Cutting algebraic curves into pseudo-segments and applications

Sharir, Micha; Zahl, Joshua

doi:10.1016/j.jcta.2017.02.006

Cited by 33 publications

(72 citation statements)

References 35 publications

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“…They then applied a variant of the Szemerédi-Trotter theorem to this set of pseudo-segments. In [18], Sharir and the author extended this cutting method from circles to general algebraic curves. This yielded an incidence theorem for points and curves in the plane that is stronger than the one given by the "partitioning + Kővári-Sós-Turán" method.…”

Section: Introductionmentioning

confidence: 99%

Breaking the 3/2 Barrier for Unit Distances in Three Dimensions

Zahl

2018

International Mathematics Research Notices

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Section: Introductionmentioning

confidence: 99%

Breaking the 3/2 Barrier for Unit Distances in Three Dimensions

Zahl

2018

International Mathematics Research Notices

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“…Note that the best known general upper bounds for these problems [16,41] are obtained by applying the Crossing Lemma to the edges in a suitable geometric or topological graph. Following the pioneering work of Dvir [17], Guth and Katz [25,26], several of these questions have been revisited from an algebraic perspective [18,28,44].…”

Section: Discussionmentioning

confidence: 99%

Planar point sets determine many pairwise crossing segments

Pach

Rubin

Tardos

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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We show that any set of n points in general position in the plane determines n 1−o(1) pairwise crossing segments. The best previously known lower bound, Ω ( √ n), was proved more than 25 years ago by Aronov, Erdős, Goddard, Kleitman, Klugerman, Pach, and Schulman. Our proof is fully constructive, and extends to dense geometric graphs.Let V be a set of n points in general position in the plane, that is, assume that no 3 points of V are collinear. A geometric graph is a graph G = (V, E) whose vertex set is V and whose edges are represented by possibly crossing straight-line segments connecting certain pairs of points in V . If every pair of points in V is connected by a segment, we have E = V 2 , and G is called a complete geometric graph. Two edges pq, p q ∈ E are said to cross if the corresponding segments share an interior point. Topological graphs are defined similarly, except that their edges can be represented by any Jordan curves that have no interior points that belong to V .Crossing patterns and intersection graphs. Finding maximum cliques or independent sets in intersection graphs of segments, rays, and other convex sets in the plane is a computationally hard problem and a classic topic in computational and combinatorial geometry [4,10,21,30,31,32]. There are many interesting Ramsey-type problems and results about the existence of large cliques or large independent sets in intersection graphs of segments [7,11,34,35,42] and, more generally, of Jordan curves ("strings") [20,24,22]. Some of these questions are intimately related to counting incidences between points and lines [41,45,46], and to bounding the complexity of k-levels in arrangements of lines in R 2 [16].It appears to be a somewhat simpler task to understand the combinatorial structure of crossings between the edges of a geometric or topological graph. Despite decades of steady progress, we have very few asymptotically tight results in this direction. Perhaps the best known and most applicable theorem of this kind is the so-called Crossing Lemma of Ajtai, Chvátal, Newborn, Szemerédi [6] and Leighton [36], which states that any topological graph G = (V, E) with |E| > 4|V | determines at least Ω |E| 3 /|V | 2 crossing pairs of edges. Recently, a similar result has been established by the authors for contact graphs of families of Jordan curves [40].According to another asymptotically tight result, for t > 1, every geometric graph G with |E| ≥ nt edges has two disjoint sets of edges, E 1 , E 2 ⊂ E, each of size Ω(t), such that every edge in E 1 crosses all edges in E 2 ; see, e.g., [20, Theorem 6]. A similar theorem holds for topological graphs, with the difference that then |E 1 |, |E 2 | = Ω (t/ log t) [22]. It is a major unsolved question to decide whether under these circumstances G must also contain a family of pairwise crossing edges, whose size tends to infinity as t → ∞. It is conjectured that one can always choose such a family consisting of almost t edges. If this stronger conjecture is true for t 1−ot(1) edges, then for t ≈ n/2, it would ...

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“…We cut each curve in S * Ω into x-monotone (open) Jordan arcs, and let W Ω be the resulting set of arcs; we have |W Ω | = O(|S * Ω |). This follows from Harnack's curve theorem and Bézout's theorem, see, e.g., [27,Section 2.2].…”

Section: The Second Decomposition Step: Random Samplingmentioning

confidence: 94%

“…(b) Note that the algorithm described above works equally well with non-vertical pairwise disjoint algebraic curves of constant degree, with only superficial modifications, mirroring the combinatorial analysis of Aronov and Sharir [10] as well as Sharir and Zahl [27]. The current analysis, however, can only guarantee quadratic running time.…”

Section: An Application: Eliminating Depth Cycles Among Linesmentioning

confidence: 97%

Constructive Polynomial Partitioning for Algebraic Curves in ℝ³ with Applications

Aronov¹,

Ezra²,

Zahl³

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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In 2015, Guth proved that, for any set of k-dimensional varieties in R d and for any positive integer D, there exists a polynomial of degree at most D whose zero-set divides R d into open connected "cells," so that only a small fraction of the given varieties intersect each cell. Guth's result generalized an earlier result of Guth and Katz for points.Guth's proof relies on a variant of the Borsuk-Ulam theorem, and for k > 0, it is unknown how to obtain an explicit representation of such a partitioning polynomial and how to construct it efficiently. In particular, it is unknown how to effectively construct such a polynomial for curves (or even lines) in R 3 .We present an efficient algorithmic construction for this setting. Given a set of n input curves and a positive integer D, we efficiently construct a decomposition of space into O(D 3 log 3 D) open cells, each of which meets O(n/D 2 ) curves from the input. The construction time is O(n 2 ). For the case of lines in 3-space we present an improved implementation, whose running time is O(n 4/3 polylog n). The constant of proportionality in both time bounds depends on D and the maximum degree of the polynomials defining the input curves.As an application, we revisit the problem of eliminating depth cycles among non-vertical lines in 3-space, recently studied by Aronov and Sharir (2018), and show an algorithm that cuts n such lines into O(n 3/2+ε ) pieces that are depth-cycle free, for any ε > 0. The algorithm runs in O(n 3/2+ε ) time, which is a considerable improvement over the previously known algorithms.

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Cutting algebraic curves into pseudo-segments and applications

Cited by 33 publications

References 35 publications

Breaking the 3/2 Barrier for Unit Distances in Three Dimensions

Breaking the 3/2 Barrier for Unit Distances in Three Dimensions

Planar point sets determine many pairwise crossing segments

Constructive Polynomial Partitioning for Algebraic Curves in ℝ³ with Applications

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Cutting algebraic curves into pseudo-segments and applications

Cited by 33 publications

References 35 publications

Breaking the 3/2 Barrier for Unit Distances in Three Dimensions

Breaking the 3/2 Barrier for Unit Distances in Three Dimensions

Planar point sets determine many pairwise crossing segments

Constructive Polynomial Partitioning for Algebraic Curves in ℝ3 with Applications

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Constructive Polynomial Partitioning for Algebraic Curves in ℝ³ with Applications