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Triangle Nearest-Neighbor Searching in 3-Space

Authors Pankaj K. Agarwal , Esther Ezra , Micha Sharir

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Subject Classification

ACM Subject Classification
  • Theory of computation
  • Theory of computation → Computational geometry
Keywords
  • line-point nearest neighbors
  • range searching
  • vertical decomposition in 3-space
  • test sets

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Abstract

We study various nearest-neighbor searching problems involving points, lines, segments and triangles in ℝ³. Among many results, we present a linear-size data structure for answering nearest-neighbor queries with lines amid n points in ℝ³, in O^*(n^{1/2}) time per query (where the O^*(⋅) notation hides subpolynomial factors). Our solution is based on parametric search, where the problem is reduced to range emptiness queries amid points in ℝ³ with cylindrical queries. For the latter problem we show that reporting all k points lying inside a cylinder query costs an additional term of O(k). We also study setups where both data and query objects are lines, segments, or triangles, and obtain improved solutions for the two extreme regimes of (near-)linear storage and of fast query time. These results also yield tradeoff bounds, where the cost of a query depends on the storage allocated to the structure. This work is a continuation of a recent work by the authors [Pankaj K. Agarwal et al., 2024].

Cite As Get BibTex

Pankaj K. Agarwal, Esther Ezra, and Micha Sharir. Triangle Nearest-Neighbor Searching in 3-Space. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 34:1-34:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.34

Author Details

Pankaj K. Agarwal
  • Department of Computer Science, Duke University, Durham, NC, USA
Esther Ezra
  • Department of Computer Science, Bar Ilan University, Ramat Gan, Israel
Micha Sharir
  • School of Computer Science, Tel Aviv University, Israel

Funding

  • Agarwal, Pankaj K.: Partially supported by NSF grants CCF-20-07556 and CCF-22-23870 and by a US-Israel Binational Science Foundation Grant 2022131.
  • Ezra, Esther: Partially supported by Israel Science Foundation Grant 800/22 and US-Israel Binational Science Foundation Grant 2022131.
  • Sharir, Micha: Partially supported by Israel Science Foundation Grant 495/23.

References

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