Quantum Algorithms for Hopcroft’s Problem

Authors Vladimirs Andrejevs , Aleksandrs Belovs, Jevgēnijs Vihrovs



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Author Details

Vladimirs Andrejevs
  • Centre for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, Latvia
Aleksandrs Belovs
  • Centre for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, Latvia
Jevgēnijs Vihrovs
  • Centre for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, Latvia

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Vladimirs Andrejevs, Aleksandrs Belovs, and Jevgēnijs Vihrovs. Quantum Algorithms for Hopcroft’s Problem. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.MFCS.2024.9

Abstract

In this work we study quantum algorithms for Hopcroft’s problem which is a fundamental problem in computational geometry. Given n points and n lines in the plane, the task is to determine whether there is a point-line incidence. The classical complexity of this problem is well-studied, with the best known algorithm running in O(n^{4/3}) time, with matching lower bounds in some restricted settings. Our results are two different quantum algorithms with time complexity Õ(n^{5/6}). The first algorithm is based on partition trees and the quantum backtracking algorithm. The second algorithm uses a quantum walk together with a history-independent dynamic data structure for storing line arrangement which supports efficient point location queries. In the setting where the number of points and lines differ, the quantum walk-based algorithm is asymptotically faster. The quantum speedups for the aforementioned data structures may be useful for other geometric problems.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • Theory of computation → Computational geometry
  • Theory of computation → Data structures design and analysis
Keywords
  • Quantum algorithms
  • Quantum walks
  • Computational Geometry

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References

  1. Scott Aaronson, Nai-Hui Chia, Han-Hsuan Lin, Chunhao Wang, and Ruizhe Zhang. On the Quantum Complexity of Closest Pair and Related Problems. In Shubhangi Saraf, editor, 35th Computational Complexity Conference (CCC 2020), volume 169 of Leibniz International Proceedings in Informatics (LIPIcs), pages 16:1-16:43, Dagstuhl, Germany, 2020. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.CCC.2020.16.
  2. Amir Abboud, Ryan Williams, and Huacheng Yu. More applications of the polynomial method to algorithm design. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, pages 218-230, USA, 2015. Society for Industrial and Applied Mathematics. URL: https://doi.org/10.5555/2722129.2722146.
  3. Amir Abboud, Virginia Vassilevska Williams, and Oren Weimann. Consequences of Faster Alignment of Sequences. In Javier Esparza, Pierre Fraigniaud, Thore Husfeldt, and Elias Koutsoupias, editors, Automata, Languages, and Programming, pages 39-51, Berlin, Heidelberg, 2014. Springer Berlin Heidelberg. URL: https://doi.org/10.1007/978-3-662-43948-7_4.
  4. Pankaj K. Agarwal. Simplex Range Searching and Its Variants: A Review, pages 1-30. Springer International Publishing, Cham, 2017. URL: https://doi.org/10.1007/978-3-319-44479-6_1.
  5. Jonathan Allcock, Jinge Bao, Aleksandrs Belovs, Troy Lee, and Miklos Santha. On the quantum time complexity of divide and conquer, 2023. URL: https://arxiv.org/abs/2311.16401.
  6. Jonathan Allcock, Jinge Bao, João F. Doriguello, Alessandro Luongo, and Miklos Santha. Constant-depth circuits for Uniformly Controlled Gates and Boolean functions with application to quantum memory circuits, 2023. URL: https://arxiv.org/abs/2308.08539.
  7. Andris Ambainis. Polynomial Degree vs. Quantum Query Complexity. In Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2003, pages 230-239, Washington, DC, USA, 2003. IEEE Computer Society. URL: https://doi.org/10.1109/SFCS.2003.1238197.
  8. Andris Ambainis. Polynomial Degree and Lower Bounds in Quantum Complexity: Collision and Element Distinctness with Small Range. Theory of Computing, 1(3):37-46, 2005. URL: https://doi.org/10.4086/toc.2005.v001a003.
  9. Andris Ambainis. Quantum Walk Algorithm for Element Distinctness. SIAM Journal on Computing, 37(1):210-239, 2007. URL: https://doi.org/10.1137/S0097539705447311.
  10. Andris Ambainis and Martins Kokainis. Quantum algorithm for tree size estimation, with applications to backtracking and 2-player games. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, pages 989-1002, New York, NY, USA, 2017. ACM. URL: https://doi.org/10.1145/3055399.3055444.
  11. Andris Ambainis and Nikita Larka. Quantum Algorithms for Computational Geometry Problems. In Steven T. Flammia, editor, 15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020), volume 158 of Leibniz International Proceedings in Informatics (LIPIcs), pages 9:1-9:10, Dagstuhl, Germany, 2020. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.TQC.2020.9.
  12. A. Bahadur, C. Dürr, T. Lafaye, and R. Kulkarni. Quantum query complexity in computational geometry revisited. In Eric J. Donkor, Andrew R. Pirich, and Howard E. Brandt, editors, Quantum Information and Computation IV, volume 6244, page 624413. International Society for Optics and Photonics, SPIE, 2006. URL: https://doi.org/10.1117/12.661591.
  13. Charles H. Bennett, Ethan Bernstein, Gilles Brassard, and Umesh Vazirani. Strengths and Weaknesses of Quantum Computing. SIAM Journal on Computing, 26(5):1510-1523, 1997. URL: https://doi.org/10.1137/S0097539796300933.
  14. Harry Buhrman, Bruno Loff, Subhasree Patro, and Florian Speelman. Limits of Quantum Speed-Ups for Computational Geometry and Other Problems: Fine-Grained Complexity via Quantum Walks. In Mark Braverman, editor, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022), volume 215 of Leibniz International Proceedings in Informatics (LIPIcs), pages 31:1-31:12, Dagstuhl, Germany, 2022. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ITCS.2022.31.
  15. Harry Buhrman, Bruno Loff, Subhasree Patro, and Florian Speelman. Memory Compression with Quantum Random-Access Gates. In François Le Gall and Tomoyuki Morimae, editors, 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022), volume 232 of Leibniz International Proceedings in Informatics (LIPIcs), pages 10:1-10:19, Dagstuhl, Germany, 2022. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.TQC.2022.10.
  16. Harry Buhrman and Robert Špalek. Quantum verification of matrix products. In Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithm, SODA 2006, pages 880-889, USA, 2006. Society for Industrial and Applied Mathematics. URL: https://arxiv.org/abs/quant-ph/0409035.
  17. Timothy M. Chan. Optimal Partition Trees. Discrete & Computational Geometry, 47:661-690, 2012. URL: https://doi.org/10.1007/s00454-012-9410-z.
  18. Timothy M. Chan and R. Ryan Williams. Deterministic APSP, Orthogonal Vectors, and More: Quickly Derandomizing Razborov-Smolensky. ACM Trans. Algorithms, 17(1), 2021. URL: https://doi.org/10.1145/3402926.
  19. Timothy M. Chan and Da Wei Zheng. Hopcroft’s Problem, Log-Star Shaving, 2D Fractional Cascading, and Decision Trees. ACM Trans. Algorithms, 2023. URL: https://doi.org/10.1145/3591357.
  20. Timothy M. Chan and Da Wei Zheng. Simplex Range Searching Revisited: How to Shave Logs in Multi-Level Data Structures, pages 1493-1511. SODA 2023. Society for Industrial and Applied Mathematics, 2023. URL: https://doi.org/10.1137/1.9781611977554.ch54.
  21. Bernard Chazelle. Cutting hyperplanes for divide-and-conquer. Discrete & Computational Geometry, 9:145-158, 1993. URL: https://doi.org/10.1007/BF02189314.
  22. Bernard Chazelle and Burton Rosenberg. Simplex range reporting on a pointer machine. Computational Geometry, 5(5):237-247, 1996. URL: https://doi.org/10.1016/0925-7721(95)00002-X.
  23. Bernard Chazelle and Emo Welzl. Quasi-optimal range searching in spaces of finite VC-dimension. Discrete & Computational Geometry, 4:467-489, 1989. URL: https://doi.org/10.1007/BF02187743.
  24. Andrew M. Childs, Shelby Kimmel, and Robin Kothari. The Quantum Query Complexity of Read-Many Formulas. In Leah Epstein and Paolo Ferragina, editors, Algorithms - ESA 2012, pages 337-348, Berlin, Heidelberg, 2012. Springer Berlin Heidelberg. URL: https://arxiv.org/abs/1112.0548.
  25. Herbert Edelsbrunner. Algorithms in Combinatorial Geometry. Springer-Verlag, Berlin, Heidelberg, 1987. URL: https://doi.org/10.1007/978-3-642-61568-9.
  26. Jeff Erickson. On the relative complexities of some geometric problems. In Proceedings of the 7th Canadian Conference on Computational Geometry, pages 85-90. Carleton University, 1995. URL: https://jeffe.cs.illinois.edu/pubs/relative.html.
  27. Jeff Erickson. New lower bounds for Hopcroft’s problem. Discrete & Computational Geometry, 16:389-418, 1996. URL: https://doi.org/10.1007/BF02712875.
  28. Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. Quantum Random Access Memory. Phys. Rev. Lett., 100:160501, 2008. URL: https://doi.org/10.1103/PhysRevLett.100.160501.
  29. Lov K. Grover. A Fast Quantum Mechanical Algorithm for Database Search. In Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, STOC 1996, pages 212-219, New York, NY, USA, 1996. Association for Computing Machinery. URL: https://doi.org/10.1145/237814.237866.
  30. Peter Høyer, Michele Mosca, and Ronald de Wolf. Quantum Search on Bounded-Error Inputs. In Automata, Languages and Programming, ICALP 2003, pages 291-299, Berlin, Heidelberg, 2003. Springer-Verlag. URL: https://doi.org/10.1007/3-540-45061-0_25.
  31. J. Mark Keil, Fraser McLeod, and Debajyoti Mondal. Quantum Speedup for Some Geometric 3SUM-Hard Problems and Beyond, 2024. URL: https://arxiv.org/abs/2404.04535.
  32. Samuel Kutin. Quantum Lower Bound for the Collision Problem with Small Range. Theory of Computing, 1(2):29-36, 2005. URL: https://doi.org/10.4086/toc.2005.v001a002.
  33. François Le Gall. Improved Quantum Algorithm for Triangle Finding via Combinatorial Arguments. In 2014 IEEE 55th Annual Symposium on Foundations of Computer Science, pages 216-225, 2014. URL: https://doi.org/10.1109/FOCS.2014.31.
  34. Frédéric Magniez, Ashwin Nayak, Jérémie Roland, and Miklos Santha. Search via Quantum Walk. SIAM Journal on Computing, 40(1):142-164, 2011. URL: https://doi.org/10.1137/090745854.
  35. Jiří Matoušek. Range searching with efficient hierarchical cuttings. Discrete & Computational Geometry, 10:157-182, 1993. URL: https://doi.org/10.1007/BF02573972.
  36. Ashley Montanaro. Quantum-Walk Speedup of Backtracking Algorithms. Theory of Computing, 14(15):1-24, 2018. URL: https://doi.org/10.4086/toc.2018.v014a015.
  37. Ketan Mulmuley and Sandeep Sen. Dynamic point location in arrangements of hyperplanes. Discrete & Computational Geometry, 8:335-360, 1992. URL: https://doi.org/10.1007/BF02293052.
  38. Kunihiko Sadakane, Noriko Sugarawa, and Takeshi Tokuyama. Quantum Computation in Computational Geometry. Interdisciplinary Information Sciences, 8(2):129-136, 2002. URL: https://doi.org/10.4036/iis.2002.129.
  39. Kunihiko Sadakane, Norito Sugawara, and Takeshi Tokuyama. Quantum Algorithms for Intersection and Proximity Problems. In Peter Eades and Tadao Takaoka, editors, Algorithms and Computation, pages 148-159, Berlin, Heidelberg, 2001. Springer Berlin Heidelberg. URL: https://doi.org/10.1007/3-540-45678-3_14.
  40. Neil Sarnak and Robert E. Tarjan. Planar point location using persistent search trees. Commun. ACM, 29(7):669-679, 1986. URL: https://doi.org/10.1145/6138.6151.
  41. Seiichiro Tani. Claw finding algorithms using quantum walk. Theoretical Computer Science, 410(50):5285-5297, 2009. Mathematical Foundations of Computer Science (MFCS 2007). URL: https://doi.org/10.1016/j.tcs.2009.08.030.
  42. Nilton Volpato and Arnaldo Moura. Tight Quantum Bounds for Computational Geometry Problems. International Journal of Quantum Information, 07(05):935-947, 2009. URL: https://doi.org/10.1142/S0219749909005572.
  43. Nilton Volpato and Arnaldo Moura. A fast quantum algorithm for the closest bichromatic pair problem, 2010. URL: https://www.ic.unicamp.br/~reltech/2010/10-03.pdf.
  44. Ryan Williams. A new algorithm for optimal 2-constraint satisfaction and its implications. Theoretical Computer Science, 348(2):357-365, 2005. Automata, Languages and Programming: Algorithms and Complexity (ICALP-A 2004). URL: https://doi.org/10.1016/j.tcs.2005.09.023.
  45. Ryan Williams. Pairwise comparison of bit vectors, 2017. URL: https://cstheory.stackexchange.com/q/37369.
  46. Ryan Williams and Huacheng Yu. Finding orthogonal vectors in discrete structures. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, pages 1867-1877, USA, 2014. Society for Industrial and Applied Mathematics. URL: https://doi.org/10.5555/2634074.2634209.
  47. Shengyu Zhang. Promised and Distributed Quantum Search. In Lusheng Wang, editor, Computing and Combinatorics, pages 430-439, Berlin, Heidelberg, 2005. Springer Berlin Heidelberg. URL: https://doi.org/10.1007/11533719_44.
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