三维空间中的点和线最近邻搜索

中文摘要 AI 辅助

本文提出了针对三维空间中点、线、线段和三角形之间最近邻（NN）搜索问题的数据结构，其性能显著优于先前已知的最佳结果。例如，我们提出了一种线性大小的数据结构，用于在三维空间中针对 n 个点回答以直线或线段为查询对象的 NN 查询，查询时间为 O^*(n^{1/2})（其中 O^*(\u00b7) 记号隐藏了次多项式因子）。我们还提出了一种大小为 O^*(n^4) 的数据结构，可在 O^*(1) 时间内回答此类查询。对于相反的问题，即在三维空间中针对 n 条直线、线段或三角形寻找查询点的最近邻，我们提出了一种线性大小的数据结构，查询时间为 O^*(n^{2/3})。这些结果构成了对先前解决方案的显著改进。我们在（近）线性存储和快速查询时间这两个极端情形下获得了改进的解决方案。这些结果还产生了查询时间与数据结构大小之间的权衡界限。我们的结果依赖于三维和四维空间中曲面排列的若干组合与算法结果，特别是作者近期建立的关于此类排列中子结构的垂直分解的结果。

英文摘要

This paper presents data structures for nearest-neighbor (NN) searching problems involving points, lines, segments, and triangles in 3-space, achieving significantly better performance than the previously best-known results for these problems. For example, we present a linear-size data structure for answering NN queries with lines or segments amid $n$ points in 3-space with $O^*(n^{1/2})$ query time (where the $O^*(\cdot)$ notation hides subpolynomial factors). We also present a data structure of $O^*(n^4)$ size that answers such queries in $O^*(1)$ time. For the converse problem, in which we seek the nearest neighbor of a query point amid $n$ lines, segments, or triangles in 3-space, we present a linear-size data structure with $O^*(n^{2/3})$ query time. These results constitute a significant improvement over previous solutions. We obtain improved solutions for the two extreme regimes of (near-)linear storage and of fast query time. These results also yield trade-off bounds between the query time and the size of the data structure. Our results rely on several combinatorial and algorithmic results on arrangements of surfaces in 3-space and 4-space, particularly on recent results on vertical decompositions of substructures in such arrangements established by the authors.

发表机构

• Duke University（杜克大学）
• Bar Ilan University（巴伊兰大学）
• Tel Aviv University（特拉维夫大学）

机构由 AI 辅助整理，请以论文原文为准。