Point and Line Nearest-Neighbor Searching in 3-Space

2026-10-02

AI Summary

This study addresses the inefficiency of nearest-neighbor search for points, lines, and triangles in three-dimensional space, where balancing storage requirements with query performance remains challenging. To overcome this limitation, the proposed method leverages combinatorial algorithms based on arrangements of three- and four-dimensional surfaces, combined with vertical decomposition techniques, to construct highly efficient data structures. This work breaks through existing theoretical bounds by achieving query times of $O^*(n^{1/2})$ or $O^*(1)$, thereby establishing an optimal trade-off between storage space and query complexity. The results significantly surpass the best previously known solutions, offering efficient query schemes under linear storage constraints as well as scalable structures that support constant-time queries.

Abstract

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.

Problem

• Nearest-neighbor searching
• 3-space
• Points and lines
• Data structures
• Computational geometry

Innovation

• nearest-neighbor searching
• data structures
• 3-space geometry
• surface arrangements
• vertical decompositions