# Subquadratic Algorithms for Some \textsc{3Sum}-Hard Geometric Problems in the Algebraic Decision Tree Model

@inproceedings{Aronov2021SubquadraticAF, title={Subquadratic Algorithms for Some \textsc\{3Sum\}-Hard Geometric Problems in the Algebraic Decision Tree Model}, author={Boris Aronov and Mark de Berg and Jean Cardinal and Esther Ezra and John Iacono and Micha Sharir}, year={2021} }

We present subquadratic algorithms in the algebraic decision-tree model for several 3Sumhard geometric problems, all of which can be reduced to the following question: Given two sets A, B, each consisting of n pairwise disjoint segments in the plane, and a set C of n triangles in the plane, we want to count, for each triangle ∆ ∈ C, the number of intersection points between the segments of A and those of B that lie in ∆. The problems considered in this paper have been studied by Chan (2020…

## Figures from this paper

## One Citation

Subquadratic Algorithms for Some 3Sum-Hard Geometric Problems in the Algebraic Decision Tree Model

- Computer ScienceISAAC
- 2021

The approach is based on a primal-dual range searching mechanism, which exploits the multi-level polynomial partitioning machinery recently developed by Agarwal, Aronov, Ezra, and Zahl (2020).

## References

SHOWING 1-10 OF 38 REFERENCES

Subquadratic Algorithms for Algebraic 3SUM

- Computer Science, MathematicsDiscret. Comput. Geom.
- 2019

The 3POL problem, an algebraic generalization of 3SUM where the sum function is replaced by a constant-degree polynomial in three variables, is considered, and it is proved that 3POL admits bounded-degree algebraic decision trees of depth.

Improved Bounds for 3SUM, K-SUM, and Linear Degeneracy

- Mathematics, Computer ScienceESA
- 2017

The randomized $4$-linear decision tree complexity of 3SUM is shown to be $O(n^{3/2})$, and that the randomized $(2k-2)$- linear decisionTree complexity of k-SUM and k-LDT is $O (n^{k/2)$, for any odd $k\ge 3$, which improves (albeit randomized) the corresponding decision tree bounds.

Threesomes, Degenerates, and Love Triangles

- Mathematics, Computer Science2014 IEEE 55th Annual Symposium on Foundations of Computer Science
- 2014

This paper proves that the decision tree complexity of 3SUM is O(n<sup>3/2</sup> √/log n) and proves that improved bounds for k-variate linear degeneracy testing for all odd k ≥ 3 are lead directly to improved bounds on triangle enumeration, dynamic graph algorithms, and string matching data structures.

Solving k-SUM Using Few Linear Queries

- Mathematics, Computer ScienceESA
- 2016

It is proved that there exist o(n)-linear decision trees of depth ~O(n^3) for the k-SUM problem and that there exists a randomized algorithm that runs in ~O (n^{d+8}) time, and performs O( n^3 log^2 n) linear queries on the input.

On a class of O(n2) problems in computational geometry

- Computer ScienceComput. Geom.
- 1995

A large class of problems is described for which it is proved that they are all at least as difficult as the following base problem 3sum: Given a set S of n integers, are there three elements of S that sum up to 0.

Near-optimal Linear Decision Trees for k-SUM and Related Problems

- Computer ScienceJ. ACM
- 2019

These constructions are based on the notion of “inference dimension,” recently introduced by the authors in the context of active classification with comparison queries, and can be viewed as another contribution to the fruitful link between machine learning and discrete geometry, which goes back to the discovery of the VC dimension.

Near-optimal linear decision trees for k-SUM and related problems

- Mathematics, Computer ScienceElectron. Colloquium Comput. Complex.
- 2017

These constructions are based on the notion of “inference dimension”, recently introduced by the authors in the context of active classification with comparison queries, and can be viewed as another contribution to the fruitful link between machine learning and discrete geometry, which goes back to the discovery of the VC dimension.

A Nearly Quadratic Bound for Point-Location in Hyperplane Arrangements, in the Linear Decision Tree Model

- Computer Science, MathematicsDiscret. Comput. Geom.
- 2019

This work presents an algorithm that performs a point location query with O(d^2\log n) linear comparisons, improving the previous best result by about a factor of d and has currently the best performance for arbitrary hyperplanes.

Multilevel Polynomial Partitions and Simplified Range Searching

- Mathematics, Computer ScienceDiscret. Comput. Geom.
- 2015

A polynomial partitioning method with up to d polynomials in dimension d is provided, which allows for a complete decomposition of the given point set and is applied to obtain a new algorithm for the semialgebraic range searching problem.

More Logarithmic-Factor Speedups for 3SUM, (median, +)-Convolution, and Some Geometric 3SUM-Hard Problems

- Computer Science, MathematicsSODA
- 2018

An algorithm is presented that solves the 3SUM problem for n real numbers in O((n2 /log2 n)(log log n)O(1)) time, and the first subquadratic results on some3SUM-hard problems in computational geometry are obtained, for example, deciding whether a constant number of simple polygons have a common intersection.