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arXiv · 2610.03451

Point and Line Nearest-Neighbor Searching in 3-Space

Pankaj K. Agarwal·Esther Ezra·Micha Sharir
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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.

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BibTeXRIS

Pankaj K. Agarwal, Esther Ezra, Micha Sharir. 2026-10-02. Point and Line Nearest-Neighbor Searching in 3-Space. https://arxiv.org/abs/2610.03451

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