LIPIcs.SoCG.2024.60.pdf
- Filesize: 0.92 MB
- 14 pages
Document
Near Optimal Locality Sensitive Orderings in Euclidean Space
Authors
Zhimeng Gao 
,
Sariel Har-Peled
-
Part of:
Volume:
40th International Symposium on Computational Geometry (SoCG 2024)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Symposium on Computational Geometry (SoCG) - License:
Creative Commons Attribution 4.0 International license
- Publication Date: 2024-06-06
File
Document Identifiers
Author Details
- Department of Computer Science and Engineering, Hong Kong University of Science and Technology, Kowloon, Hong Kong
Cite AsGet BibTex
Zhimeng Gao and Sariel Har-Peled. Near Optimal Locality Sensitive Orderings in Euclidean Space. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 60:1-60:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SoCG.2024.60
https://doi.org/10.4230/LIPIcs.SoCG.2024.60
Abstract
For a parameter ε ∈ (0,1), a set of ε-locality-sensitive orderings (LSOs) has the property that for any two points, p,q ∈ [0,1)^d, there exist an order in the set such that all the points between p and q (in the order) are ε-close to either p or q. Since the original construction of LSOs can not be (significantly) improved, we present a construction of modified LSOs, that yields a smaller set, while preserving their usefulness. Specifically, the resulting set of LSOs has size M = O(ℰ^{d-1} log ℰ), where ℰ = 1/ε. This improves over previous work by a factor of ℰ, and is optimal up to a factor of log ℰ.
As a consequence we get a flotilla of improved dynamic geometric algorithms, such as maintaining bichromatic closest pair, and spanners, among others. In particular, for geometric dynamic spanners the new result matches (up to the aforementioned log ℰ factor) the lower bound, Specifically, this is a near-optimal simple dynamic data-structure for maintaining spanners under insertions and deletions.
Subject Classification
ACM Subject Classification
- Theory of computation → Computational geometry
Keywords
- Orderings
- approximation
Metrics
- Access Statistics
-
Total Accesses (updated on a weekly basis)
0PDF Downloads
References
-
Brian Alspach. The wonderful Walecki construction. Bull. Inst. Combin. Appl., 52:7-20, 2008.
- Kevin Buchin, Sariel Har-Peled, and Dániel Oláh. A spanner for the day after. Discrete Comput. Geom., 64(4):1167-1191, 2020. URL: https://doi.org/10.1007/s00454-020-00228-6.
- Kevin Buchin, Sariel Har-Peled, and Dániel Oláh. Sometimes reliable spanners of almost linear size. J. Comput. Geom., 13(1):178-196, 2022. URL: https://doi.org/10.20382/jocg.v13i1a6.
- P. B. Callahan and S. R. Kosaraju. A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields. J. Assoc. Comput. Mach., 42(1):67-90, 1995. URL: https://doi.org/10.1145/200836.200853.
- T. M. Chan. Approximate nearest neighbor queries revisited. Discrete Comput. Geom., 20(3):359-373, 1998. URL: https://doi.org/10.1007/PL00009390.
- Timothy M. Chan, Sariel Har-Peled, and Mitchell Jones. On locality-sensitive orderings and their applications. SIAM J. Comput., 49(3):583-600, 2020. URL: https://doi.org/10.1137/19M1246493.
- Arnold Filtser. Labeled nearest neighbor search and metric spanners via locality sensitive orderings. In Erin W. Chambers and Joachim Gudmundsson, editors, Proc. 39th Int. Annu. Sympos. Comput. Geom. (SoCG), volume 258 of LIPIcs, pages 33:1-33:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.SoCG.2023.33.
- Arnold Filtser and Hung Le. Locality-sensitive orderings and applications to reliable spanners. In Proc. 54rd Annu. ACM Sympos. Theory Comput. (STOC), STOC 2022, pages 1066-1079, New York, NY, USA, 2022. Association for Computing Machinery. URL: https://doi.org/10.1145/3519935.3520042.
- Zhimeng Gao and Sariel Har-Peled. Almost optimal locality sensitive orderings in euclidean space. CoRR, abs/2310.12792, 2023. URL: https://doi.org/10.48550/arXiv.2310.12792.
- S. Har-Peled. Geometric Approximation Algorithms, volume 173 of Math. Surveys & Monographs. Amer. Math. Soc., Boston, MA, USA, 2011. URL: https://doi.org/10.1090/surv/173.
- Sariel Har-Peled and Benjamin Raichel. Net and prune: A linear time algorithm for Euclidean distance problems. J. Assoc. Comput. Mach., 62(6):44:1-44:35, December 2015. URL: https://doi.org/10.1145/2831230.
- Sándor Kisfaludi-Bak and Geert van Wordragen. A quadtree for hyperbolic space. CoRR, abs/2305.01356, 2023. URL: https://doi.org/10.48550/arXiv.2305.01356.
- Hung Le and Shay Solomon. Truly optimal euclidean spanners, 2021. URL: https://arxiv.org/abs/1904.12042.