Randomized Query Composition and Product Distributions

Author Swagato Sanyal

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Swagato Sanyal
  • Department of Computer Science and Engineering , Indian Institute of Technology Kharagpur, India

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Swagato Sanyal. Randomized Query Composition and Product Distributions. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 56:1-56:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


Let 𝖱_ε denote randomized query complexity for error probability ε, and R: = 𝖱_{1/3}. In this work we investigate whether a perfect composition theorem 𝖱(f∘gⁿ) = Ω(𝖱(f)⋅ 𝖱(g)) holds for a relation f ⊆ {0,1}ⁿ × 𝒮 and a total inner function g:{0,1}^m → {0,1}. Composition theorems of the form 𝖱(f∘gⁿ) = Ω(𝖱(f)⋅ 𝖬(g)) are known for various measures 𝖬. Such measures include the sabotage complexity RS defined by Ben-David and Kothari (ICALP 2015), the max-conflict complexity defined by Gavinsky, Lee, Santha and Sanyal (ICALP 2019), and the linearized complexity measure defined by Ben-David, Blais, Göös and Maystre (FOCS 2022). The above measures are asymptotically non-decreasing in the above order. However, for total Boolean functions no asymptotic separation is known between any two of them. Let 𝖣^{prod} denote the maximum distributional query complexity with respect to any product (over variables) distribution . In this work we show that for any total Boolean function g, the sabotage complexity RS(g) = Ω̃(𝖣^{prod}(g)). This gives the composition theorem 𝖱(f∘gⁿ) = Ω̃(𝖱(f)⋅ 𝖣^{prod}(g)). In light of the minimax theorem which states that 𝖱(g) is the maximum distributional complexity of g over any distribution, our result makes progress towards answering the composition question. We prove our result by means of a complexity measure 𝖱_ε^{prod} that we define for total Boolean functions. Informally, 𝖱_ε^{prod}(g) is the minimum complexity of any randomized decision tree with unlabelled leaves with the property that, for every product distribution μ over the inputs, the average bias of its leaves is at least ((1-ε)-ε)/2 = 1/2-ε. It follows by standard arguments that 𝖱_{1/3}^{prod}(g) = Ω(𝖣^{prod}(g)). We show that 𝖱_{1/3}^{prod} is equivalent to the sabotage complexity up to a logarithmic factor. Ben-David and Kothari asked whether RS(g) = Θ(𝖱(g)) for total functions g. We generalize their question and ask if for any error ε, 𝖱_ε^{prod}(g) = Θ̃(𝖱_ε(g)). We observe that the work by Ben-David, Blais, Göös and Maystre (FOCS 2022) implies that for a perfect composition theorem 𝖱_{1/3}(f∘gⁿ) = Ω(𝖱_{1/3}(f)⋅𝖱_{1/3}(g)) to hold for any relation f and total function g, a necessary condition is that 𝖱_{1/3}(g) = O(1/(ε)⋅ 𝖱_{1/2-ε}(g)) holds for any total function g. We show that 𝖱_ε^{prod}(g) admits a similar error-reduction 𝖱_{1/3}^{prod}(g) = Õ(1/(ε)⋅𝖱_{1/2-ε}^{prod}(g)). Note that from the definition of 𝖱_ε^{prod} it is not immediately clear that 𝖱_ε^{prod} admits any error-reduction at all. We ask if our bound RS(g) = Ω̃(𝖣^{prod}(g)) is tight. We answer this question in the negative, by showing that for the NAND tree function, sabotage complexity is polynomially larger than 𝖣^{prod}. Our proof yields an alternative and different derivation of the tight lower bound on the bounded error randomized query complexity of the NAND tree function (originally proved by Santha in 1985), which may be of independent interest. Our result shows that sometimes, 𝖱_{1/3}^{prod} and sabotage complexity may be useful in producing an asymptotically larger lower bound on 𝖱(f∘gⁿ) than Ω̃(𝖱(f)⋅ 𝖣^{prod}(g)). In addition, this gives an explicit polynomial separation between 𝖱 and 𝖣^{prod} which, to our knowledge, was not known prior to our work.

Subject Classification

ACM Subject Classification
  • Theory of computation → Oracles and decision trees
  • Theory of computation → Communication complexity
  • Mathematics of computing → Graph theory
  • Randomized query complexity
  • Decision Tree
  • Boolean function complexity
  • Analysis of Boolean functions


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  1. Anurag Anshu, Dmitry Gavinsky, Rahul Jain, Srijita Kundu, Troy Lee, Priyanka Mukhopadhyay, Miklos Santha, and Swagato Sanyal. A composition theorem for randomized query complexity. In 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, page 1, 2018. Google Scholar
  2. Andrew Bassilakis, Andrew Drucker, Mika Göös, Lunjia Hu, Weiyun Ma, and Li-Yang Tan. The power of many samples in query complexity. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2020. Google Scholar
  3. Shalev Ben-David and Eric Blais. A tight composition theorem for the randomized query complexity of partial functions. In 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), pages 240-246. IEEE, 2020. Google Scholar
  4. Shalev Ben-David, Eric Blais, Mika Göös, and Gilbert Maystre. Randomised composition and small-bias minimax. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pages 624-635. IEEE, 2022. Google Scholar
  5. Sourav Chakraborty, Chandrima Kayal, Rajat Mittal, Manaswi Paraashar, Swagato Sanyal, and Nitin Saurabh. On the composition of randomized query complexity and approximate degree. In Nicole Megow and Adam D. Smith, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2023, September 11-13, 2023, Atlanta, Georgia, USA, volume 275 of LIPIcs, pages 63:1-63:23. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. Google Scholar
  6. Dmitry Gavinsky, Troy Lee, Miklos Santha, and Swagato Sanyal. A composition theorem for randomized query complexity via max-conflict complexity. In 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019,, volume 132 of LIPIcs, pages 64:1-64:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. Google Scholar
  7. Mika Göös, TS Jayram, Toniann Pitassi, and Thomas Watson. Randomized communication versus partition number. ACM Transactions on Computation Theory (TOCT), 10(1):1-20, 2018. Google Scholar
  8. Prahladh Harsha, Rahul Jain, and Jaikumar Radhakrishnan. Partition bound is quadratically tight for product distributions. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2016. Google Scholar
  9. Rahul Jain, Hartmut Klauck, Srijita Kundu, Troy Lee, Miklos Santha, Swagato Sanyal, and Jevgēnijs Vihrovs. Quadratically tight relations for randomized query complexity. Theory of Computing Systems, 64(1):101-119, 2020. Google Scholar
  10. Gillat Kol. Interactive compression for product distributions. In Proceedings of the forty-eighth annual ACM symposium on Theory of Computing, pages 987-998, 2016. Google Scholar
  11. Ashley Montanaro. A composition theorem for decision tree complexity. Chicago Journal Of Theoretical Computer Science, 6:1-8, 2014. Google Scholar
  12. Ryan O'Donnell. Analysis of boolean functions. Cambridge University Press, 2014. Google Scholar
  13. Judea Pearl. The solution for the branching factor of the alpha-beta pruning algorithm and its optimality. Communications of the ACM, 25(8):559-564, 1982. Google Scholar
  14. Michael Saks and Avi Wigderson. Probabilistic boolean decision trees and the complexity of evaluating game trees. In 27th Annual Symposium on Foundations of Computer Science (sfcs 1986), pages 29-38. IEEE, 1986. Google Scholar
  15. M Santha. On the monte carlo boolean decision tree complexity of read-once formulae. In 1991 Proceedings of the Sixth Annual Structure in Complexity Theory Conference, pages 180-181. IEEE Computer Society, 1991. Google Scholar
  16. Ben-David Shalev and Robin Kothari. Randomized query complexity of sabotaged and composed functions. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2016. Google Scholar
  17. Clifford Smyth. Reimer’s inequality and tardos' conjecture. In Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, pages 218-221, 2002. Google Scholar
  18. Marc Snir. Lower bounds on probabilistic linear decision trees. Theoretical Computer Science, 38:69-82, 1985. Google Scholar
  19. Avishay Tal. Properties and applications of boolean function composition. In Proceedings of the 4th conference on Innovations in Theoretical Computer Science, pages 441-454, 2013. Google Scholar
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