### Abstract

The 3SUM problem asks if an input n-set of real numbers contains a triple whose sum is zero. We consider the 3POL problem, a natural generalization of 3SUM where we replace the sum function by a constant-degree polynomial in three variables. The motivations are threefold. Raz, Sharir, and de Zeeuw gave an O(n^{11}/^{6}) upper bound on the number of solutions of trivariate polynomial equations when the solutions are taken from the cartesian product of three n-sets of real numbers. We give algorithms for the corresponding problem of counting such solutions. Grønlund and Pettie recently designed subquadratic algorithms for 3SUM. We generalize their results to 3POL. 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 prove that there exist bounded-degree algebraic decision trees of depth O(n^{12/7+ϵ}) that solve 3POL, and that 3POL can be solved in O(n^{2}(log log n)3/2/(log n)1/2) time in the real-RAM model. Among the possible applications of those results, we show how to solve GPT in sub-quadratic time when the input points lie on o((log n)1/6/(log log n)1/2) constant-degree polynomial curves. This constitutes the first step towards closing the major open question of whether GPT can be solved in subquadratic time. To obtain these results, we generalize important tools - such as batch range searching and dominance reporting - to a polynomial setting. We expect these new tools to be useful in other applications.

Original language | English (US) |
---|---|

Title of host publication | 33rd International Symposium on Computational Geometry, SoCG 2017 |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

Pages | 131-1315 |

Number of pages | 1185 |

Volume | 77 |

ISBN (Electronic) | 9783959770385 |

DOIs | |

State | Published - Jun 1 2017 |

Event | 33rd International Symposium on Computational Geometry, SoCG 2017 - Brisbane, Australia Duration: Jul 4 2017 → Jul 7 2017 |

### Other

Other | 33rd International Symposium on Computational Geometry, SoCG 2017 |
---|---|

Country | Australia |

City | Brisbane |

Period | 7/4/17 → 7/7/17 |

### Fingerprint

### Keywords

- 3SUM
- Dominance reporting
- General position testing
- Polynomial curves
- Range searching
- Subquadratic algorithms

### ASJC Scopus subject areas

- Software

### Cite this

*33rd International Symposium on Computational Geometry, SoCG 2017*(Vol. 77, pp. 131-1315). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.SoCG.2017.13

**Subquadratic algorithms for algebraic generalizations of 3SUM.** / Barba, Luis; Cardinal, Jean; Iacono, John; Langerman, Stefan; Ooms, Aurélien; Solomon, Noam.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*33rd International Symposium on Computational Geometry, SoCG 2017.*vol. 77, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, pp. 131-1315, 33rd International Symposium on Computational Geometry, SoCG 2017, Brisbane, Australia, 7/4/17. https://doi.org/10.4230/LIPIcs.SoCG.2017.13

}

TY - GEN

T1 - Subquadratic algorithms for algebraic generalizations of 3SUM

AU - Barba, Luis

AU - Cardinal, Jean

AU - Iacono, John

AU - Langerman, Stefan

AU - Ooms, Aurélien

AU - Solomon, Noam

PY - 2017/6/1

Y1 - 2017/6/1

N2 - The 3SUM problem asks if an input n-set of real numbers contains a triple whose sum is zero. We consider the 3POL problem, a natural generalization of 3SUM where we replace the sum function by a constant-degree polynomial in three variables. The motivations are threefold. Raz, Sharir, and de Zeeuw gave an O(n11/6) upper bound on the number of solutions of trivariate polynomial equations when the solutions are taken from the cartesian product of three n-sets of real numbers. We give algorithms for the corresponding problem of counting such solutions. Grønlund and Pettie recently designed subquadratic algorithms for 3SUM. We generalize their results to 3POL. 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 prove that there exist bounded-degree algebraic decision trees of depth O(n12/7+ϵ) that solve 3POL, and that 3POL can be solved in O(n2(log log n)3/2/(log n)1/2) time in the real-RAM model. Among the possible applications of those results, we show how to solve GPT in sub-quadratic time when the input points lie on o((log n)1/6/(log log n)1/2) constant-degree polynomial curves. This constitutes the first step towards closing the major open question of whether GPT can be solved in subquadratic time. To obtain these results, we generalize important tools - such as batch range searching and dominance reporting - to a polynomial setting. We expect these new tools to be useful in other applications.

AB - The 3SUM problem asks if an input n-set of real numbers contains a triple whose sum is zero. We consider the 3POL problem, a natural generalization of 3SUM where we replace the sum function by a constant-degree polynomial in three variables. The motivations are threefold. Raz, Sharir, and de Zeeuw gave an O(n11/6) upper bound on the number of solutions of trivariate polynomial equations when the solutions are taken from the cartesian product of three n-sets of real numbers. We give algorithms for the corresponding problem of counting such solutions. Grønlund and Pettie recently designed subquadratic algorithms for 3SUM. We generalize their results to 3POL. 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 prove that there exist bounded-degree algebraic decision trees of depth O(n12/7+ϵ) that solve 3POL, and that 3POL can be solved in O(n2(log log n)3/2/(log n)1/2) time in the real-RAM model. Among the possible applications of those results, we show how to solve GPT in sub-quadratic time when the input points lie on o((log n)1/6/(log log n)1/2) constant-degree polynomial curves. This constitutes the first step towards closing the major open question of whether GPT can be solved in subquadratic time. To obtain these results, we generalize important tools - such as batch range searching and dominance reporting - to a polynomial setting. We expect these new tools to be useful in other applications.

KW - 3SUM

KW - Dominance reporting

KW - General position testing

KW - Polynomial curves

KW - Range searching

KW - Subquadratic algorithms

UR - http://www.scopus.com/inward/record.url?scp=85029939658&partnerID=8YFLogxK

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U2 - 10.4230/LIPIcs.SoCG.2017.13

DO - 10.4230/LIPIcs.SoCG.2017.13

M3 - Conference contribution

AN - SCOPUS:85029939658

VL - 77

SP - 131

EP - 1315

BT - 33rd International Symposium on Computational Geometry, SoCG 2017

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

ER -