On Multidimensional and Monotone k-SUM

Authors Chloe Ching-Yun Hsu, Chris Umans

Thumbnail PDF


  • Filesize: 0.49 MB
  • 13 pages

Document Identifiers

Author Details

Chloe Ching-Yun Hsu
Chris Umans

Cite AsGet BibTex

Chloe Ching-Yun Hsu and Chris Umans. On Multidimensional and Monotone k-SUM. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 50:1-50:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


The well-known k-SUM conjecture is that integer k-SUM requires time Omega(n^{\ceil{k/2}-o(1)}). Recent work has studied multidimensional k-SUM in F_p^d, where the best known algorithm takes time \tilde O(n^{\ceil{k/2}}). Bhattacharyya et al. [ICS 2011] proved a min(2^{\Omega(d)},n^{\Omega(k)}) lower bound for k-SUM in F_p^d under the Exponential Time Hypothesis. We give a more refined lower bound under the standard k-SUM conjecture: for sufficiently large p, k-SUM in F_p^d requires time Omega(n^{k/2-o(1)}) if k is even, and Omega(n^{\ceil{k/2}-2k(log k)/(log p)-o(1)}) if k is odd. For a special case of the multidimensional problem, bounded monotone d-dimensional 3SUM, Chan and Lewenstein [STOC 2015] gave a surprising \tilde O(n^{2-2/(d+13)}) algorithm using additive combinatorics. We show this algorithm is essentially optimal. To be more precise, bounded monotone d-dimensional 3SUM requires time Omega(n^{2-\frac{4}{d}-o(1)}) under the standard 3SUM conjecture, and time Omega(n^{2-\frac{2}{d}-o(1)}) under the so-called strong 3SUM conjecture. Thus, even though one might hope to further exploit the structural advantage of monotonicity, no substantial improvements beyond those obtained by Chan and Lewenstein are possible for bounded monotone d-dimensional 3SUM.
  • 3SUM
  • kSUM
  • monotone 3SUM
  • strong 3SUM conjecture


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


  1. Amir Abboud and Kevin Lewi. Exact weight subgraphs and the k-SUM conjecture. In International Colloquium on Automata, Languages, and Programming, pages 1-12. Springer, 2013. Google Scholar
  2. Amir Abboud, Kevin Lewi, and Ryan Williams. Losing weight by gaining edges. In European Symposium on Algorithms, pages 1-12. Springer, 2014. Google Scholar
  3. Nir Ailon and Bernard Chazelle. Lower bounds for linear degeneracy testing. Journal of the ACM (JACM), 52(2):157-171, 2005. Google Scholar
  4. Amihood Amir, Timothy M Chan, Moshe Lewenstein, and Noa Lewenstein. On hardness of jumbled indexing. In Automata, Languages, and Programming, pages 114-125. Springer, 2014. Google Scholar
  5. Ilya Baran, Erik D Demaine, and Mihai Pătraşcu. Subquadratic algorithms for 3SUM. Algorithmica, 50(4):584-596, 2008. Google Scholar
  6. Arnab Bhattacharyya, Piotr Indyk, David P Woodruff, and Ning Xie. The complexity of linear dependence problems in vector spaces. In ICS, pages 496-508, 2011. Google Scholar
  7. Timothy M Chan and Moshe Lewenstein. Clustered integer 3SUM via additive combinatorics. In Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, pages 31-40. ACM, 2015. Google Scholar
  8. Rod G Downey, Michael R Fellows, Alexander Vardy, and Geoff Whittle. The parametrized complexity of some fundamental problems in coding theory. SIAM Journal on Computing, 29(2):545-570, 1999. Google Scholar
  9. Jeff Erickson. Lower bounds for linear satisfiability problems. In Chicago Journal of Theoretical Computer Science, volume 8, 1999. Google Scholar
  10. Ari Freund. Improved subquadratic 3sum. Algorithmica, pages 1-19, 2015. Google Scholar
  11. Anka Gajentaan and Mark H Overmars. On a class of O(n²) problems in computational geometry. Computational Geometry, 5(3):165-185, 1995. Google Scholar
  12. Omer Gold and Micha Sharir. Improved bounds for 3sum, k-sum, and linear degeneracy. arXiv preprint arXiv:1512.05279, 2015. Google Scholar
  13. Isaac Goldstein, Tsvi Kopelowitz, Moshe Lewenstein, and Ely Porat. How hard is it to find (honest) witnesses? In LIPIcs-Leibniz International Proceedings in Informatics, volume 57. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2016. Google Scholar
  14. Allan Grønlund and Seth Pettie. Threesomes, degenerates, and love triangles. In Foundations of Computer Science (FOCS), 2014 IEEE 55th Annual Symposium on, pages 621-630. IEEE, 2014. Google Scholar
  15. Zahra Jafargholi and Emanuele Viola. 3SUM, 3XOR, triangles. Algorithmica, 74(1):326-343, 2016. Google Scholar
  16. Tsvi Kopelowitz, Seth Pettie, and Ely Porat. Higher lower bounds from the 3sum conjecture. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1272-1287. Society for Industrial and Applied Mathematics, 2016. Google Scholar
  17. Mihai Pătraşcu. Towards polynomial lower bounds for dynamic problems. In Proceedings of the Forty-Second ACM Symposium on Theory of computing, pages 603-610. ACM, 2010. Google Scholar
  18. Mihai Pătraşcu and Ryan Williams. On the possibility of faster SAT algorithms. In Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1065-1075. SIAM, 2010. Google Scholar
  19. Virginia Vassilevska and Ryan Williams. Finding, minimizing, and counting weighted subgraphs. In Proceedings of the Forty-First Annual ACM Symposium on Theory of Computing, pages 455-464. ACM, 2009. Google Scholar
  20. Emanuele Viola. Reducing 3XOR to listing triangles, an exposition. In Electronic Colloquium on Computational Complexity (ECCC), volume 18, page 113, 2011. Google Scholar