Extended Learning Graphs for Triangle Finding

Authors Titouan Carette, Mathieu Laurière, Frédéric Magniez

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Titouan Carette
Mathieu Laurière
Frédéric Magniez

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Titouan Carette, Mathieu Laurière, and Frédéric Magniez. Extended Learning Graphs for Triangle Finding. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 20:1-20:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


We present new quantum algorithms for Triangle Finding improving its best previously known quantum query complexities for both dense and sparse instances. For dense graphs on n vertices, we get a query complexity of O(n^(5/4)) without any of the extra logarithmic factors present in the previous algorithm of Le Gall [FOCS'14]. For sparse graphs with m >= n^(5/4) edges, we get a query complexity of O(n^(11/12) m^(1/6) sqrt(log n)), which is better than the one obtained by Le Gall and Nakajima [ISAAC'15] when m >= n^(3/2). We also obtain an algorithm with query complexity O(n^(5/6) (m log n)^(1/6) + d_2 sqrt(n)) where d_2 is the variance of the degree distribution. Our algorithms are designed and analyzed in a new model of learning graphs that we call extended learning graphs. In addition, we present a framework in order to easily combine and analyze them. As a consequence we get much simpler algorithms and analyses than previous algorithms of Le Gall based on the MNRS quantum walk framework [SICOMP'11].
  • Quantum query complexity
  • learning graphs
  • triangle finding


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  1. S. Aaronson, S. Ben-David, and R. Kothari. Separations in query complexity using cheat sheets. In Proceedings of 48th ACM Symposium on Theory of Computing, pages 863-876, 2016. URL: http://dx.doi.org/10.1145/2897518.2897644.
  2. A. Ambainis. Quantum search with variable times. Theory of Computing Systems, 47(3):786-807, 2010. URL: http://dx.doi.org/10.1007/s00224-009-9219-1.
  3. A. Ambainis, K. Balodis, A. Belovs, T. Lee, M. Santha, and J. Smotrovs. Separations in query complexity based on pointer functions. In Proceedings of 48th ACM Symposium on Theory of Computing, pages 800-813, 2016. URL: http://dx.doi.org/10.1145/2897518.2897524.
  4. R. Beals, H. Buhrman, R. Cleve, M. Mosca, and R. de Wolf. Quantum lower bounds by polynomials. Journal of the ACM, 48(4):778-797, 2001. Google Scholar
  5. A. Belovs. Learning-graph-based quantum algorithm for k-distinctness. In Prooceedings of 53rd IEEE Symposium on Foundations of Computer Science, pages 207-216, 2012. Google Scholar
  6. A. Belovs. Span programs for functions with constant-sized 1-certificates. In Proceedings of 44th Symposium on Theory of Computing Conference, pages 77-84, 2012. Google Scholar
  7. A. Belovs, A. Childs, S. Jeffery, R. Kothari, and F. Magniez. Time-efficient quantum walks for 3-distinctness. In Proceedings of 40th International Colloquium on Automata, Languages and Programming, pages 105-122, 2013. Google Scholar
  8. A. Belovs and T. Lee. Quantum algorithm for k-distinctness with prior knowledge on the input. Technical Report arXiv:1108.3022, arXiv, 2011. Google Scholar
  9. A. Belovs and A. Rosmanis. On the power of non-adaptive learning graphs. In Proceedings of 28th IEEE Conference on Computational Complexity, pages 44-55, 2013. Google Scholar
  10. H. Buhrman, C. Dürr, M. Heiligman, P. Høyer, F. Magniez, M. Santha, and R. de Wolf. Quantum algorithms for element distinctness. SIAM Journal on Computing, 34(6):1324-1330, 2005. Google Scholar
  11. F. Le Gall. Improved quantum algorithm for triangle finding via combinatorial arguments. In Proceedings of 55th IEEE Foundations of Computer Science, pages 216-225, 2014. URL: http://dx.doi.org/10.1109/FOCS.2014.31.
  12. F. Le Gall and N. Shogo. Quantum algorithm for triangle finding in sparse graphs. In Proc. of 26th International Symposium Algorithms and Computation, pages 590-600, 2015. URL: http://dx.doi.org/10.1007/978-3-662-48971-0_50.
  13. L. Grover. A fast quantum mechanical algorithm for database search. In Proceedings of 28th ACM Symposium on the Theory of Computing, pages 212-219, 1996. Google Scholar
  14. P. Høyer, T. Lee, and R. Špalek. Negative weights make adversaries stronger. In Proceedings of 39th ACM Symposium on Theory of Computing, pages 526-535, 2007. Google Scholar
  15. P. Høyer and R. Špalek. Lower bounds on quantum query complexity. Bulletin of the European Association for Theoretical Computer Science, 87, 2005. Google Scholar
  16. T. Lee, F. Magniez, and M. Santha. Improved quantum query algorithms for triangle finding and associativity testing. Algorithmica, 2015. To appear. Google Scholar
  17. T. Lee, R. Mittal, B. Reichardt, R. Špalek, and M. Szegedy. Quantum query complexity of state conversion. In Proceedings of 52nd IEEE Symposium on Foundations of Computer Science, pages 344-353, 2011. Google Scholar
  18. F. Magniez, A. Nayak, J. Roland, and M. Santha. Search via quantum walk. SIAM Journal on Computing, 40(1):142-164, 2011. Google Scholar
  19. F. Magniez, M. Santha, and M. Szegedy. Quantum algorithms for the triangle problem. SIAM Journal on Computing, 37(2):413-424, 2007. Google Scholar
  20. N. Nisan. Crew prams and decision trees. SIAM Journal on Computing, 20(6):999-1007, 1991. URL: http://dx.doi.org/10.1137/0220062.
  21. B. Reichardt. Reflections for quantum query algorithms. In Proceedings of 22nd ACM-SIAM Symposium on Discrete Algorithms, pages 560-569, 2011. Google Scholar
  22. P. Shor. Algorithms for quantum computation: Discrete logarithm and factoring. SIAM Journal on Computing, 26(5):1484-1509, 1997. Google Scholar