Document Open Access Logo

Approximate Dynamic Nearest Neighbor Searching in a Polygonal Domain

Authors Joost van der Laan, Frank Staals , Lorenzo Theunissen

PDF

File

Thumbnail PDF
PDF
LIPIcs.SoCG.2026.69.pdf
  • Filesize: 1.11 MB
  • 16 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • dynamic data structure
  • nearest neighbor search
  • polygonal domain

Metrics

Abstract

We present efficient data structures for approximate nearest neighbor searching and approximate 2-point shortest path queries in a two-dimensional polygonal domain P with n vertices. Our goal is to store a dynamic set of m point sites S in P so that we can efficiently find a site s ∈ S closest to an arbitrary query point q. We will allow both insertions and deletions in the set of sites S. However, as even just computing the distance between an arbitrary pair of points q,s ∈ P requires a substantial amount of space, we allow for approximating the distances. Given a parameter ε > 0, we build an O(n/(ε)log n) space data structure that can compute a 1+ε-approximation of the distance between q and s in O((1/ε²)log n) time. Building on this, we then obtain an O((n+m)/ε log n + m/ε log m) space data structure that allows us to report a site s ∈ S so that the distance between query point q and s is at most (1+ε)-times the distance between q and its true nearest neighbor in O((1/ε²)log n + 1/(ε)log n log m + (1/ε)log² m) time. Our data structure supports updates in O((1/ε²)log n + (1/ε)log n log m + (1/ε)log² m) amortized time.

Cite As Get BibTex

Joost van der Laan, Frank Staals, and Lorenzo Theunissen. Approximate Dynamic Nearest Neighbor Searching in a Polygonal Domain. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 69:1-69:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.69

Author Details

Joost van der Laan
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands
Frank Staals
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands
Lorenzo Theunissen
  • Delft University of Technology, The Netherlands

References

  1. Pankaj K. Agarwal, Lars Arge, and Frank Staals. Improved dynamic geodesic nearest neighbor searching in a simple polygon. In Bettina Speckmann and Csaba D. Tóth, editors, 34th International Symposium on Computational Geometry, SoCG 2018, June 11-14, 2018, Budapest, Hungary, volume 99 of LIPIcs, pages 4:1-4:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.SOCG.2018.4.
  2. Pankaj K. Agarwal and Jiří Matoušek. Dynamic Half-Space Range Reporting and its Applications. Algorithmica, 13(4):325-345, 1995. URL: https://doi.org/10.1007/BF01293483.
  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. Mark de Berg, Otfried Cheong, Marc van Kreveld, and Mark Overmars. Computational Geometry: Algorithms and Applications. Springer-Verlag TELOS, Santa Clara, CA, USA, 3rd ed. edition, 2008. Google Scholar
  5. Sergio Cabello, Delia Garijo, Antonia Kalb, Fabian Klute, Irene Parada, and Rodrigo I. Silveira. Algorithms for distance problems in continuous graphs. In Pat Morin and Eunjin Oh, editors, 19th International Symposium on Algorithms and Data Structures, WADS 2025, August 11-15, 2025, York University, Toronto, Canada, volume 349 of LIPIcs, pages 13:1-13:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. URL: https://doi.org/10.4230/LIPIcs.WADS.2025.13.
  6. Timothy M. Chan. A dynamic data structure for 3-d convex hulls and 2-d nearest neighbor queries. J. ACM, 57(3):16:1-16:15, 2010. URL: https://doi.org/10.1145/1706591.1706596.
  7. Timothy M. Chan. Dynamic generalized closest pair: Revisiting eppstein’s technique. In Martin Farach-Colton and Inge Li Gørtz, editors, 3rd Symposium on Simplicity in Algorithms, SOSA 2020, Salt Lake City, UT, USA, January 6-7, 2020, pages 33-37. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611976014.6.
  8. Timothy M. Chan. Dynamic geometric data structures via shallow cuttings. Discret. Comput. Geom., 64(4):1235-1252, 2020. URL: https://doi.org/10.1007/S00454-020-00229-5.
  9. Bernard Chazelle and Leonidas J. Guibas. Visibility and intersection problems in plane geometry. Discrete & Computational Geometry, 4:551-581, 1989. URL: https://doi.org/10.1007/BF02187747.
  10. K. Clarkson. Approximation algorithms for shortest path motion planning. In Proceedings of the Nineteenth Annual ACM Symposium on Theory of Computing, STOC '87, pages 56-65, New York, NY, USA, 1987. Association for Computing Machinery. URL: https://doi.org/10.1145/28395.28402.
  11. Sarita de Berg, Tillmann Miltzow, and Frank Staals. Towards space efficient two-point shortest path queries in a polygonal domain. In Wolfgang Mulzer and Jeff M. Phillips, editors, 40th International Symposium on Computational Geometry, SoCG 2024, June 11-14, 2024, Athens, Greece, volume 293 of LIPIcs, pages 17:1-17:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.SOCG.2024.17.
  12. Sarita de Berg, Frank Staals, and Marc J. van Kreveld. The complexity of geodesic spanners. J. Comput. Geom., 15(1):21-65, 2024. URL: https://doi.org/10.20382/JOCG.V15I1A2.
  13. Leonidas J. Guibas and John Hershberger. Optimal shortest path queries in a simple polygon. J. Comput. Syst. Sci., 39(2):126-152, 1989. URL: https://doi.org/10.1016/0022-0000(89)90041-X.
  14. John Hershberger and Subhash Suri. An optimal algorithm for euclidean shortest paths in the plane. SIAM Journal on Computing, 28(6):2215-2256, 1999. URL: https://doi.org/10.1137/S0097539795289604.
  15. Haim Kaplan, Wolfgang Mulzer, Liam Roditty, Paul Seiferth, and Micha Sharir. Dynamic planar voronoi diagrams for general distance functions and their algorithmic applications. Discret. Comput. Geom., 64(3):838-904, 2020. URL: https://doi.org/10.1007/S00454-020-00243-7.
  16. Chih-Hung Liu. Nearly optimal planar k nearest neighbors queries under general distance functions. SIAM J. Comput., 51(3):723-765, 2022. URL: https://doi.org/10.1137/20M1388371.
  17. Eunjin Oh and Hee-Kap Ahn. Voronoi diagrams for a moderate-sized point-set in a simple polygon. Discret. Comput. Geom., 63(2):418-454, 2020. URL: https://doi.org/10.1007/S00454-019-00063-4.
  18. Neil Sarnak and Robert Endre Tarjan. Planar point location using persistent search trees. Commun. ACM, 29(7):669-679, 1986. URL: https://doi.org/10.1145/6138.6151.
  19. Mikkel Thorup. Compact oracles for approximate distances around obstacles in the plane. In Lars Arge, Michael Hoffmann, and Emo Welzl, editors, Algorithms - ESA 2007, pages 383-394, Berlin, Heidelberg, 2007. Springer Berlin Heidelberg. URL: https://doi.org/10.1007/978-3-540-75520-3_35.
  20. Joost van der Laan, Frank Staals, and Lorenzo Theunissen. Approximate dynamic nearest neighbor searching in a polygonal domain, 2026. URL: https://arxiv.org/abs/2603.11775.
  21. Haitao Wang. A new algorithm for euclidean shortest paths in the plane. J. ACM, 70(2):11:1-11:62, 2023. URL: https://doi.org/10.1145/3580475.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2026 Schloss Dagstuhl – LZI GmbH Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact