Randomized and Quantum Query Complexities of Finding a King in a Tournament

Authors Nikhil S. Mande , Manaswi Paraashar, Nitin Saurabh

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Nikhil S. Mande
  • University of Liverpool, UK
Manaswi Paraashar
  • University of Copenhagen, Denmark
Nitin Saurabh
  • Indian Institute of Technology Hyderabad, India

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Nikhil S. Mande, Manaswi Paraashar, and Nitin Saurabh. Randomized and Quantum Query Complexities of Finding a King in a Tournament. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 30:1-30:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


A tournament is a complete directed graph. It is well known that every tournament contains at least one vertex v such that every other vertex is reachable from v by a path of length at most 2. All such vertices v are called kings of the underlying tournament. Despite active recent research in the area, the best-known upper and lower bounds on the deterministic query complexity (with query access to directions of edges) of finding a king in a tournament on n vertices are from over 20 years ago, and the bounds do not match: the best-known lower bound is Ω(n^{4/3}) and the best-known upper bound is O(n^{3/2}) [Shen, Sheng, Wu, SICOMP'03]. Our contribution is to show tight bounds (up to logarithmic factors) of Θ̃(n) and Θ̃(√n) in the randomized and quantum query models, respectively. We also study the randomized and quantum query complexities of finding a maximum out-degree vertex in a tournament.

Subject Classification

ACM Subject Classification
  • Theory of computation → Oracles and decision trees
  • Theory of computation → Quantum complexity theory
  • Query complexity
  • quantum computing
  • randomized query complexity
  • tournament solutions
  • search problems


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