Document Open Access Logo

Optimal Euclidean Tree Covers

Authors Hsien-Chih Chang, Jonathan Conroy, Hung Le, Lazar Milenković, Shay Solomon, Cuong Than

PDF

File

Thumbnail PDF
PDF
LIPIcs.SoCG.2024.37.pdf
  • Filesize: 0.73 MB
  • 15 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Tree cover
  • spanner
  • Steiner point
  • routing
  • bounded-degree
  • quadtree
  • net-tree

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

A (1+e)-stretch tree cover of a metric space is a collection of trees, where every pair of points has a (1+e)-stretch path in one of the trees. The celebrated Dumbbell Theorem [Arya et al. STOC'95] states that any set of n points in d-dimensional Euclidean space admits a (1+e)-stretch tree cover with O_d(e^{-d} ⋅ log(1/e)) trees, where the O_d notation suppresses terms that depend solely on the dimension d. The running time of their construction is O_d(n log n ⋅ log(1/e)/e^d + n ⋅ e^{-2d}). Since the same point may occur in multiple levels of the tree, the maximum degree of a point in the tree cover may be as large as Ω(log Φ), where Φ is the aspect ratio of the input point set.
In this work we present a (1+e)-stretch tree cover with O_d(e^{-d+1} ⋅ log(1/e)) trees, which is optimal (up to the log(1/e) factor). Moreover, the maximum degree of points in any tree is an absolute constant for any d. As a direct corollary, we obtain an optimal {routing scheme} in low-dimensional Euclidean spaces. We also present a (1+e)-stretch Steiner tree cover (that may use Steiner points) with O_d(e^{(-d+1)/2} ⋅ log(1/e)) trees, which too is optimal. The running time of our two constructions is linear in the number of edges in the respective tree covers, ignoring an additive O_d(n log n) term; this improves over the running time underlying the Dumbbell Theorem.

Cite As Get BibTex

Hsien-Chih Chang, Jonathan Conroy, Hung Le, Lazar Milenković, Shay Solomon, and Cuong Than. Optimal Euclidean Tree Covers. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 37:1-37:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SoCG.2024.37

Author Details

Hsien-Chih Chang
  • Department of Computer Science, Dartmouth College, Hanover, NH, USA
Jonathan Conroy
  • Department of Computer Science, Dartmouth College, Hanover, NH, USA
Hung Le
  • Manning CICS, UMass Amherst, MA, USA
Lazar Milenković
  • Tel Aviv University, Israel
Shay Solomon
  • Tel Aviv University, Israel
Cuong Than
  • Manning CICS, UMass Amherst, MA, USA

Funding

Hung Le and Cuong Than are supported by the NSF CAREER Award No. CCF-2237288 and an NSF Grant No. CCF-2121952. Shay Solomon is funded by the European Union (ERC, DynOpt, 101043159). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. Shay Solomon is also supported by the Israel Science Foundation(ISF) grant No.1991/1. Shay Solomon and Lazar Milenković are supported by a grant from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel, and the United States National Science Foundation(NSF).

References

  1. Pankaj K Agarwal, Hsien-Chih Chang, Sharath Raghvendra, and Allen Xiao. Deterministic, near-linear ε-approximation algorithm for geometric bipartite matching. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, pages 1052-1065, 2022. Google Scholar
  2. Noga Alon, Richard M. Karp, David Peleg, and Douglas West. A graph-theoretic game and its application to the k-server problem. SIAM Journal on Computing, 24(1):78-100, 1995. URL: https://doi.org/10.1137/s0097539792224474.
  3. Sunil Arya, Gautam Das, David M Mount, Jeffrey S Salowe, and Michiel Smid. Euclidean spanners: short, thin, and lanky. In Proceedings of the twenty-seventh annual ACM symposium on Theory of computing, pages 489-498, 1995. Google Scholar
  4. Baruch Awerbuch and David Peleg. Routing with polynomial communication-space trade-off. SIAM J. Discret. Math., 5(2):151-162, May 1992. URL: https://doi.org/10.1137/0405013.
  5. Yair Bartal, Ora Nova Fandina, and Ofer Neiman. Covering metric spaces by few trees. Journal of Computer and System Sciences, 130:26-42, 2022. Google Scholar
  6. Sujoy Bhore and Csaba D Tóth. Euclidean steiner spanners: Light and sparse. SIAM Journal on Discrete Mathematics, 36(3):2411-2444, 2022. Google Scholar
  7. T.-H. Hubert Chan, Anupam Gupta, Bruce M. Maggs, and Shuheng Zhou. On hierarchical routing in doubling metrics. ACM Transactions on Algorithms (TALG), 12(4):1-22, 2016. Google Scholar
  8. Timothy M. Chan. Approximate nearest neighbor queries revisited. Discret. Comput. Geom., 20(3):359-373, 1998. Google Scholar
  9. Hsien-Chih Chang, Jonathan Conroy, Hung Le, Lazar Milenkovic, and Shay Solomon. Covering planar metrics (and beyond): O(1) trees suffice. arXiv preprint arXiv:2306.06215, 2023. Accepted to FOCS 2023. Google Scholar
  10. Hsien-Chih Chang, Jonathan Conroy, Hung Le, Lazar Milenkovic, Shay Solomon, and Cuong Than. Shortcut partitions in minor-free graphs: Steiner point removal, distance oracles, tree covers, and more. arXiv preprint arXiv:2308.00555, 2023. Accepted to SODA 2024. Google Scholar
  11. Feodor F Dragan, Chenyu Yan, and Irina Lomonosov. Collective tree spanners of graphs. SIAM Journal on Discrete Mathematics, 20(1):240-260, 2006. Google Scholar
  12. Jeff Erickson, Ivor Van Der Hoog, and Tillmann Miltzow. Smoothing the gap between np and er. SIAM Journal on Computing, pages FOCS20-102 - FOCS20-138, 2022. Google Scholar
  13. Steven Fortune and Christopher J. Van Wyk. Efficient exact arithmetic for computational geometry. In Proceedings of the Ninth Annual Symposium on Computational Geometry, SCG '93, pages 163-172, New York, NY, USA, 1993. Association for Computing Machinery. URL: https://doi.org/10.1145/160985.161015.
  14. Zhimeng Gao and Sariel Har-Peled. Almost optimal locality sensitive orderings in Euclidean space. CoRR, abs/2310.12792, 2023. Google Scholar
  15. L. Gottlieb and L. Roditty. Improved algorithms for fully dynamic geometric spanners and geometric routing. In Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'08, pages 591-600, 2008. URL: https://doi.org/10.5555/1347082.1347148.
  16. A. Gupta, A. Kumar, and R. Rastogi. Traveling with a Pez dispenser (or, routing issues in MPLS). In Proceedings 42nd IEEE Symposium on Foundations of Computer Science,FOCS' 01, 2001. URL: https://doi.org/10.1109/sfcs.2001.959889.
  17. Anupam Gupta, Robert Krauthgamer, and James R. Lee. Bounded geometries, fractals, and low-distortion embeddings. In FOCS, pages 534-543. IEEE Computer Society, 2003. Google Scholar
  18. Omri Kahalon, Hung Le, Lazar Milenković, and Shay Solomon. Can't see the forest for the trees: navigating metric spaces by bounded hop-diameter spanners. In Proceedings of the 2022 ACM Symposium on Principles of Distributed Computing, pages 151-162, 2022. Google Scholar
  19. Hung Le and Shay Solomon. Truly optimal Euclidean spanners. SIAM Journal on Computing, pages FOCS19-135 - FOCS19-199, 2022. Google Scholar
  20. Chen Li, Sylvain Pion, and Chee-Keng Yap. Recent progress in exact geometric computation. The Journal of Logic and Algebraic Programming, 64(1):85-111, 2005. Google Scholar
  21. Manor Mendel and Assaf Naor. Ramsey partitions and proximity data structures. In 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS' 06), 2006. URL: https://doi.org/10.1109/focs.2006.65.
  22. Giri Narasimhan and Michiel Smid. Geometric spanner networks. Cambridge University Press, 2007. Google Scholar
  23. David Salesin, Jorge Stolfi, and Leonidas Guibas. Epsilon geometry: building robust algorithms from imprecise computations. In Proceedings of the fifth annual symposium on Computational geometry, pages 208-217, 1989. Google Scholar
  24. Michiel Smid. Notes on binary dumbbell trees. Unpublished notes, 2012. URL: https://people.scs.carleton.ca/~michiel/dumbbelltrees.pdf.
  25. Mikkel Thorup and Uri Zwick. Approximate distance oracles. Journal of the ACM (JACM), 52(1):1-24, 2005. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact