Document Open Access Logo

New Sampling Lower Bounds via the Separator

Author Emanuele Viola

PDF
Thumbnail PDF

File

LIPIcs.CCC.2023.26.pdf
  • Filesize: 0.76 MB
  • 23 pages

Document Identifiers

Author Details

Emanuele Viola
  • Khoury College of Computer Sciences, Northeastern University, Boston, MA, USA

Funding

  • Viola, Emanuele: Supported by NSF CCF award 2114116.

Acknowledgements

We thank the anonymous reviewers for detailed and helpful feedback.

Cite AsGet BibTex

Emanuele Viola. New Sampling Lower Bounds via the Separator. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 26:1-26:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.CCC.2023.26

Abstract

Suppose that a target distribution can be approximately sampled by a low-depth decision tree, or more generally by an efficient cell-probe algorithm. It is shown to be possible to restrict the input to the sampler so that its output distribution is still not too far from the target distribution, and at the same time many output coordinates are almost pairwise independent. This new tool is then used to obtain several new sampling lower bounds and separations, including a separation between AC0 and low-depth decision trees, and a hierarchy theorem for sampling. It is also used to obtain a new proof of the Patrascu-Viola data-structure lower bound for Rank, thereby unifying sampling and data-structure lower bounds.

Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
Keywords
  • Sampling
  • data structures
  • lower bounds
  • cell probe
  • decision forest
  • AC0
  • rank
  • predecessor

Metrics

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

Related Versions

References

  1. Miklós Ajtai. Σ_1^1-formulae on finite structures. Annals of Pure and Applied Logic, 24(1):1-48, 1983. Google Scholar
  2. Andris Ambainis, Leonard J. Schulman, Amnon Ta-Shma, Umesh V. Vazirani, and Avi Wigderson. The quantum communication complexity of sampling. SIAM J. on Computing, 32(6):1570-1585, 2003. Google Scholar
  3. Sanjeev Arora and Boaz Barak. Computational Complexity. Cambridge University Press, 2009. A modern approach. Google Scholar
  4. Paul Beame, Russell Impagliazzo, and Srikanth Srinivasan. Approximating AC^0 by small height decision trees and a deterministic algorithm for #AC^0SAT. In Proceedings of the 27th Conference on Computational Complexity, CCC 2012, Porto, Portugal, June 26-29, 2012, pages 117-125. IEEE Computer Society, 2012. URL: https://doi.org/10.1109/CCC.2012.40.
  5. Chris Beck, Russell Impagliazzo, and Shachar Lovett. Large deviation bounds for decision trees and sampling lower bounds for AC0-circuits. Electronic Colloquium on Computational Complexity (ECCC), 19:42, 2012. Google Scholar
  6. Chris Beck, Russell Impagliazzo, and Shachar Lovett. Large deviation bounds for decision trees and sampling lower bounds for AC0-circuits. In IEEE Symp. on Foundations of Computer Science (FOCS), pages 101-110, 2012. Google Scholar
  7. Avraham Ben-Aroya, Dean Doron, and Amnon Ta-Shma. Explicit two-source extractors for near-logarithmic min-entropy. Electronic Colloquium on Computational Complexity (ECCC), 23:88, 2016. URL: http://eccc.hpi-web.de/report/2016/088, URL: https://arxiv.org/abs/TR16-088.
  8. Itai Benjamini, Gil Cohen, and Igor Shinkar. Bi-lipschitz bijection between the boolean cube and the hamming ball. In IEEE Symp. on Foundations of Computer Science (FOCS), 2014. Google Scholar
  9. Eshan Chattopadhyay, Jesse Goodman, and David Zuckerman. The space complexity of sampling. Electron. Colloquium Comput. Complex., page 106, 2021. URL: https://eccc.weizmann.ac.il/report/2021/106, URL: https://arxiv.org/abs/TR21-106.
  10. Eshan Chattopadhyay and David Zuckerman. Explicit two-source extractors and resilient functions. In ACM Symp. on the Theory of Computing (STOC), pages 670-683, 2016. Google Scholar
  11. Gil Cohen. Making the most of advice: New correlation breakers and their applications. In IEEE Symp. on Foundations of Computer Science (FOCS), pages 188-196, 2016. URL: https://doi.org/10.1109/FOCS.2016.28.
  12. Gil Cohen and Leonard J. Schulman. Extractors for near logarithmic min-entropy. Electronic Colloquium on Computational Complexity (ECCC), 23:14, 2016. Google Scholar
  13. Thomas Cover and Joy Thomas. Elements of Information Theory (Wiley Series in Telecommunications and Signal Processing). Wiley-Interscience, 2006. Google Scholar
  14. Imre Csiszar and Janos Korner. Information Theory: Coding Theorems for Discrete Memoryless Systems. Academic Press, Inc., 1982. Google Scholar
  15. Anindya De and Thomas Watson. Extractors and lower bounds for locally samplable sources. In Workshop on Randomization and Computation (RANDOM), 2011. Google Scholar
  16. P. Erdős and R. Rado. Intersection theorems for systems of sets. J. London Math. Soc., 35:85-90, 1960. URL: https://doi.org/10.1112/jlms/s1-35.1.85.
  17. Merrick L. Furst, James B. Saxe, and Michael Sipser. Parity, circuits, and the polynomial-time hierarchy. Mathematical Systems Theory, 17(1):13-27, 1984. Google Scholar
  18. Anna Gál and Peter Bro Miltersen. The cell probe complexity of succinct data structures. Theoretical Computer Science, 379(3):405-417, 2007. Google Scholar
  19. Alexander Golynski. Cell probe lower bounds for succinct data structures. In 20th ACM-SIAM Symp. on Discrete Algorithms (SODA), pages 625-634, 2009. Google Scholar
  20. Mika Göös and Thomas Watson. A lower bound for sampling disjoint sets. ACM Trans. Comput. Theory, 12(3):20:1-20:13, 2020. URL: https://doi.org/10.1145/3404858.
  21. Aryeh Grinberg, Ronen Shaltiel, and Emanuele Viola. Indistinguishability by adaptive procedures with advice, and lower bounds on hardness amplification proofs. In IEEE Symp. on Foundations of Computer Science (FOCS), 2018. Available at URL: https://www.ccs.neu.edu/home/viola/papers/adaptivemajority.pdf.
  22. Johan Håstad. Computational limitations of small-depth circuits. MIT Press, 1987. Google Scholar
  23. Johan Håstad. On the correlation of parity and small-depth circuits. SIAM J. on Computing, 43(5):1699-1708, 2014. Google Scholar
  24. Russell Impagliazzo, William Matthews, and Ramamohan Paturi. A satisfiability algorithm for AC^0. In ACM-SIAM Symp. on Discrete Algorithms (SODA), pages 961-972, 2012. Google Scholar
  25. Stasys Jukna. Boolean Function Complexity: Advances and Frontiers. Springer, 2012. Google Scholar
  26. Xin Li. Improved two-source extractors, and affine extractors for polylogarithmic entropy. In IEEE Symp. on Foundations of Computer Science (FOCS), 2016. Google Scholar
  27. Mingmou Liu and Huacheng Yu. Lower bound for succinct range minimum query. In Konstantin Makarychev, Yury Makarychev, Madhur Tulsiani, Gautam Kamath, and Julia Chuzhoy, editors, ACM Symp. on the Theory of Computing (STOC), pages 1402-1415. ACM, 2020. URL: https://doi.org/10.1145/3357713.3384260.
  28. Shachar Lovett and Emanuele Viola. Bounded-depth circuits cannot sample good codes. Computational Complexity, 21(2):245-266, 2012. Google Scholar
  29. Peter Bro Miltersen. Cell probe complexity - A survey, 1999. Invited talk/paper at Advances in Data Structures (Pre-conference workshop of FSTTCS'99). Google Scholar
  30. Ryan O'Donnell. Analysis of Boolean Functions. Cambridge University Press, 2014. Google Scholar
  31. Mihai Pǎtraşcu. Succincter. In 49th IEEE Symp. on Foundations of Computer Science (FOCS). IEEE, 2008. Google Scholar
  32. Mihai Patrascu and Mikkel Thorup. Time-space trade-offs for predecessor search. In Jon M. Kleinberg, editor, ACM Symp. on the Theory of Computing (STOC), pages 232-240. ACM, 2006. URL: https://doi.org/10.1145/1132516.1132551.
  33. Mihai Pǎtraşcu and Emanuele Viola. Cell-probe lower bounds for succinct partial sums. In 21th ACM-SIAM Symp. on Discrete Algorithms (SODA), pages 117-122, 2010. Google Scholar
  34. Alexander A. Razborov. Pseudorandom generators hard for k-DNF resolution and polynomial calculus resolution. Ann. of Math., 181(2):415-472, 2015. URL: https://doi.org/10.4007/annals.2015.181.2.1.
  35. Nathan Segerlind, Sam Buss, and Russell Impagliazzo. A switching lemma for small restrictions and lower bounds for k-DNF resolution. SIAM J. on Computing, 33(5):1171-1200, 2004. Google Scholar
  36. Mikkel Thorup. Mihai patrascu: Obituary and open problems. Bulletin of the EATCS, 109:7-13, 2013. URL: http://albcom.lsi.upc.edu/ojs/index.php/beatcs/article/view/163/176.
  37. Emanuele Viola. The Complexity of Distributions, Fall 2018 talk at the Simons Institute. URL: https://www.youtube.com/watch?v=O78b085HE3w.
  38. Emanuele Viola. Bit-probe lower bounds for succinct data structures. SIAM J. on Computing, 41(6):1593-1604, 2012. Google Scholar
  39. Emanuele Viola. The complexity of distributions. SIAM J. on Computing, 41(1):191-218, 2012. Google Scholar
  40. Emanuele Viola. Extractors for turing-machine sources. In Workshop on Randomization and Computation (RANDOM), 2012. Google Scholar
  41. Emanuele Viola. Extractors for circuit sources. SIAM J. on Computing, 43(2):355-972, 2014. Google Scholar
  42. Emanuele Viola. Lower bounds for data structures with space close to maximum imply circuit lower bounds. Theory of Computing, 15:1-9, 2019. URL: https://theoryofcomputing.org/articles/v015a018/v015a018.pdf.
  43. Emanuele Viola. Sampling lower bounds: boolean average-case and permutations. SIAM J. on Computing, 49(1), 2020. Available at URL: https://www.ccs.neu.edu/home/viola/papers/sampling-lower-bounds.pdf.
  44. Emanuele Viola, 2022. URL: https://emanueleviola.wordpress.com/2022/09/14/myth-creation-the-switching-lemma/.
  45. Andrew Yao. Separating the polynomial-time hierarchy by oracles. In 26th IEEE Symp. on Foundations of Computer Science (FOCS), pages 1-10, 1985. Google Scholar
  46. Andrew Chi-Chih Yao. Should tables be sorted? J. of the ACM, 28(3):615-628, 1981. URL: https://doi.org/10.1145/322261.322274.
  47. Huacheng Yu. Optimal succinct rank data structure via approximate nonnegative tensor decomposition. In Moses Charikar and Edith Cohen, editors, ACM Symp. on the Theory of Computing (STOC), pages 955-966. ACM, 2019. URL: https://doi.org/10.1145/3313276.3316352.
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
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact