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A Fast Algorithm for Computing a Planar Support for Non-Piercing Rectangles

Authors Ambar Pal , Rajiv Raman , Saurabh Ray, Karamjeet Singh

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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Algorithms
  • Hypergraphs
  • Computational Geometry
  • Visualization

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Abstract

For a hypergraph ℋ = (X,ℰ) a support is a graph G on X such that for each E ∈ ℰ, the induced subgraph of G on the elements in E is connected. If G is planar, we call it a planar support. A set of axis parallel rectangles ℛ forms a non-piercing family if for any R₁, R₂ ∈ ℛ, R₁⧵R₂ is connected.
Given a set P of n points in ℝ² and a set ℛ of m non-piercing axis-aligned rectangles, we give an algorithm for computing a planar support for the hypergraph (P,ℛ) in O(nlog² n + (n+m)log m) time, where each R ∈ ℛ defines a hyperedge consisting of all points of P contained in R.

Cite As Get BibTex

Ambar Pal, Rajiv Raman, Saurabh Ray, and Karamjeet Singh. A Fast Algorithm for Computing a Planar Support for Non-Piercing Rectangles. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 53:1-53:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ISAAC.2024.53

Author Details

Ambar Pal
  • Johns Hopkins University, Baltime, MD, USA
Rajiv Raman
  • Indraprastha Institute of Information Technology Delhi, New Delhi, India
Saurabh Ray
  • NYU Abu Dhabi, UAE
Karamjeet Singh
  • Indraprastha Institute of Information Technology Delhi, New Delhi, India

References

  1. Emmanuelle Anceaume, Maria Gradinariu, Ajoy Kumar Datta, Gwendal Simon, and Antonino Virgillito. A semantic overlay for self-peer-to-peer publish/subscribe. In 26th IEEE International Conference on Distributed Computing Systems (ICDCS'06), pages 22-22. IEEE, 2006. Google Scholar
  2. Daniel Antunes, Claire Mathieu, and Nabil H. Mustafa. Combinatorics of local search: An optimal 4-local Hall’s theorem for planar graphs. In 25th Annual European Symposium on Algorithms, ESA 2017, September 4-6, 2017, Vienna, Austria, pages 8:1-8:13, 2017. URL: https://doi.org/10.4230/LIPICS.ESA.2017.8.
  3. Roberto Baldoni, Roberto Beraldi, Vivien Quema, Leonardo Querzoni, and Sara Tucci-Piergiovanni. Tera: topic-based event routing for peer-to-peer architectures. In Proceedings of the 2007 inaugural international conference on Distributed event-based systems, pages 2-13, 2007. Google Scholar
  4. Roberto Baldoni, Roberto Beraldi, Leonardo Querzoni, and Antonino Virgillito. Efficient publish/subscribe through a self-organizing broker overlay and its application to SIENA. The Computer Journal, 50(4):444-459, 2007. URL: https://doi.org/10.1093/COMJNL/BXM002.
  5. Aniket Basu Roy, Sathish Govindarajan, Rajiv Raman, and Saurabh Ray. Packing and covering with non-piercing regions. Discrete & Computational Geometry, 2018. Google Scholar
  6. Sergey Bereg, Krzysztof Fleszar, Philipp Kindermann, Sergey Pupyrev, Joachim Spoerhase, and Alexander Wolff. Colored non-crossing Euclidean Steiner forest. In Algorithms and Computation: 26th International Symposium, ISAAC 2015, Nagoya, Japan, December 9-11, 2015, Proceedings, pages 429-441. Springer, 2015. URL: https://doi.org/10.1007/978-3-662-48971-0_37.
  7. Sergey Bereg, Minghui Jiang, Boting Yang, and Binhai Zhu. On the red/blue spanning tree problem. Theoretical computer science, 412(23):2459-2467, 2011. URL: https://doi.org/10.1016/J.TCS.2010.10.038.
  8. Ulrik Brandes, Sabine Cornelsen, Barbara Pampel, and Arnaud Sallaberry. Blocks of hypergraphs: applied to hypergraphs and outerplanarity. In Combinatorial Algorithms: 21st International Workshop, IWOCA 2010, London, UK, July 26-28, 2010, Revised Selected Papers 21, pages 201-211. Springer, 2011. URL: https://doi.org/10.1007/978-3-642-19222-7_21.
  9. Ulrik Brandes, Sabine Cornelsen, Barbara Pampel, and Arnaud Sallaberry. Path-based supports for hypergraphs. Journal of Discrete Algorithms, 14:248-261, 2012. URL: https://doi.org/10.1016/J.JDA.2011.12.009.
  10. Gerth Stølting Brodal and Konstantinos Tsakalidis. Dynamic planar range maxima queries. In International Colloquium on Automata, Languages, and Programming, pages 256-267. Springer, 2011. URL: https://doi.org/10.1007/978-3-642-22006-7_22.
  11. Kevin Buchin, Marc J van Kreveld, Henk Meijer, Bettina Speckmann, and KAB Verbeek. On planar supports for hypergraphs. Journal of Graph Algorithms and Applications, 15(4):533-549, 2011. URL: https://doi.org/10.7155/JGAA.00237.
  12. Timothy M. Chan and Sariel Har-Peled. Approximation algorithms for maximum independent set of pseudo-disks. Discret. Comput. Geom., 48(2):373-392, 2012. URL: https://doi.org/10.1007/S00454-012-9417-5.
  13. Raphaël Chand and Pascal Felber. Semantic peer-to-peer overlays for publish/subscribe networks. In Euro-Par 2005 Parallel Processing: 11th International Euro-Par Conference, Lisbon, Portugal, August 30-September 2, 2005. Proceedings 11, pages 1194-1204. Springer, 2005. URL: https://doi.org/10.1007/11549468_130.
  14. Gregory Chockler, Roie Melamed, Yoav Tock, and Roman Vitenberg. Constructing scalable overlays for pub-sub with many topics. In Proceedings of the twenty-sixth annual ACM symposium on Principles of distributed computing, pages 109-118, 2007. URL: https://doi.org/10.1145/1281100.1281118.
  15. Vincent Cohen-Addad and Claire Mathieu. Effectiveness of local search for geometric optimization. In Proceedings of the Thirty-first International Symposium on Computational Geometry, SoCG '15, pages 329-343, Dagstuhl, Germany, 2015. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPICS.SOCG.2015.329.
  16. Ding-Zhu Du. An optimization problem on graphs. Discrete applied mathematics, 14(1):101-104, 1986. URL: https://doi.org/10.1016/0166-218X(86)90010-7.
  17. Ding-Zhu Du and Dean F Kelley. On complexity of subset interconnection designs. Journal of Global Optimization, 6(2):193-205, 1995. URL: https://doi.org/10.1007/BF01096768.
  18. Ding-Zhu Du and Zevi Miller. Matroids and subset interconnection design. SIAM journal on discrete mathematics, 1(4):416-424, 1988. URL: https://doi.org/10.1137/0401042.
  19. Frédéric Havet, Dorian Mazauric, Viet-Ha Nguyen, and Rémi Watrigant. Overlaying a hypergraph with a graph with bounded maximum degree. Discrete Applied Mathematics, 319:394-406, 2022. URL: https://doi.org/10.1016/J.DAM.2022.05.022.
  20. Jun Hosoda, Juraj Hromkovič, Taisuke Izumi, Hirotaka Ono, Monika Steinová, and Koichi Wada. On the approximability and hardness of minimum topic connected overlay and its special instances. Theoretical Computer Science, 429:144-154, 2012. URL: https://doi.org/10.1016/J.TCS.2011.12.033.
  21. Ferran Hurtado, Matias Korman, Marc van Kreveld, Maarten Löffler, Vera Sacristán, Akiyoshi Shioura, Rodrigo I Silveira, Bettina Speckmann, and Takeshi Tokuyama. Colored spanning graphs for set visualization. Computational Geometry, 68:262-276, 2018. URL: https://doi.org/10.1016/J.COMGEO.2017.06.006.
  22. David S Johnson and Henry O Pollak. Hypergraph planarity and the complexity of drawing Venn diagrams. Journal of graph theory, 11(3):309-325, 1987. URL: https://doi.org/10.1002/JGT.3190110306.
  23. Ephraim Korach and Michal Stern. The clustering matroid and the optimal clustering tree. Mathematical Programming, 98:385-414, 2003. URL: https://doi.org/10.1007/S10107-003-0410-X.
  24. Erik Krohn, Matt Gibson, Gaurav Kanade, and Kasturi Varadarajan. Guarding terrains via local search. Journal of Computational Geometry, 5(1):168-178, 2014. URL: https://doi.org/10.20382/JOCG.V5I1A9.
  25. Nabil H Mustafa and Saurabh Ray. Improved results on geometric hitting set problems. Discrete & Computational Geometry, 44(4):883-895, 2010. URL: https://doi.org/10.1007/S00454-010-9285-9.
  26. Melih Onus and Andréa W Richa. Minimum maximum-degree publish-subscribe overlay network design. IEEE/ACM Transactions on Networking, 19(5):1331-1343, 2011. URL: https://doi.org/10.1109/TNET.2011.2144999.
  27. Rajiv Raman and Saurabh Ray. Constructing planar support for non-piercing regions. Discrete & Computational Geometry, 64(3):1098-1122, 2020. URL: https://doi.org/10.1007/S00454-020-00216-W.
  28. Rajiv Raman and Saurabh Ray. On the geometric set multicover problem. Discret. Comput. Geom., 68(2):566-591, 2022. URL: https://doi.org/10.1007/s00454-022-00402-y.
  29. Rajiv Raman and Karamjeet Singh. On hypergraph supports, 2024. URL: https://arxiv.org/abs/2303.16515.
  30. AA Voloshina and VZ Feinberg. Planarity of hypergraphs. In Doklady Akademii Nauk Belarusi, volume 28, pages 309-311. Akademii Nauk Belarusi F Scorina Pr 66, room 403, Minsk, Byelarus 220072, 1984. Google Scholar
  31. TRS Walsh. Hypermaps versus bipartite maps. Journal of Combinatorial Theory, Series B, 18(2):155-163, 1975. Google Scholar
  32. Alexander Aleksandrovich Zykov. Hypergraphs. Russian Mathematical Surveys, 29(6):89, 1974. Google Scholar
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