几何支配集——三线问题的最小形式

IF 0.4 4区 计算机科学 Q4 MATHEMATICS
Oswin Aichholzer , David Eppstein , Eva-Maria Hainzl
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引用次数: 2

摘要

我们考虑众所周知的三线问题的一个最小化变体,即几何控制集问题:n×n网格中的最小点数是多少,使得每个网格点与该集中的两个点位于一条公共线上?我们给出了Ω(n2/3)点的下界,并提供了大小为2°n/2°的构造上界。如果要求支配集的点处于一般位置,我们为尺寸达到12×12的网格提供了最优解。对于任意n,一般位置的点的当前最佳上界仍然是明显的2n。最后,我们讨论了离散环面上的问题,其中我们证明了O((nlog)的上界⁡n) 1/2)。对于n偶数或3的倍数,我们甚至可以显示4的常数上界。我们还提到了一些悬而未决的问题以及该问题的一些进一步变化。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Geometric dominating sets - a minimum version of the No-Three-In-Line Problem

We consider a minimizing variant of the well-known No-Three-In-Line Problem, the Geometric Dominating Set Problem: What is the smallest number of points in an n×n grid such that every grid point lies on a common line with two of the points in the set? We show a lower bound of Ω(n2/3) points and provide a constructive upper bound of size 2n/2. If the points of the dominating sets are required to be in general position we provide optimal solutions for grids of size up to 12×12. For arbitrary n the currently best upper bound for points in general position remains the obvious 2n. Finally, we discuss the problem on the discrete torus where we prove an upper bound of O((nlogn)1/2). For n even or a multiple of 3, we can even show a constant upper bound of 4. We also mention a number of open questions and some further variations of the problem.

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来源期刊
CiteScore
1.60
自引率
16.70%
发文量
43
审稿时长
>12 weeks
期刊介绍: Computational Geometry is a forum for research in theoretical and applied aspects of computational geometry. The journal publishes fundamental research in all areas of the subject, as well as disseminating information on the applications, techniques, and use of computational geometry. Computational Geometry publishes articles on the design and analysis of geometric algorithms. All aspects of computational geometry are covered, including the numerical, graph theoretical and combinatorial aspects. Also welcomed are computational geometry solutions to fundamental problems arising in computer graphics, pattern recognition, robotics, image processing, CAD-CAM, VLSI design and geographical information systems. Computational Geometry features a special section containing open problems and concise reports on implementations of computational geometry tools.
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