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From Donkeys to Kings in Tournaments

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

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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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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.

Cite As Get BibTex

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

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

Funding

  • Abboud, Amir: This work is part of the project CONJEXITY, which has received funding from the European Research Council (ERC) under the European Union’s Horizon Europe research and innovation program (grant agreement No. 101078482). Supported by an Alon scholarship and a research grant from the Center for New Scientists at the Weizmann Institute of Science.
  • Naor, Moni: Incumbent of the Judith Kleeman Professorial Chair. Supported in part by grants from the Israel Science Foundation (no.2686/20), by the Simons Foundation Collaboration on the Theory of Algorithmic Fairness and by the Israeli Council for Higher Education (CHE) via the Weizmann Data Science Research Center.
  • Solomon, Tomer: This material is based upon work supported by DARPA under Agreement No. HR00110C0086. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the United States Government or DARPA.

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