Direct Sum and Partitionability Testing over General Groups

Authors Andrej Bogdanov , Gautam Prakriya

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Author Details

Andrej Bogdanov
  • Department of Computer Science and Engineering and Institute of Theoretical Computer Science and Communications, Chinese University of Hong Kong, China
Gautam Prakriya
  • Institute of Theoretical Computer Science and Communications, Chinese University of Hong Kong, China


We thank Chandra Nair and Yan Nan Wang for valuable discussions on the hypercontractivity of the binary erasure channel and an anonymous reviewer for bringing to our attention the works [Ben-Sasson et al., 2003] and [Amir Shpilka and Avi Wigderson, 2006].

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Andrej Bogdanov and Gautam Prakriya. Direct Sum and Partitionability Testing over General Groups. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 33:1-33:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


A function f(x₁, … , x_n) from a product domain 𝒟₁ × ⋯ × 𝒟_n to an abelian group 𝒢 is a direct sum if it is of the form f₁(x₁) + ⋯ + f_n(x_n). We present a new 4-query direct sum test with optimal (up to constant factors) soundness error. This generalizes a result of Dinur and Golubev (RANDOM 2019) which is tailored to the target group 𝒢 = ℤ₂. As a special case, we obtain an optimal affinity test for 𝒢-valued functions on domain {0, 1}ⁿ under product measure. Our analysis relies on the hypercontractivity of the binary erasure channel. We also study the testability of function partitionability over product domains into disjoint components. A 𝒢-valued f(x₁, … , x_n) is k-direct sum partitionable if it can be written as a sum of functions over k nonempty disjoint sets of inputs. A function f(x₁, … , x_n) with unstructured product range ℛ^k is direct product partitionable if its outputs depend on disjoint sets of inputs. We show that direct sum partitionability and direct product partitionability are one-sided error testable with O((n - k)(log n + 1/ε) + 1/ε) adaptive queries and O((n/ε) log²(n/ε)) nonadaptive queries, respectively. Both bounds are tight up to the logarithmic factors for constant ε even with respect to adaptive, two-sided error testers. We also give a non-adaptive one-sided error tester for direct sum partitionability with query complexity O(kn² (log n)² / ε).

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
  • Theory of computation → Randomness, geometry and discrete structures
  • Direct Sum Test
  • Function Partitionability
  • Hypercontractive Inequality
  • Property Testing


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  1. Rudolf Ahlswede and Peter Gacs. Spreading of sets in product spaces and hypercontraction of the markov operator. Ann. Probab., 4(6):925-939, December 1976. URL:
  2. Eli Ben-Sasson, Madhu Sudan, Salil Vadhan, and Avi Wigderson. Randomness-efficient low degree tests and short PCPs via epsilon-biased sets. In Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing, STOC '03, pages 612-621, New York, NY, USA, 2003. Association for Computing Machinery. URL:
  3. Eric Blais. Testing juntas nearly optimally. In Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, Bethesda, MD, USA, May 31 - June 2, 2009, pages 151-158, 2009. URL:
  4. M. Blum, M. Luby, and R. Rubinfeld. Self-testing/correcting with applications to numerical problems. In Proceedings of the Twenty-Second Annual ACM Symposium on Theory of Computing, pages 73-83, New York, NY, USA, 1990. Association for Computing Machinery. URL:
  5. Andrej Bogdanov and Gautam Prakriya. Direct sum and partitionability testing over general groups. Electronic Colloquium on Computational Complexity, 2020. URL:
  6. Andrej Bogdanov and Baoxiang Wang. Learning and testing variable partitions. In Thomas Vidick, editor, 11th Innovations in Theoretical Computer Science Conference, ITCS 2020, January 12-14, 2020, Seattle, Washington, USA, volume 151 of LIPIcs, pages 37:1-37:22. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL:
  7. Aline Bonami. Étude des coefficients de Fourier des fonctions de l^p(g). Annales de l'Institut Fourier, 20(2):335-402, 1970. URL:
  8. Yotam Dikstein and Irit Dinur. Agreement testing theorems on layered set systems. In David Zuckerman, editor, 60th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2019, Baltimore, Maryland, USA, November 9-12, 2019, pages 1495-1524. IEEE Computer Society, 2019. URL:
  9. Irit Dinur and Konstantin Golubev. Direct sum testing: The general case. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2019, September 20-22, 2019, Massachusetts Institute of Technology, Cambridge, MA, USA, pages 40:1-40:11, 2019. URL:
  10. Irit Dinur and David Steurer. Direct product testing. In IEEE 29th Conference on Computational Complexity, CCC 2014, Vancouver, BC, Canada, June 11-13, 2014, pages 188-196. IEEE Computer Society, 2014. URL:
  11. Chandra Nair and Yan Nan Wang. Evaluating hypercontractivity parameters using information measures. In 2016 IEEE International Symposium on Information Theory (ISIT), pages 570-574, 2016. Google Scholar
  12. Ryan O'Donnell. Analysis of Boolean Functions. Cambridge University Press, USA, 2014. Google Scholar
  13. Rocco A. Servedio, Li-Yang Tan, and John Wright. Adaptivity Helps for Testing Juntas. In David Zuckerman, editor, 30th Conference on Computational Complexity (CCC 2015), volume 33 of Leibniz International Proceedings in Informatics (LIPIcs), pages 264-279, Dagstuhl, Germany, 2015. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL:
  14. Amir Shpilka and Avi Wigderson. Derandomizing homomorphism testing in general groups. SIAM J. Comput., 36(4):1215-1230, 2006. URL:
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