One-Way Communication Complexity of Partial XOR Functions

Authors Vladimir V. Podolskii , Dmitrii Sluch



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Vladimir V. Podolskii
  • Tufts University, Medford, MA, USA
Dmitrii Sluch
  • Nebius, Tel Aviv, Israel

Acknowledgements

We would like to thank the anonymous reviewers for useful comments that helped us improve the presentation.

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Vladimir V. Podolskii and Dmitrii Sluch. One-Way Communication Complexity of Partial XOR Functions. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 116:1-116:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.116

Abstract

Boolean function F(x,y) for x,y ∈ {0,1}ⁿ is an XOR function if F(x,y) = f(x⊕ y) for some function f on n input bits, where ⊕ is a bit-wise XOR. XOR functions are relevant in communication complexity, partially for allowing the Fourier analytic technique. For total XOR functions, it is known that deterministic communication complexity of F is closely related to parity decision tree complexity of f. Montanaro and Osbourne (2009) observed that one-way communication complexity D_{cc}^{→}(F) of F is exactly equal to non-adaptive parity decision tree complexity NADT^{⊕}(f) of f. Hatami et al. (2018) showed that unrestricted communication complexity of F is polynomially related to parity decision tree complexity of f. We initiate the study of a similar connection for partial functions. We show that in the case of one-way communication complexity whether these measures are equal, depends on the number of undefined inputs of f. More precisely, if D_{cc}^{→}(F) = t and f is undefined on at most O((2^{n-t})/(√{n-t})) inputs, then NADT^{⊕}(f) = t. We also provide stronger bounds in extreme cases of small and large complexity. We show that the restriction on the number of undefined inputs in these results is unavoidable. That is, for a wide range of values of D_{cc}^{→}(F) and NADT^{⊕}(f) (from constant to n-2) we provide partial functions (with more than Ω((2^{n-t})/(√{n-t})) undefined inputs, where t = D_{cc}^{→}) for which D_{cc}^{→}(F) < NADT^{⊕}(f). In particular, we provide a function with an exponential gap between the two measures. Our separation results translate to the case of two-way communication complexity as well, in particular showing that the result of Hatami et al. (2018) cannot be generalized to partial functions. Previous results for total functions heavily rely on the Boolean Fourier analysis and thus, the technique does not translate to partial functions. For the proofs of our results we build a linear algebraic framework instead. Separation results are proved through the reduction to covering codes.

Subject Classification

ACM Subject Classification
  • Theory of computation → Communication complexity
  • Theory of computation → Oracles and decision trees
  • Theory of computation → Error-correcting codes
Keywords
  • Partial functions
  • XOR functions
  • communication complexity
  • decision trees
  • covering codes

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References

  1. Anurag Anshu, Naresh Goud Boddu, and Dave Touchette. Quantum log-approximate-rank conjecture is also false. In David Zuckerman, editor, 60th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2019, Baltimore, Maryland, USA, November 9-12, 2019, pages 982-994. IEEE Computer Society, 2019. URL: https://doi.org/10.1109/FOCS.2019.00063.
  2. Arkadev Chattopadhyay, Ankit Garg, and Suhail Sherif. Towards stronger counterexamples to the log-approximate-rank conjecture. In Mikolaj Bojanczyk and Chandra Chekuri, editors, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2021, December 15-17, 2021, Virtual Conference, volume 213 of LIPIcs, pages 13:1-13:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2021.13.
  3. Arkadev Chattopadhyay and Nikhil S. Mande. A lifting theorem with applications to symmetric functions. In Satya V. Lokam and R. Ramanujam, editors, 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2017, December 11-15, 2017, Kanpur, India, volume 93 of LIPIcs, pages 23:1-23:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2017.23.
  4. Arkadev Chattopadhyay, Nikhil S. Mande, Swagato Sanyal, and Suhail Sherif. Lifting to parity decision trees via stifling. In Yael Tauman Kalai, editor, 14th Innovations in Theoretical Computer Science Conference, ITCS 2023, January 10-13, 2023, MIT, Cambridge, Massachusetts, USA, volume 251 of LIPIcs, pages 33:1-33:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.ITCS.2023.33.
  5. Arkadev Chattopadhyay, Nikhil S. Mande, and Suhail Sherif. The log-approximate-rank conjecture is false. J. ACM, 67(4):23:1-23:28, 2020. URL: https://doi.org/10.1145/3396695.
  6. Gérard D. Cohen, Iiro S. Honkala, Simon Litsyn, and Antoine Lobstein. Covering Codes, volume 54 of North-Holland mathematical library. North-Holland, 2005. Google Scholar
  7. Susanna F. de Rezende, Or Meir, Jakob Nordström, Toniann Pitassi, Robert Robere, and Marc Vinyals. Lifting with simple gadgets and applications to circuit and proof complexity. In Sandy Irani, editor, 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020, pages 24-30. IEEE, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00011.
  8. Uma Girish, Ran Raz, and Avishay Tal. Quantum versus randomized communication complexity, with efficient players. Comput. Complex., 31(2):17, 2022. URL: https://doi.org/10.1007/s00037-022-00232-7.
  9. Uma Girish, Makrand Sinha, Avishay Tal, and Kewen Wu. Fourier growth of communication protocols for XOR functions. CoRR, abs/2307.13926, 2023. URL: https://doi.org/10.48550/arXiv.2307.13926.
  10. Parikshit Gopalan, Ryan O'Donnell, Rocco A. Servedio, Amir Shpilka, and Karl Wimmer. Testing fourier dimensionality and sparsity. SIAM J. Comput., 40(4):1075-1100, July 2011. URL: https://doi.org/10.1137/100785429.
  11. Lianna Hambardzumyan, Hamed Hatami, and Pooya Hatami. Dimension-free bounds and structural results in communication complexity. Israel Journal of Mathematics, 253:555-616, 2023. URL: https://doi.org/10.1007/s11856-022-2365-8.
  12. L. H. Harper. Global Methods for Combinatorial Isoperimetric Problems. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2004. URL: https://doi.org/10.1017/CBO9780511616679.
  13. Hamed Hatami, Kaave Hosseini, and Shachar Lovett. Structure of protocols for XOR functions. SIAM J. Comput., 47(1):208-217, 2018. URL: https://doi.org/10.1137/17M1136869.
  14. Stasys Jukna. Boolean Function Complexity - Advances and Frontiers, volume 27 of Algorithms and combinatorics. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-24508-4.
  15. Sampath Kannan, Elchanan Mossel, Swagato Sanyal, and Grigory Yaroslavtsev. Linear sketching over f_2. In Rocco A. Servedio, editor, 33rd Computational Complexity Conference, CCC 2018, June 22-24, 2018, San Diego, CA, USA, volume 102 of LIPIcs, pages 8:1-8:37. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.CCC.2018.8.
  16. Eyal Kushilevitz and Noam Nisan. Communication Complexity. Cambridge University Press, 1996. URL: https://doi.org/10.1017/CBO9780511574948.
  17. Bruno Loff and Sagnik Mukhopadhyay. Lifting theorems for equality. In Rolf Niedermeier and Christophe Paul, editors, 36th International Symposium on Theoretical Aspects of Computer Science, STACS 2019, March 13-16, 2019, Berlin, Germany, volume 126 of LIPIcs, pages 50:1-50:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.STACS.2019.50.
  18. Nikhil S. Mande, Swagato Sanyal, and Suhail Sherif. One-way communication complexity and non-adaptive decision trees. In Petra Berenbrink and Benjamin Monmege, editors, 39th International Symposium on Theoretical Aspects of Computer Science, STACS 2022, March 15-18, 2022, Marseille, France (Virtual Conference), volume 219 of LIPIcs, pages 49:1-49:24. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.STACS.2022.49.
  19. Ashley Montanaro and Tobias Osborne. On the communication complexity of XOR functions. CoRR, abs/0909.3392, 2009. URL: https://arxiv.org/abs/0909.3392.
  20. A. Rao and A. Yehudayoff. Communication Complexity: and Applications. Cambridge University Press, 2020. URL: https://books.google.com/books?id=emw8PgAACAAJ.
  21. Swagato Sanyal. Fourier sparsity and dimension. Theory Comput., 15:1-13, 2019. URL: https://doi.org/10.4086/toc.2019.v015a011.
  22. Alexander A. Sherstov, Andrey A. Storozhenko, and Pei Wu. An optimal separation of randomized and quantum query complexity. SIAM J. Comput., 52(2):525-567, 2023. URL: https://doi.org/10.1137/22m1468943.
  23. Makrand Sinha and Ronald de Wolf. Exponential separation between quantum communication and logarithm of approximate rank. In David Zuckerman, editor, 60th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2019, Baltimore, Maryland, USA, November 9-12, 2019, pages 966-981. IEEE Computer Society, 2019. URL: https://doi.org/10.1109/FOCS.2019.00062.
  24. J.H. Spencer and L. Florescu. Asymptopia. Student mathematical library. American Mathematical Society, 2014. URL: https://books.google.com/books?id=uBMLugEACAAJ.
  25. Hing Yin Tsang, Chung Hoi Wong, Ning Xie, and Shengyu Zhang. Fourier sparsity, spectral norm, and the log-rank conjecture. In 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, 26-29 October, 2013, Berkeley, CA, USA, pages 658-667. IEEE Computer Society, 2013. URL: https://doi.org/10.1109/FOCS.2013.76.
  26. Shengyu Zhang. Efficient quantum protocols for XOR functions. In Chandra Chekuri, editor, Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 1878-1885. SIAM, 2014. URL: https://doi.org/10.1137/1.9781611973402.136.
  27. Zhiqiang Zhang and Yaoyun Shi. On the parity complexity measures of boolean functions. Theor. Comput. Sci., 411(26-28):2612-2618, 2010. URL: https://doi.org/10.1016/j.tcs.2010.03.027.