论几何设置中避免诱导双斜时的事件数

IF 0.6 3区 数学 Q4 COMPUTER SCIENCE, THEORY & METHODS Discrete & Computational Geometry Pub Date : 2024-05-23 DOI:10.1007/s00454-024-00648-8
Timothy M. Chan, Sariel Har-Peled
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引用次数: 0

摘要

给定一组点(P)和一组区域(O),一个入射是一对((p,\mathcalligra {o})\in P\times\mathcal {O}\),使得(p\in \mathcalligra {o}\)。我们在组合几何中的一个经典问题上得到了许多新结果:在某些限制条件下)发生数是多少?我们证明了在\(\mathbb {R}^d\)中,如果没有k个盒子包含k个公共点,即如果点和盒子之间的入射图不包含\(K_{k,k}\)作为子图,那么n个点和\(\mathbb {R}^d\)中n个轴平行的盒子之间的入射次数的界限是\(O\bigl ( k n(\log n/\log \log n)^{d-1} \bigr )\) 。与 Basit 等人(Forum Math Sigma 9:59, 2021)之前的工作相比,这个新约束在 \(d >2\) 时提高了 \(\log ^d n\) 的系数。此外,它还符合查泽尔(J ACM 37(2):200-212,1990)的工作中对(k=2)所暗示的下限,从而解决了点和盒的问题。我们还研究了问题的其他几个变体。对于半空间,使用浅切,我们得到了二维和三维的线性约束。我们还提出了低联合复杂度形状的线性(或接近线性)约束,如伪圆盘和胖三角形。
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On the Number of Incidences When Avoiding an Induced Biclique in Geometric Settings

Given a set of points \(P\) and a set of regions \(\mathcal {O}\), an incidence is a pair \((p,\mathcalligra {o}) \in P\times \mathcal {O}\) such that \(p\in \mathcalligra {o}\). We obtain a number of new results on a classical question in combinatorial geometry: What is the number of incidences (under certain restrictive conditions)? We prove a bound of \(O\bigl ( k n(\log n/\log \log n)^{d-1} \bigr )\) on the number of incidences between n points and n axis-parallel boxes in \(\mathbb {R}^d\), if no k boxes contain k common points, that is, if the incidence graph between the points and the boxes does not contain \(K_{k,k}\) as a subgraph. This new bound improves over previous work, by Basit et al. (Forum Math Sigma 9:59, 2021), by more than a factor of \(\log ^d n\) for \(d >2\). Furthermore, it matches a lower bound implied by the work of Chazelle (J ACM 37(2):200–212, 1990), for \(k=2\), thus settling the question for points and boxes. We also study several other variants of the problem. For halfspaces, using shallow cuttings, we get a linear bound in two and three dimensions. We also present linear (or near linear) bounds for shapes with low union complexity, such as pseudodisks and fat triangles.

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来源期刊
Discrete & Computational Geometry
Discrete & Computational Geometry 数学-计算机:理论方法
CiteScore
1.80
自引率
12.50%
发文量
99
审稿时长
6-12 weeks
期刊介绍: Discrete & Computational Geometry (DCG) is an international journal of mathematics and computer science, covering a broad range of topics in which geometry plays a fundamental role. It publishes papers on such topics as configurations and arrangements, spatial subdivision, packing, covering, and tiling, geometric complexity, polytopes, point location, geometric probability, geometric range searching, combinatorial and computational topology, probabilistic techniques in computational geometry, geometric graphs, geometry of numbers, and motion planning.
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