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Convexity Helps Iterated Search in 3D

Authors Peyman Afshani , Yakov Nekrich , Frank Staals

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  • Theory of computation → Data structures design and analysis
Keywords
  • Data structures
  • range searching

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Abstract

Inspired by the classical fractional cascading technique [Bernard Chazelle and Leonidas J. Guibas, 1986; Bernard Chazelle and Leonidas J. Guibas, 1986], we introduce new techniques to speed up the following type of iterated search in 3D: The input is a graph 𝐆 with bounded degree together with a set H_v of 3D hyperplanes associated with every vertex of v of 𝐆. The goal is to store the input such that given a query point q ∈ ℝ³ and a connected subgraph 𝐇 ⊂ 𝐆, we can decide if q is below or above the lower envelope of H_v for every v ∈ 𝐇. We show that using linear space, it is possible to answer queries in roughly O(log n + |𝐇|√{log n}) time which improves trivial bound of O(|𝐇|log n) obtained by using planar point location data structures. Our data structure can in fact answer more general queries (it combines with shallow cuttings) and it even works when 𝐇 is given one vertex at a time. We show that this has a number of new applications and in particular, we give improved solutions to a set of natural data structure problems that up to our knowledge had not seen any improvements.
We believe this is a very surprising result because obtaining similar results for the planar point location problem was known to be impossible [Chazelle and Liu, 2004].

Cite As Get BibTex

Peyman Afshani, Yakov Nekrich, and Frank Staals. Convexity Helps Iterated Search in 3D. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 3:1-3:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.SoCG.2025.3

Author Details

Peyman Afshani
  • Department of Computer Science, Aarhus University, Denmark
Yakov Nekrich
  • Department of Computer Science, Michigan Technological University, Houghton, MI, USA
Frank Staals
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands

Funding

  • Afshani, Peyman: Supported by DFF (Danmarks Frie Forskningsfond) of Danish Council for Independent Research under grant ID 10.46540/3103-00334B.

References

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