Document Open Access Logo

An O(n log n)-Time Approximation Scheme for Geometric Many-To-Many Matching

Authors Sayan Bandyapadhyay , Jie Xue

PDF
Thumbnail PDF

File

LIPIcs.SoCG.2024.12.pdf
  • Filesize: 0.81 MB
  • 15 pages

Document Identifiers

Author Details

Sayan Bandyapadhyay
  • Department of Computer Science, Portland State University, OR, USA
Jie Xue
  • Department of Computer Science, New York University Shanghai, China

Cite AsGet BibTex

Sayan Bandyapadhyay and Jie Xue. An O(n log n)-Time Approximation Scheme for Geometric Many-To-Many Matching. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 12:1-12:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SoCG.2024.12

Abstract

Geometric matching is an important topic in computational geometry and has been extensively studied over decades. In this paper, we study a geometric-matching problem, known as geometric many-to-many matching. In this problem, the input is a set S of n colored points in ℝ^d, which implicitly defines a graph G = (S,E(S)) where E(S) = {(p,q): p,q ∈ S have different colors}, and the goal is to compute a minimum-cost subset E^* ⊆ E(S) of edges that cover all points in S. Here the cost of E^* is the sum of the costs of all edges in E^*, where the cost of a single edge e is the Euclidean distance (or more generally, the L_p-distance) between the two endpoints of e. Our main result is a (1+ε)-approximation algorithm with an optimal running time O_ε(n log n) for geometric many-to-many matching in any fixed dimension, which works under any L_p-norm. This is the first near-linear approximation scheme for the problem in any d ≥ 2. Prior to this work, only the bipartite case of geometric many-to-many matching was considered in ℝ¹ and ℝ², and the best known approximation scheme in ℝ² takes O_ε(n^{1.5} ⋅ poly(log n)) time.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Design and analysis of algorithms
Keywords
  • many-to-many matching
  • geometric optimization
  • approximation algorithms

Metrics

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

Related Versions

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 (STOC), pages 1052-1065, 2022. Google Scholar
  2. Pankaj K Agarwal, Herbert Edelsbrunner, Otfried Schwarzkopf, and Emo Welzl. Euclidean minimum spanning trees and bichromatic closest pairs. Discrete & Computational Geometry, 6:407-422, 1991. Google Scholar
  3. Alok Aggarwal, Herbert Edelsbrunner, Prahakar Raghavan, and Prasoon Tiwari. Optimal time bounds for some proximity problems in the plane. Information Processing Letters, 42(1):55-60, 1992. Google Scholar
  4. Sanjeev Arora. Polynomial time approximation schemes for Euclidean TSP and other geometric problems. In Proceedings of the 37th Annual Symposium on Foundations of Computer Science (FOCS), pages 2-11. IEEE, 1996. Google Scholar
  5. Sanjeev Arora. Nearly linear time approximation schemes for Euclidean TSP and other geometric problems. In Proceedings of the 38th Annual Symposium on Foundations of Computer Science (FOCS), pages 554-563. IEEE, 1997. Google Scholar
  6. Sunil Arya, Gautam Das, David M Mount, Jeffrey S Salowe, and Michiel Smid. Euclidean spanners: short, thin, and lanky. In Proceedings of the 27th Annual ACM Symposium on Theory of Computing (STOC), pages 489-498, 1995. Google Scholar
  7. Brenda S Baker. Approximation algorithms for NP-complete problems on planar graphs. Journal of the ACM, 41(1):153-180, 1994. Google Scholar
  8. Sayan Bandyapadhyay, William Lochet, Daniel Lokshtanov, Saket Saurabh, and Jie Xue. Euclidean bottleneck steiner tree is fixed-parameter tractable. In Proceedings of the 35th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 699-711. SIAM, 2024. Google Scholar
  9. Sayan Bandyapadhyay, Anil Maheshwari, and Michiel Smid. Exact and approximation algorithms for many-to-many point matching in the plane. In 32nd International Symposium on Algorithms and Computation (ISAAC). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. Google Scholar
  10. Hiba Belhadi, Karima Akli-Astouati, Youcef Djenouri, and Jerry Chun-Wei Lin. Data mining-based approach for ontology matching problem. Applied Intelligence, 50:1204-1221, 2020. Google Scholar
  11. Amir Ben-Dor, Richard M Karp, Benno Schwikowski, and Ron Shamir. The restriction scaffold problem. In Proceedings of the 6th Annual International Conference on Computational Biology, pages 58-66, 2002. Google Scholar
  12. Marcus Brazil, Ronald L Graham, Doreen A Thomas, and Martin Zachariasen. On the history of the Euclidean steiner tree problem. Archive for history of exact sciences, 68:327-354, 2014. Google Scholar
  13. Timothy M Chan, Sariel Har-Peled, and Mitchell Jones. On locality-sensitive orderings and their applications. SIAM Journal on Computing, 49(3):583-600, 2020. Google Scholar
  14. Justin Colannino, Mirela Damian, Ferran Hurtado, Stefan Langerman, Henk Meijer, Suneeta Ramaswami, Diane L. Souvaine, and Godfried Toussaint. Efficient many-to-many point matching in one dimension. Graphs and Combinatorics, 23:169-178, 2007. Google Scholar
  15. Justin Colannino and Godfried Toussaint. Faster algorithms for computing distances between one-dimensional point sets. Proceedings of the XI Encuentros de Geometria Computacional, pages 189-198, 2005. Google Scholar
  16. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. Google Scholar
  17. Mark de Berg, Hans L Bodlaender, Sándor Kisfaludi-Bak, and Sudeshna Kolay. An ETH-tight exact algorithm for Euclidean TSP. In Proceedings of the 59th Annual Symposium on Foundations of Computer Science (FOCS), pages 450-461. IEEE, 2018. Google Scholar
  18. Thomas Eiter and Heikki Mannila. Distance measures for point sets and their computation. Acta informatica, 34(2):109-133, 1997. Google Scholar
  19. Jeff Erickson. On the relative complexities of some geometric problems. In Proceedings of the 7th Canadian Conference on Computational Geometry (CCCG), volume 95, pages 85-90, 1995. Google Scholar
  20. S. M. Ferdous, Alex Pothen, and Arif Khan. New approximation algorithms for minimum weighted edge cover. In Fredrik Manne, Peter Sanders, and Sivan Toledo, editors, Proceedings of the 8th SIAM Workshop on Combinatorial Scientific Computing, CSC 2018, Bergen, Norway, June 6-8, 2018, pages 97-108. SIAM, 2018. Google Scholar
  21. Michael L Fredman and Robert Endre Tarjan. Fibonacci heaps and their uses in improved network optimization algorithms. Journal of the ACM, 34(3):596-615, 1987. Google Scholar
  22. Harold N Gabow and Robert E Tarjan. Faster scaling algorithms for network problems. SIAM Journal on Computing, 18(5):1013-1036, 1989. Google Scholar
  23. Steven A Gallagher. A 4/3-approximation for Minimum Weight Edge Cover. PhD thesis, Purdue University, 2019. Google Scholar
  24. Sariel Har-Peled, Piotr Indyk, and Anastasios Sidiropoulos. Euclidean spanners in high dimensions. In Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 804-809. SIAM, 2013. Google Scholar
  25. Piotr Indyk. A near linear time constant factor approximation for Euclidean bichromatic matching (cost). In Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), volume 7, pages 39-42. SIAM, 2007. Google Scholar
  26. Haim Kaplan, Wolfgang Mulzer, Liam Roditty, Paul Seiferth, and Micha Sharir. Dynamic planar voronoi diagrams for general distance functions and their algorithmic applications. Discrete & Computational Geometry, 64:838-904, 2020. Google Scholar
  27. Judith Keijsper and Rudi Pendavingh. An efficient algorithm for minimum-weight bibranching. Journal of Combinatorial Theory, Series B, 73(2):130-145, 1998. Google Scholar
  28. Harold W Kuhn. Variants of the Hungarian method for assignment problems. Naval Research Logistics Quarterly, 3(4):253-258, 1956. Google Scholar
  29. Hung Le and Shay Solomon. Truly optimal Euclidean spanners. SIAM Journal on Computing, pages FOCS19-135 - FOCS19-199, 2022. Google Scholar
  30. William B March, Parikshit Ram, and Alexander G Gray. Fast Euclidean minimum spanning tree: algorithm, analysis, and applications. In Proceedings of the 16th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 603-612, 2010. Google Scholar
  31. Yaodong Ni. Fuzzy minimum weight edge covering problem. Applied mathematical modelling, 32(7):1327-1337, 2008. Google Scholar
  32. Robert Z Norman and Michael O Rabin. An algorithm for a minimum cover of a graph. Proceedings of the American Mathematical Society, 10(2):315-319, 1959. Google Scholar
  33. Sharath Raghvendra and Pankaj K. Agarwal. A near-linear time ε-approximation algorithm for geometric bipartite matching. Journal of the ACM, 67(3):18:1-18:19, 2020. Google Scholar
  34. Fatemeh Rajabi-Alni and Alireza Bagheri. An O(n²) algorithm for the limited-capacity many-to-many point matching in one dimension. Algorithmica, 76(2):381-400, 2016. Google Scholar
  35. Fatemeh Rajabi-Alni and Alireza Bagheri. A fast and efficient algorithm for many-to-many matching of points with demands in one dimension. arXiv preprint, 2019. URL: https://arxiv.org/abs/1904.05184.
  36. Fatemeh Rajabi-Alni and Alireza Bagheri. A faster algorithm for the limited-capacity many-to-many point matching in one dimension. arXiv preprint, 2019. URL: https://arxiv.org/abs/1904.03015.
  37. Petar Ristoski, Petar Petrovski, Peter Mika, and Heiko Paulheim. A machine learning approach for product matching and categorization. Semantic web, 9(5):707-728, 2018. Google Scholar
  38. Michael Ian Shamos and Dan Hoey. Closest-point problems. In 16th Annual Symposium on Foundations of Computer Science (FOCS), pages 151-162. IEEE, 1975. Google Scholar
  39. Warren D Smith. How to find steiner minimal trees in Euclidean d-space. Algorithmica, 7:137-177, 1992. Google Scholar
  40. Godfried Toussaint. The geometry of musical rhythm. In Japanese Conference on Discrete and Computational Geometry, pages 198-212. Springer, 2004. Google Scholar
  41. Godfried T Toussaint et al. A comparison of rhythmic similarity measures. In ISMIR, 2004. Google Scholar
  42. Pravin M Vaidya. Geometry helps in matching. SIAM Journal on Computing, 18(6):1201-1225, 1989. Google Scholar
  43. Pravin M Vaidya. An O(n log n) algorithm for the all-nearest-neighbors problem. Discrete & Computational Geometry, 4:101-115, 1989. Google Scholar
  44. Kasturi R Varadarajan. A divide-and-conquer algorithm for min-cost perfect matching in the plane. In Proceedings of the 39th Annual Symposium on Foundations of Computer Science (FOCS), pages 320-329. IEEE, 1998. Google Scholar
  45. Kasturi R Varadarajan and Pankaj K Agarwal. Approximation algorithms for bipartite and non-bipartite matching in the plane. In Proceedings of the 10th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), volume 99, pages 805-814. SIAM, 1999. Google Scholar
  46. Lee J White. Minimum covers of fixed cardinality in weighted graphs. SIAM Journal on Applied Mathematics, 21(1):104-113, 1971. Google Scholar
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact