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Subquadratic Algorithms for Algebraic 3SUM

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Abstract

The 3SUM problem asks if an input n-set of real numbers contains a triple whose sum is zero. We qualify such a triple of degenerate because the probability of finding one in a random input is zero. We consider the 3POL problem, an algebraic generalization of 3SUM where we replace the sum function by a constant-degree polynomial in three variables. The motivations are threefold. Raz et al. gave an \(O(n^{11/6})\) upper bound on the number of degenerate triples for the 3POL problem. We give algorithms for the corresponding problem of counting them. Grønlund and Pettie designed subquadratic algorithms for 3SUM. We prove that 3POL admits bounded-degree algebraic decision trees of depth \(O(n^{12/7+\varepsilon })\), and we prove that 3POL can be solved in \(O(n^2 {(\log \log n)}^{3/2} / {(\log n)}^{1/2})\) time in the real-RAM model, generalizing their results. Finally, we shed light on the General Position Testing (GPT) problem: “Given n points in the plane, do three of them lie on a line?”, a key problem in computational geometry: we show how to solve GPT in subquadratic time when the input points lie on a small number of constant-degree polynomial curves. Many other geometric degeneracy testing problems reduce to 3POL.

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Notes

  1. Throughout this document, \(\varepsilon \) denotes a positive real number that can be made as small as desired.

  2. Chan [16] shows that an additional logarithmic factor can be shaved by augmenting the real-RAM model with constant time nonstandard operations on \({\varTheta }(\log n)\) bits words. His improvements extend to 3POL.

  3. The main result in Gold and Sharir’s paper [31] is a randomized\(O(n^{k/2})\)-depth \((2k-2)\)-linear decision tree for \(k\)-SUM, for all odd \(k \ge 3\).

  4. Because our results do not depend on the meaning of group related form, we do not bother defining it here. We refer the reader to Raz et al. [54] for the exact definition.

  5. Note that it is easy to modify the algorithm to count or report the solutions. In the latter case, the algorithm becomes output sensitive.

  6. Note that vertical and horizontal lines fall in both categories.

  7. The original algorithm relies on hierarchical cuttings which cannot be implemented in the bounded-degree ADT model.

  8. In the real-RAM and word-RAM models.

  9. Not including the independent monomial, namely, 1.

  10. Note that Raz et al. [54] use the same points and curves.

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Acknowledgements

The authors wish to thank the anonymous referees for many useful suggestions. Jean Cardinal is supported by the “Action de Recherche Concertée” (ARC) COPHYMA, convention number 4.110.H.000023. John Iacono’s research was partially completed while on sabbatical at the Algorithms Research Group of the Département d’Informatique at the Université libre de Bruxelles with support from a Fulbright Research Fellowship, the Fonds de la Recherche Scientifique—FNRS, and NSF Grants CNS-1229185, CCF-1319648, and CCF-1533564. Stefan Langerman is Directeur de recherches du F.R.S.-FNRS. Aurélien Ooms is supported by the Fund for Research Training in Industry and Agriculture (FRIA). Noam Solomon is supported by Grant 892/13 from the Israel Science Foundation.

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Authors and Affiliations

  1. Department of Computer Science, ETH Zürich, Zürich, Switzerland

    Luis Barba

  2. Département d’Informatique, Université libre de Bruxelles (ULB), Brussels, Belgium

    Jean Cardinal, Stefan Langerman & Aurélien Ooms

  3. Department of Computer Science and Engineering, New York University (NYU), New York, NY, USA

    John Iacono

  4. Center of Mathematical Sciences and Applications, Harvard University, Cambridge, MA, USA

    Noam Solomon

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  1. Luis Barba
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Barba, L., Cardinal, J., Iacono, J. et al. Subquadratic Algorithms for Algebraic 3SUM. Discrete Comput Geom 61, 698–734 (2019). https://doi.org/10.1007/s00454-018-0040-y

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