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Improved Time-Space Tradeoffs for 3SUM-Indexing

Authors Itai Dinur , Alexander Golovnev

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LIPIcs.ICALP.2026.78.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Data structures design and analysis
Keywords
  • Data Structures
  • 3SUM
  • Function Inversion

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Abstract

3SUM-Indexing is a preprocessing variant of the 3SUM problem that has recently received a lot of attention. The best known time-space tradeoff for the problem is T S³ = n⁶ (up to logarithmic factors), where n is the number of input integers, S is the length of the preprocessed data structure, and T is the running time of the query algorithm. This tradeoff was achieved in [Kopelowitz and Porat, 2019; Golovnev et al., 2020] using the Fiat-Naor generic algorithm for Function Inversion. Consequently, [Golovnev et al., 2020] asked whether this algorithm can be improved by leveraging the structure of 3SUM-Indexing. 
In this paper, we exploit the structure of 3SUM-Indexing to give a time-space tradeoff of T S = n^{2.5}, which is better than the best known one in the range n^{3/2} ≪ S ≪ n^{7/4}. We further extend this improvement to the kSUM-Indexing problem - a generalization of 3SUM-Indexing - and to the related kXOR-Indexing problem, where addition is replaced with XOR. Additionally, we improve the best known time-space tradeoffs for the Jumbled Indexing problem, which is a well-known data structure problem related to 3SUM-Indexing. 
Our improvement comes from an alternative way to apply the Fiat-Naor algorithm to 3SUM-Indexing. Specifically, we exploit the structure of the function to be inverted by decomposing it into "sub-functions" with certain properties. This allows us to apply an improvement to the Fiat-Naor algorithm (which is not directly applicable to 3SUM-Indexing), obtained in [Golovnev et al., 2023] in a much larger range of parameters. We believe that our techniques may be useful in additional application-dependent optimizations of the Fiat-Naor algorithm.

Cite As Get BibTex

Itai Dinur and Alexander Golovnev. Improved Time-Space Tradeoffs for 3SUM-Indexing. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 78:1-78:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.78

Author Details

Itai Dinur
  • Ben-Gurion University, Be'er-Sheva, Israel
  • Georgetown University, Washington, DC, USA
Alexander Golovnev
  • Georgetown University, Washington, DC, USA

Funding

  • Golovnev, Alexander: Supported by the National Science Foundation CAREER award (grant CCF-2338730).

Acknowledgements

We thank the anonymous referees for their very helpful comments.

References

  1. Josh Alman, Yizhi Huang, and Kevin Yeo. Fine-grained complexity in a world without cryptography. In EUROCRYPT, 2025. Google Scholar
  2. Boris Aronov, Jean Cardinal, Justin Dallant, and John Iacono. A general technique for searching in implicit sets via function inversion. In SOSA, 2024. Google Scholar
  3. Boris Aronov, Esther Ezra, Micha Sharir, and Guy Zigdon. Time and space efficient collinearity indexing. Computational Geometry, 110:101963, 2023. URL: https://doi.org/10.1016/J.COMGEO.2022.101963.
  4. Philip Bille, Inge Li Gørtz, Moshe Lewenstein, Solon P. Pissis, Eva Rotenberg, and Teresa Anna Steiner. Gapped string indexing in subquadratic space and sublinear query time. In STACS, 2024. URL: https://doi.org/10.4230/LIPIcs.STACS.2024.16.
  5. Alex Biryukov and Adi Shamir. Cryptanalytic time/memory/data tradeoffs for stream ciphers. In ASIACRYPT, 2000. Google Scholar
  6. Alex Biryukov, Adi Shamir, and David Wagner. Real time cryptanalysis of A5/1 on a PC. In FSE, 2001. Google Scholar
  7. Timothy M. Chan and Moshe Lewenstein. Clustered integer 3SUM via additive combinatorics. In STOC, 2015. Google Scholar
  8. Eldon Chung and Kasper Green Larsen. Stronger 3SUM-Indexing lower bounds. In SODA, 2023. Google Scholar
  9. Ferdinando Cicalese, Gabriele Fici, and Zsuzsanna Lipták. Searching for jumbled patterns in strings. In Prague Stringology Conference, 2009. Google Scholar
  10. Henry Corrigan-Gibbs and Dmitry Kogan. The function-inversion problem: Barriers and opportunities. In TCC, 2019. Google Scholar
  11. Erik D. Demaine and Salil P. Vadhan. Some notes on 3SUM, December 2001. Unpublished manuscript. Google Scholar
  12. Pavel Dvořák, Michal Koucký, Karel Král, and Veronika Slívová. Data structures lower bounds and popular conjectures. In ESA, 2021. URL: https://doi.org/10.4230/LIPIcs.ESA.2021.39.
  13. Paul Erdős. On a new method in elementary number theory which leads to an elementary proof of the prime number theorem. Proceedings of the National Academy of Sciences, 35(7):374-384, 1949. Google Scholar
  14. Amos Fiat and Moni Naor. Rigorous time/space tradeoffs for inverting functions. In STOC, 1991. Google Scholar
  15. Anka Gajentaan and Mark H. Overmars. On a class of O(n²) problems in computational geometry. Computational geometry, 5(3):165-185, 1995. URL: https://doi.org/10.1016/0925-7721(95)00022-2.
  16. Isaac Goldstein, Tsvi Kopelowitz, Moshe Lewenstein, and Ely Porat. Conditional lower bounds for space/time tradeoffs. In WADS, 2017. Google Scholar
  17. Alexander Golovnev, Siyao Guo, Thibaut Horel, Sunoo Park, and Vinod Vaikuntanathan. Data structures meet cryptography: 3SUM with preprocessing. In STOC, 2020. Google Scholar
  18. Alexander Golovnev, Siyao Guo, Spencer Peters, and Noah Stephens-Davidowitz. Revisiting time-space tradeoffs for function inversion. In CRYPTO, 2023. Google Scholar
  19. Martin Hellman. A cryptanalytic time-memory trade-off. IEEE Transactions on Information Theory, 26(4):401-406, 1980. URL: https://doi.org/10.1109/TIT.1980.1056220.
  20. Shuichi Hirahara, Rahul Ilango, and Ryan Williams. Beating brute force for compression problems. In STOC, 2024. Google Scholar
  21. Tomasz Kociumaka, Jakub Radoszewski, and Wojciech Rytter. Efficient indexes for jumbled pattern matching with constant-sized alphabet. In ESA, 2013. Google Scholar
  22. Tsvi Kopelowitz and Ely Porat. The strong 3SUM-INDEXING conjecture is false. arXiv, 2019. URL: https://arxiv.org/abs/1907.11206.
  23. Noam Mazor and Rafael Pass. The non-uniform perebor conjecture for time-bounded Kolmogorov complexity is false. In ITCS, 2024. URL: https://doi.org/10.4230/LIPIcs.ITCS.2024.80.
  24. Samuel McCauley. Improved space-efficient approximate nearest neighbor search using function inversion. In ESA, 2024. URL: https://doi.org/10.4230/LIPIcs.ESA.2024.88.
  25. Arvind Narayanan and Vitaly Shmatikov. Fast dictionary attacks on passwords using time-space tradeoff. In CCS, 2005. Google Scholar
  26. Philippe Oechslin. Making a faster cryptanalytic time-memory trade-off. In CRYPTO, 2003. Google Scholar
  27. Virginia Vassilevska Williams. On some fine-grained questions in algorithms and complexity. In ICM 2018, 2018. Google Scholar
  28. Virginia Vassilevska Williams and Ryan Williams. Fixed-parameter and fine-grained complexity, 2024. Lecture Notes. Google Scholar
  29. Andrew Chi-Chih Yao. Coherent functions and program checkers. In STOC, 1990. Google Scholar
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