tournament Graphs
  • Kings
  • Query Complexity
  • Fine Grained Complexity

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