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Incremental Planar Nearest Neighbor Queries with Optimal Query Time

Authors John Iacono , Yakov Nekrich

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

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Abstract

In this paper we show that two-dimensional nearest neighbor queries can be answered in optimal O(log n) time while supporting insertions in O(log^{1+ε} n) time. No previous data structure was known that supports O(log n)-time queries and polylog-time insertions. In order to achieve logarithmic queries our data structure uses a new technique related to fractional cascading that leverages the inherent geometry of this problem. Our method can be also used in other semi-dynamic scenarios.

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John Iacono and Yakov Nekrich. Incremental Planar Nearest Neighbor Queries with Optimal Query Time. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 59:1-59:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.SoCG.2025.59

Author Details

John Iacono
  • Université libre de Bruxelles, Belgium
Yakov Nekrich
  • Michigan Technological University, Houghton, MI, USA

Funding

  • Iacono, John: Research supported by the Fonds de la Recherche Scientifique - FNRS.
  • Nekrich, Yakov: Supported by the National Science Foundation under NSF grant 2203278.

References

  1. Pankaj K. Agarwal and Jirí Matousek. Dynamic half-space range reporting and its applications. Algorithmica, 13(4):325-345, 1995. URL: https://doi.org/10.1007/BF01293483.
  2. Sarah R. Allen, Luis Barba, John Iacono, and Stefan Langerman. Incremental Voronoi diagrams. Discrete and Computational Geometry, 58(4):822-848, 2017. URL: https://doi.org/10.1007/s00454-017-9943-2.
  3. Jon Louis Bentley and James B. Saxe. Decomposable searching problems I: static-to-dynamic transformation. Journal of Algorithms, 1(4):301-358, 1980. URL: https://doi.org/10.1016/0196-6774(80)90015-2.
  4. Timothy M. Chan. A dynamic data structure for 3-d convex hulls and 2-d nearest neighbor queries. Journal of the ACM, 57(3):16:1-16:15, 2010. URL: https://doi.org/10.1145/1706591.1706596.
  5. Timothy M. Chan. Dynamic geometric data structures via shallow cuttings. Discrete and Computational Geometry, 64(4):1235-1252, 2020. URL: https://doi.org/10.1007/s00454-020-00229-5.
  6. Timothy M. Chan and Konstantinos Tsakalidis. Optimal deterministic algorithms for 2-d and 3-d shallow cuttings. Discrete and Computational Geometry, 56(4):866-881, 2016. URL: https://doi.org/10.1007/s00454-016-9784-4.
  7. Bernard Chazelle and Leonidas J. Guibas. Fractional cascading: I. A data structuring technique. Algorithmica, 1(2):133-162, 1986. URL: https://doi.org/10.1007/BF01840440.
  8. Bernard Chazelle and Ding Liu. Lower bounds for intersection searching and fractional cascading in higher dimension. Journal of Computing & Systems Sciences, 68(2):269-284, 2004. URL: https://doi.org/10.1016/j.jcss.2003.07.003.
  9. Kenneth L. Clarkson, Kurt Mehlhorn, and Raimund Seidel. Four results on randomized incremental constructions. Computational Geometry, 3:185-212, 1993. URL: https://doi.org/10.1016/0925-7721(93)90009-U.
  10. Mark de Berg, Otfried Cheong, Marc J. van Kreveld, and Mark H. Overmars. Computational geometry: algorithms and applications, 3rd Edition. Springer, 2008. URL: https://doi.org/10.1007/978-3-540-77974-2.
  11. Olivier Devillers, Stefan Meiser, and Monique Teillaud. Fully dynamic Delaunay triangulation in logarithmic expected time per operation. Computational Geometry, 2:55-80, 1992. URL: https://doi.org/10.1016/0925-7721(92)90025-N.
  12. David P. Dobkin and Subhash Suri. Maintenance of geometric extrema. Journal of the ACM, 38(2):275-298, 1991. URL: https://doi.org/10.1145/103516.103518.
  13. Greg N. Frederickson. Fast algorithms for shortest paths in planar graphs, with applications. SIAM Journal on Computing, 16(6):1004-1022, 1987. URL: https://doi.org/10.1137/0216064.
  14. Michael T. Goodrich. Planar separators and parallel polygon triangulation. Journal of Computer and Systems Sciences, 51(3):374-389, 1995. URL: https://doi.org/10.1006/jcss.1995.1076.
  15. John Iacono and Yakov Nekrich. Incremental planar nearest neighbor queries with optimal query time, 2025. URL: https://arxiv.org/abs/2504.07366.
  16. Haim Kaplan, Wolfgang Mulzer, Liam Roditty, Paul Seiferth, and Micha Sharir. Dynamic planar Voronoi diagrams for general distance functions and their algorithmic applications. Discrete and Computational Geometry, 64(3):838-904, 2020. URL: https://doi.org/10.1007/s00454-020-00243-7.
  17. David G. Kirkpatrick. Optimal search in planar subdivisions. SIAM Journal on Computing, 12(1):28-35, 1983. URL: https://doi.org/10.1137/0212002.
  18. Philip N. Klein, Shay Mozes, and Christian Sommer. Structured recursive separator decompositions for planar graphs in linear time. In Symposium on Theory of Computing (STOC), pages 505-514, 2013. URL: https://doi.org/10.1145/2488608.2488672.
  19. Joseph S.B. Mitchell and Wolfgang Mulzer. Proximity algorithms. In Jacob E. Goodman, Joseph O'Rourke, and Csaba D. Tóth, editors, Handbook of Discrete and Computational Geometry, chapter 32, pages 849-874. CRC Press, 2017. Google Scholar
  20. Ketan Mulmuley. Computational geometry - An introduction through randomized algorithms. Prentice Hall, 1994. Google Scholar
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