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

Authors Nikhil S. Mande , Manaswi Paraashar, Nitin Saurabh

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Subject Classification

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

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Abstract

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.

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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) https://doi.org/10.4230/LIPIcs.FSTTCS.2023.30

Author Details

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

References

  1. Scott Aaronson, Shalev Ben-David, Robin Kothari, Shravas Rao, and Avishay Tal. Degree vs. approximate degree and quantum implications of huang’s sensitivity theorem. In STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 1330-1342. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451047.
  2. Scott Aaronson and Patrick Rall. Quantum approximate counting, simplified. In Symposium on simplicity in algorithms, pages 24-32, 2020. Google Scholar
  3. Miklós Ajtai, Vitaly Feldman, Avinatan Hassidim, and Jelani Nelson. Sorting and selection with imprecise comparisons. ACM Transactions on Algorithms (TALG), 12(2):1-19, 2015. Google Scholar
  4. Ramachandran Balasubramanian, Venkatesh Raman, and G Srinivasaragavan. Finding scores in tournaments. Journal of Algorithms, 24(2):380-394, 1997. Google Scholar
  5. Charles H. Bennett, Ethan Bernstein, Gilles Brassard, and Umesh V. Vazirani. Strengths and weaknesses of quantum computing. SIAM J. Comput., 26(5):1510-1523, 1997. URL: https://doi.org/10.1137/S0097539796300933.
  6. Arindam Biswas, Varunkumar Jayapaul, Venkatesh Raman, and Srinivasa Rao Satti. Finding kings in tournaments. Discret. Appl. Math., 322:240-252, 2022. URL: https://doi.org/10.1016/j.dam.2022.08.014.
  7. Michel Boyer, Gilles Brassard, Peter Høyer, and Alain Tapp. Tight bounds on quantum searching. Fortschritte der Physik: Progress of Physics, 46(4-5):493-505, 1998. Google Scholar
  8. Gilles Brassard, Peter Hoyer, Michele Mosca, and Alain Tapp. Quantum amplitude amplification and estimation. Contemporary Mathematics, 305:53-74, 2002. Google Scholar
  9. Amit Chakrabarti and Subhash Khot. Improved lower bounds on the randomized complexity of graph properties. In Automata, Languages and Programming: 28th International Colloquium, ICALP 2001 Crete, Greece, July 8-12, 2001 Proceedings 28, pages 285-296. Springer, 2001. Google Scholar
  10. Andrew M Childs and Robin Kothari. Quantum query complexity of minor-closed graph properties. SIAM Journal on Computing, 41(6):1426-1450, 2012. Google Scholar
  11. Palash Dey. Query complexity of tournament solutions. In Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence, pages 2992-2998. AAAI Press, 2017. URL: http://aaai.org/ocs/index.php/AAAI/AAAI17/paper/view/14180.
  12. Christoph Dürr, Mark Heiligman, Peter Høyer, and Mehdi Mhalla. Quantum query complexity of some graph problems. SIAM Journal on Computing, 35(6):1310-1328, 2006. Google Scholar
  13. Christoph Dürr and Peter Høyer. A quantum algorithm for finding the minimum. arXiv preprint quant-ph/9607014, 1996. Google Scholar
  14. Lov K. Grover. A fast quantum mechanical algorithm for database search. In Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing (STOC), pages 212-219. ACM, 1996. URL: https://doi.org/10.1145/237814.237866.
  15. Péter Hajnal. An Ω(n^4/3) lower bound on the randomized complexity of graph properties. Combinatorica, 11:131-143, 1991. Google Scholar
  16. Hao Huang. Induced subgraphs of hypercubes and a proof of the sensitivity conjecture. Annals of Mathematics, 190(3):949-955, 2019. Google Scholar
  17. Oded Lachish, Felix Reidl, and Chhaya Trehan. When you come at the king you best not miss. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2022, volume 250 of LIPIcs, pages 25:1-25:12. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2022.25.
  18. HG Landau. On dominance relations and the structure of animal societies: Iii the condition for a score structure. The bulletin of mathematical biophysics, 15:143-148, 1953. Google Scholar
  19. Stephen B Maurer. The king chicken theorems. Mathematics Magazine, 53(2):67-80, 1980. Google Scholar
  20. John W Moon. Topics on tournaments in graph theory. Courier Dover Publications, 2015. Google Scholar
  21. Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information (10th Anniversary edition). Cambridge University Press, 2016. URL: https://www.cambridge.org/de/academic/subjects/physics/quantum-physics-quantum-information-and-quantum-computation/quantum-computation-and-quantum-information-10th-anniversary-edition?format=HB.
  22. K.B. Reid. Every vertex a king. Discrete Mathematics, 38(1):93-98, 1982. URL: https://doi.org/10.1016/0012-365X(82)90173-X.
  23. Ronald L Rivest and Jean Vuillemin. On recognizing graph properties from adjacency matrices. Theoretical Computer Science, 3(3):371-384, 1976. Google Scholar
  24. Arnold L Rosenberg. On the time required to recognize properties of graphs: A problem. ACM SIGACT News, 5(4):15-16, 1973. Google Scholar
  25. Jian Shen, Li Sheng, and Jie Wu. Searching for sorted sequences of kings in tournaments. SIAM J. Comput., 32(5):1201-1209, 2003. URL: https://doi.org/10.1137/S0097539702410053.
  26. Ronald de Wolf. Quantum computing: Lecture notes, 2019. arXiv:1907.09415, version 5. URL: https://arxiv.org/abs/1907.09415.
  27. Andrew Chi-Chih Yao. Probabilistic computations: Toward a unified measure of complexity. In 18th Annual Symposium on Foundations of Computer Science (SFCS 1977), pages 222-227. IEEE Computer Society, 1977. Google Scholar
  28. Andrew Chi-Chih Yao. Lower bounds to randomized algorithms for graph properties. In 28th Annual Symposium on Foundations of Computer Science (sfcs 1987), pages 393-400. IEEE, 1987. Google Scholar
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