Randomized vs. Deterministic Separation in Time-Space Tradeoffs of Multi-Output Functions

Authors Huacheng Yu , Wei Zhan



PDF
Thumbnail PDF

File

LIPIcs.ITCS.2024.99.pdf
  • Filesize: 0.8 MB
  • 15 pages

Document Identifiers

Author Details

Huacheng Yu
  • Princeton University, NJ, USA
Wei Zhan
  • University of Chicago, IL, USA

Cite As Get BibTex

Huacheng Yu and Wei Zhan. Randomized vs. Deterministic Separation in Time-Space Tradeoffs of Multi-Output Functions. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 99:1-99:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ITCS.2024.99

Abstract

We prove the first polynomial separation between randomized and deterministic time-space tradeoffs of multi-output functions. In particular, we present a total function that on the input of n elements in [n], outputs O(n) elements, such that:  
- There exists a randomized oblivious algorithm with space O(log n), time O(nlog n) and one-way access to randomness, that computes the function with probability 1-O(1/n); 
- Any deterministic oblivious branching program with space S and time T that computes the function must satisfy T²S ≥ Ω(n^{2.5}/log n).  This implies that logspace randomized algorithms for multi-output functions cannot be black-box derandomized without an Ω̃(n^{1/4}) overhead in time.
Since previously all the polynomial time-space tradeoffs of multi-output functions are proved via the Borodin-Cook method, which is a probabilistic method that inherently gives the same lower bound for randomized and deterministic branching programs, our lower bound proof is intrinsically different from previous works.
We also examine other natural candidates for proving such separations, and show that any polynomial separation for these problems would resolve the long-standing open problem of proving n^{1+Ω(1)} time lower bound for decision problems with polylog(n) space.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • Time-space tradeoffs
  • Randomness
  • Borodin-Cook method

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

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 Samir Khuller and Virginia Vassilevska Williams, editors, STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 1330-1342. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451047.
  2. Karl R. Abrahamson. Time-space tradeoffs for algebraic problems on general sequential machines. J. Comput. Syst. Sci., 43(2):269-289, 1991. URL: https://doi.org/10.1016/0022-0000(91)90014-V.
  3. László Babai, Noam Nisan, and Mario Szegedy. Multiparty protocols, pseudorandom generators for logspace, and time-space trade-offs. J. Comput. Syst. Sci., 45(2):204-232, 1992. URL: https://doi.org/10.1016/0022-0000(92)90047-M.
  4. Nikhil Bansal, Shashwat Garg, Jesper Nederlof, and Nikhil Vyas. Faster space-efficient algorithms for subset sum, k-sum, and related problems. SIAM J. Comput., 47(5):1755-1777, 2018. URL: https://doi.org/10.1137/17M1158203.
  5. Paul Beame. A general sequential time-space tradeoff for finding unique elements. SIAM J. Comput., 20(2):270-277, 1991. URL: https://doi.org/10.1137/0220017.
  6. Paul Beame, Raphaël Clifford, and Widad Machmouchi. Element distinctness, frequency moments, and sliding windows. In 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, 26-29 October, 2013, Berkeley, CA, USA, pages 290-299. IEEE Computer Society, 2013. URL: https://doi.org/10.1109/FOCS.2013.39.
  7. Paul Beame and Widad Machmouchi. Making branching programs oblivious requires superlogarithmic overhead. In Proceedings of the 26th Annual IEEE Conference on Computational Complexity, CCC 2011, pages 12-22. IEEE Computer Society, 2011. URL: https://doi.org/10.1109/CCC.2011.35.
  8. Allan Borodin and Stephen A. Cook. A time-space tradeoff for sorting on a general sequential model of computation. SIAM J. Comput., 11(2):287-297, 1982. URL: https://doi.org/10.1137/0211022.
  9. Amit Chakrabarti and Yining Chen. Time-space tradeoffs for the memory game. arXiv preprint, 2017. URL: https://arxiv.org/abs/1712.01330.
  10. Lijie Chen, Ce Jin, R. Ryan Williams, and Hongxun Wu. Truly low-space element distinctness and subset sum via pseudorandom hash functions. In Joseph (Seffi) Naor and Niv Buchbinder, editors, Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA 2022, Virtual Conference / Alexandria, VA, USA, January 9 - 12, 2022, pages 1661-1678. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.67.
  11. Lijie Chen and Roei Tell. Simple and fast derandomization from very hard functions: eliminating randomness at almost no cost. In Samir Khuller and Virginia Vassilevska Williams, editors, STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 283-291. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451059.
  12. Itai Dinur. Tight time-space lower bounds for finding multiple collision pairs and their applications. In Anne Canteaut and Yuval Ishai, editors, Advances in Cryptology - EUROCRYPT 2020 - 39th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Zagreb, Croatia, May 10-14, 2020, Proceedings, Part I, volume 12105 of Lecture Notes in Computer Science, pages 405-434. Springer, 2020. URL: https://doi.org/10.1007/978-3-030-45721-1_15.
  13. Venkatesan Guruswami, Christopher Umans, and Salil P. Vadhan. Unbalanced expanders and randomness extractors from parvaresh-vardy codes. J. ACM, 56(4):20:1-20:34, 2009. URL: https://doi.org/10.1145/1538902.1538904.
  14. Yassine Hamoudi and Frédéric Magniez. Quantum time-space tradeoff for finding multiple collision pairs. In Min-Hsiu Hsieh, editor, 16th Conference on the Theory of Quantum Computation, Communication and Cryptography, TQC 2021, July 5-8, 2021, Virtual Conference, volume 197 of LIPIcs, pages 1:1-1:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.TQC.2021.1.
  15. William M. Hoza. Typically-correct derandomization for small time and space. In Amir Shpilka, editor, 34th Computational Complexity Conference, CCC 2019, July 18-20, 2019, New Brunswick, NJ, USA, volume 137 of LIPIcs, pages 9:1-9:39. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.CCC.2019.9.
  16. Hartmut Klauck. Quantum time-space tradeoffs for sorting. In Lawrence L. Larmore and Michel X. Goemans, editors, Proceedings of the 35th Annual ACM Symposium on Theory of Computing, June 9-11, 2003, San Diego, CA, USA, pages 69-76. ACM, 2003. URL: https://doi.org/10.1145/780542.780553.
  17. Hartmut Klauck, Robert Špalek, and Ronald de Wolf. Quantum and classical strong direct product theorems and optimal time-space tradeoffs. SIAM J. Comput., 36(5):1472-1493, 2007. URL: https://doi.org/10.1137/05063235X.
  18. Xin Lyu and Weihao Zhu. Time-space tradeoffs for element distinctness and set intersection via pseudorandomness. In Nikhil Bansal and Viswanath Nagarajan, editors, Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms, SODA 2023, Florence, Italy, January 22-25, 2023, pages 5243-5281. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.ch190.
  19. Dylan M. McKay and R. Ryan Williams. Quadratic Time-Space Lower Bounds for Computing Natural Functions with a Random Oracle. In Avrim Blum, editor, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019), volume 124 of Leibniz International Proceedings in Informatics (LIPIcs), pages 56:1-56:20, Dagstuhl, Germany, 2018. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik. URL: https://doi.org/10.4230/LIPIcs.ITCS.2019.56.
  20. Ashley Montanaro. Nonadaptive quantum query complexity. Inf. Process. Lett., 110(24):1110-1113, 2010. URL: https://doi.org/10.1016/j.ipl.2010.09.009.
  21. John E. Savage. Models of computation - exploring the power of computing. Addison-Wesley, 1998. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail