Abstract
We describe subquadratic algorithms, in the algebraic decision-tree model of computation, for detecting whether there exists a triple of points, belonging to three respective sets A, B, and C of points in the plane, that satisfy a pair of polynomial equations. In particular, this has an application to detect collinearity among three sets A, B, C of n points each, in the complex plane, when each of the sets A, B, C lies on some constant-degree algebraic curve. In another development, we present a subquadratic algorithm, in the algebraic decision-tree model, for the following problem: Given a pair of sets A, B each consisting of n pairwise disjoint line segments in the plane, and a third set C of arbitrary line segments in the plane, determine whether \(A\times B\times C\) contains a triple of concurrent segments. This is one of four 3sum-hard geometric problems recently studied by Chan (2020). The results reported in this extended abstract are based on the recent studies of the author with Aronov and Sharir (2020, 2021).
Work partially supported by NSF CAREER under grant CCF:AF-1553354 and by Grant 824/17 from the Israel Science Foundation.
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Notes
- 1.
The work in [5] also shows how to detect whether A, B, C satisfy a single polynomial equation under the condition that two of the sets lie on two respective one-dimensional curves and the third is placed arbitrarily in the plane. We do not report this particular development in this extended abstract.
- 2.
A two-dimensional algebraic surface S in \({\mathbb {R}}^4\) has good fibers if, for every point \(p \in {\mathbb {R}}^2\), the fibers \((\{p\} \times {\mathbb {R}}^2) \cap S\) and \(({\mathbb {R}^2} \times \{p\}) \cap S\) are finite.
- 3.
The number of cells is in fact \(O(n/g)^2)\), but the analysis in [5] uses hierarchical polynomial partitioning in order to speed up computation, which slightly increases the number of cells to \(O((n/g)^{2 + \varepsilon })\). We skip this variant in this extended abstract.
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Ezra, E. (2021). On 3SUM-hard Problems in the Decision Tree Model. In: De Mol, L., Weiermann, A., Manea, F., Fernández-Duque, D. (eds) Connecting with Computability. CiE 2021. Lecture Notes in Computer Science(), vol 12813. Springer, Cham. https://doi.org/10.1007/978-3-030-80049-9_16
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