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Incremental k-Lowest Planes and Planar k-Nearest Neighbor with Optimal Query Time

Authors John Iacono , Yakov Nekrich , Martin P. Seybold

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LIPIcs.ICALP.2026.113.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Data structures design and analysis
Keywords
  • Data Structures
  • Dynamic Data Structures
  • k Nearest-Neighbor Queries

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Abstract

In a set of planes in ℝ³, the k-lowest planes query asks for the k lowest planes pierced by a vertical line q. In this paper we describe a semi-dynamic insertion-only data structure that answers k-lowest planes queries in optimal O(log n+k) time. Our data structure uses O(n) space, where n is the number of stored planes, and supports insertions in O(log⁸ n) amortized time. This result provides the first query optimal data structures for several fundamental problems: - An insertion-only structure that answers 3D halfspace range reporting queries on a set of n points in O(log n+k) time, where k is the number of reported points. - An insertion-only structure that answers 3D vertical ray shooting queries on a set of n planes in O(log n+k) time, where k is the number of reported planes. - An insertion-only structure that answers planar k-nearest neighbor queries in O(log n + k) time for any prescribed k (specified at query time). - An insertion-only structure that answers planar circular range reporting queries in time O(log n + k), where k is the number of reported points. For all of the above problems the query bound O(log n + k) is optimal, even in the static scenario. All of the above structures use linear O(n) space and support insertions in O(log⁸ n) amortized time. 
We also obtain a query optimal static structure for a 4D problem. That is, one can compute in near-linear time a data structure that answers weighted halfspace range reporting queries in O(log n + k) time, where k is the number of reported points. The structure uses near-linear space.

Cite As Get BibTex

John Iacono, Yakov Nekrich, and Martin P. Seybold. Incremental k-Lowest Planes and Planar k-Nearest Neighbor with Optimal Query Time. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 113:1-113:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.113

Author Details

John Iacono
  • Université libre de Bruxelles, Belgium
Yakov Nekrich
  • Michigan Technological University, Houghton, MI, USA
Martin P. Seybold
  • Faculty of Computer Science, Theory and Applications of Algorithms, University of Vienna, Austria
  • Université libre de Bruxelles, Belgium

Funding

Work supported by the Fonds de la Recherche Scientifique - FNRS. Supported by the National Science Foundation under NSF grant 2203278. This research was funded in whole or in part by the Austrian Science Fund (FWF) 10.55776/PAT2190825.

Acknowledgements

Many thanks to all contributors of Vorosketch and Ipe.

References

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