在点集中的深层小组
Stefan Langerman1, Marcelo Mydlarz2, Emo Welzl3
1Département d'informatique, Université Libre de Bruxelles, Brussels, Belgium.
概括
这项研究介绍了k-deep点群,即点的子集,其中点的对是k-deep. 它在点集中为k-deep的集群大小设定了界限,特别关注减半集群.
科学领域:
- 计算几何学的计算几何学
- 组合几何组合几何学
- 离散数学 离散数学 离散数学
背景情况:
- 介绍了一个集合P内的k-深点对的概念.
- 定义一个k-deep的群组为P的子集,其中所有对都是k-deep.
研究的目的:
- 为了在一组n个点中建立k-deep小组的大小的严格界限.
- 调查减半小组的特性和存在 (k-deep小组的特殊情况).
主要方法:
- 对点集及其子集的几何性质的分析.
- 对k-deep群体大小的下限和上限的推导.
- 检查具体情况,包括一般位置和凸起位置的点.
主要成果:
- 在一般位置点集中的k-深点群的大小的下界是n/2的.
- 在任何点集合中,一个k-deep点群的大小的上限是n^2/4点群.
- 对于"k = n/2" (分半小组) 的具体界限,包括对给定分半小组大小的总点数的必要条件.
结论:
- 该研究为k-deep集团大小提供了严格的界限,证明了它们的存在和局限性.
- 将小组减少一半的结果提供了对其结构和与整体点集的关系的见解.
- 这些发现有助于理解点集中的几何结构.
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