Document Open Access Logo

Fully Dynamic Maximum Independent Sets of Disks in Polylogarithmic Update Time

Authors Sujoy Bhore , Martin Nöllenburg , Csaba D. Tóth , Jules Wulms

PDF
Thumbnail PDF

File

LIPIcs.SoCG.2024.19.pdf
  • Filesize: 1.03 MB
  • 16 pages

Document Identifiers

Author Details

Sujoy Bhore
  • Department of Computer Science and Engineering, Indian Institute of Technology Bombay, Mumbai, India
Martin Nöllenburg
  • Institute of Logic and Computation, Algorithms and Complexity Group, TU Wien, Austria
Csaba D. Tóth
  • Department of Mathematics, California State University Northridge, Los Angeles, CA, USA
  • Department of Computer Science, Tufts University, Medford, MA, USA
Jules Wulms
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands

Funding

  • Tóth, Csaba D.: Research was partially supported by the NSF award DMS-2154347.

Cite AsGet BibTex

Sujoy Bhore, Martin Nöllenburg, Csaba D. Tóth, and Jules Wulms. Fully Dynamic Maximum Independent Sets of Disks in Polylogarithmic Update Time. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 19:1-19:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SoCG.2024.19

Abstract

A fundamental question is whether one can maintain a maximum independent set (MIS) in polylogarithmic update time for a dynamic collection of geometric objects in Euclidean space. For a set of intervals, it is known that no dynamic algorithm can maintain an exact MIS in sublinear update time. Therefore, the typical objective is to explore the trade-off between update time and solution size. Substantial efforts have been made in recent years to understand this question for various families of geometric objects, such as intervals, hypercubes, hyperrectangles, and fat objects. We present the first fully dynamic approximation algorithm for disks of arbitrary radii in the plane that maintains a constant-factor approximate MIS in polylogarithmic expected amortized update time. Moreover, for a fully dynamic set of n unit disks in the plane, we show that a 12-approximate MIS can be maintained with worst-case update time O(log n), and optimal output-sensitive reporting. This result generalizes to fat objects of comparable sizes in any fixed dimension d, where the approximation ratio depends on the dimension and the fatness parameter. Further, we note that, even for a dynamic set of disks of unit radius in the plane, it is impossible to maintain O(1+ε)-approximate MIS in truly sublinear update time, under standard complexity assumptions. Our results build on two recent technical tools: (i) The MIX algorithm by Cardinal et al. (ESA 2021) that can smoothly transition from one independent set to another; hence it suffices to maintain a family of independent sets where the largest one is an O(1)-approximate MIS. (ii) A dynamic nearest/farthest neighbor data structure for disks by Kaplan et al. (DCG 2020) and Liu (SICOMP 2022), which generalizes the dynamic convex hull data structure by Chan (JACM 2010), and quickly yields a "replacement" disk (if any) when a disk in one of our independent sets is deleted.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Dynamic algorithm
  • Independent set
  • Geometric intersection graph

Metrics

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

Related Versions

References

  1. Pankaj K Agarwal, Marc Van Kreveld, and Subhash Suri. Label placement by maximum independent set in rectangles. Comput. Geom., 11(3-4):209-218, 1998. URL: https://doi.org/10.1016/S0925-7721(98)00028-5.
  2. Jochen Alber and Jirí Fiala. Geometric separation and exact solutions for the parameterized independent set problem on disk graphs. J. Algorithms, 52(2):134-151, 2004. URL: https://doi.org/10.1016/j.jalgor.2003.10.001.
  3. Franz Aurenhammer, Rolf Klein, and Der-Tsai Lee. Voronoi Diagrams and Delaunay Triangulations. World Scientific, 2013. URL: https://doi.org/10.1142/8685.
  4. Reuven Bar-Yehuda, Magnús M Halldórsson, Joseph Naor, Hadas Shachnai, and Irina Shapira. Scheduling split intervals. SIAM J. Computing, 36(1):1-15, 2006. URL: https://doi.org/10.1137/S0097539703437843.
  5. Sarita de Berg and Frank Staals. Dynamic data structures for k-nearest neighbor queries. Comput. Geom., 111:101976, 2023. URL: https://doi.org/10.1016/j.comgeo.2022.101976.
  6. Piotr Berman, Bhaskar DasGupta, S. Muthukrishnan, and Suneeta Ramaswami. Efficient approximation algorithms for tiling and packing problems with rectangles. J. Algorithms, 41(2):443-470, 2001. URL: https://doi.org/10.1006/jagm.2001.1188.
  7. Piotr Berman and Toshihiro Fujito. On approximation properties of the independent set problem for low degree graphs. Theory Comput. Syst., 32:115-132, 1999. URL: https://doi.org/10.1007/s002240000113.
  8. Sujoy Bhore, Jean Cardinal, John Iacono, and Grigorios Koumoutsos. Dynamic geometric independent set. In Abst. of 23rd Thailand-Japan Conference on Discrete and Computational Geometry, Graphs, and Games (TJCDCG'21), 2021. URL: https://doi.org/10.48550/arXiv.2007.08643.
  9. Sujoy Bhore and Timothy M. Chan. Fully dynamic geometric vertex cover and matching. CoRR, abs/2402.07441, 2024. URL: https://doi.org/10.48550/arXiv.2402.07441.
  10. Sujoy Bhore, Fabian Klute, and Jelle J. Oostveen. On streaming algorithms for geometric independent set and clique. In Proc. 20th Workshop on Approximation and Online Algorithms (WAOA'22), volume 13538 of LNCS, pages 211-224. Springer, 2022. URL: https://doi.org/10.1007/978-3-031-18367-6_11.
  11. Sujoy Bhore, Guangping Li, and Martin Nöllenburg. An algorithmic study of fully dynamic independent sets for map labeling. ACM J. Exp. Algorithmics, 27(1):1-36, 2022. URL: https://doi.org/10.1145/3514240.
  12. Sujoy Bhore, Martin Nöllenburg, Csaba D. Tóth, and Jules Wulms. Fully dynamic maximum independent sets of disks in polylogarithmic update time. CoRR, abs/2308.00979, 2023. URL: https://doi.org/10.48550/arXiv.2308.00979.
  13. Jean Cardinal, John Iacono, and Grigorios Koumoutsos. Worst-case efficient dynamic geometric independent set. In Proc. 29th European Symposium on Algorithms (ESA'21), volume 204 of LIPIcs, pages 25:1-25:15, 2021. See also arXiv:2108.08050. URL: https://doi.org/10.4230/LIPIcs.ESA.2021.25.
  14. Timothy M. Chan. Polynomial-time approximation schemes for packing and piercing fat objects. J. Algorithms, 46(2):178-189, 2003. URL: https://doi.org/10.1016/S0196-6774(02)00294-8.
  15. 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.
  16. 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.
  17. 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.
  18. Brent N. Clark, Charles J. Colbourn, and David S. Johnson. Unit disk graphs. Discret. Math., 86(1-3):165-177, 1990. URL: https://doi.org/10.1016/0012-365X(90)90358-O.
  19. Spencer Compton, Slobodan Mitrovic, and Ronitt Rubinfeld. New partitioning techniques and faster algorithms for approximate interval scheduling. In Proc. 50th International Colloquium on Automata, Languages, and Programming (ICALP'23), volume 261 of LIPIcs, pages 45:1-45:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.ICALP.2023.45.
  20. Alon Efrat, Matthew J. Katz, Frank Nielsen, and Micha Sharir. Dynamic data structures for fat objects and their applications. Comput. Geom., 15(4):215-227, 2000. URL: https://doi.org/10.1016/S0925-7721(99)00059-0.
  21. Thomas Erlebach, Klaus Jansen, and Eike Seidel. Polynomial-time approximation schemes for geometric intersection graphs. SIAM J. Computing, 34(6):1302-1323, 2005. URL: https://doi.org/10.1137/S0097539702402676.
  22. Waldo Gálvez, Arindam Khan, Mathieu Mari, Tobias Mömke, Madhusudhan Reddy Pittu, and Andreas Wiese. A 3-approximation algorithm for maximum independent set of rectangles. In Proc. 33rd Symposium on Discrete Algorithms (SODA'22), pages 894-905, 2022. URL: https://doi.org/10.1137/1.9781611977073.38.
  23. Alexander Gavruskin, Bakhadyr Khoussainov, Mikhail Kokho, and Jiamou Liu. Dynamic algorithms for monotonic interval scheduling problem. Theor. Comput. Sci., 562:227-242, 2015. URL: https://doi.org/10.1016/j.tcs.2014.09.046.
  24. Sariel Har-Peled. Geometric Approximation Algorithms, volume 173 of Mathematical Surveys and Monographs. AMS, 2011. URL: https://bookstore.ams.org/surv-173/.
  25. Monika Henzinger, Stefan Neumann, and Andreas Wiese. Dynamic approximate maximum independent set of intervals, hypercubes and hyperrectangles. In Proc. 36th Symposium on Computational Geometry (SoCG'20), volume 164 of LIPIcs, pages 51:1-51:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.SoCG.2020.51.
  26. Dorit S. Hochbaum and Wolfgang Maass. Approximation schemes for covering and packing problems in image processing and VLSI. J. ACM, 32(1):130-136, 1985. URL: https://doi.org/10.1145/2455.214106.
  27. Harry B. Hunt III, Madhav V. Marathe, Venkatesh Radhakrishnan, S. S. Ravi, Daniel J. Rosenkrantz, and Richard Edwin Stearns. NC-approximation schemes for NP- and PSPACE-hard problems for geometric graphs. J. Algorithms, 26(2):238-274, 1998. URL: https://doi.org/10.1006/jagm.1997.0903.
  28. 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.
  29. Richard M. Karp. Reducibility among combinatorial problems. In Complexity of Computer Computations, The IBM Research Symposia Series, pages 85-103. Springer, Boston, MA, 1972. URL: https://doi.org/10.1007/978-1-4684-2001-2_9.
  30. Sanjeev Khanna, Shan Muthukrishnan, and Mike Paterson. On approximating rectangle tiling and packing. In Proc. 9th Symposium on Discrete Algorithms (SODA'98), pages 384-393, 1998. URL: https://doi.org/10.5555/314613.314768.
  31. Kerstin Kirchner and Gerhard Wengerodt. Die dichteste Packung von 36 Kreisen in einem Quadrat. Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, 25:147-160, 1987. Google Scholar
  32. 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.
  33. Madhav V. Marathe, Heinz Breu, Harry B. Hunt III, Sekharipuram S. Ravi, and Daniel J. Rosenkrantz. Simple heuristics for unit disk graphs. Networks, 25(2):59-68, 1995. URL: https://doi.org/10.1002/net.3230250205.
  34. Dániel Marx. Efficient approximation schemes for geometric problems? In Proc. 13th European Symposium on Algorithms (ESA'05), volume 3669 of LNCS, pages 448-459. Springer, 2005. URL: https://doi.org/10.1007/11561071_41.
  35. Dániel Marx. On the optimality of planar and geometric approximation schemes. In Proc. 48th Symposium on Foundations of Computer Science (FOCS'07), pages 338-348, 2007. URL: https://doi.org/10.1109/FOCS.2007.26.
  36. Joseph S.B. Mitchell. Approximating maximum independent set for rectangles in the plane. In Proc. 62nd Symposium on Foundations of Computer Science (FOCS'21), pages 339-350, 2022. URL: https://doi.org/10.1109/FOCS52979.2021.00042.
  37. Micha Sharir and Pankaj K. Agarwal. Davenport-Schinzel Sequences and their Geometric Applications. Cambridge University Press, 1995. Google Scholar
  38. Péter Gábor Szabó and Eckard Specht. Packing up to 200 equal circles in a square. In Models and Algorithms for Global Optimization: Essays Dedicated to Antanas Žilinskas on the Occasion of His 60th Birthday, pages 141-156. Springer, Boston, 2007. URL: https://doi.org/10.1007/978-0-387-36721-7_9.
  39. David Zuckerman. Linear degree extractors and the inapproximability of max clique and chromatic number. Theory Comput., 3(1):103-128, 2007. URL: https://doi.org/10.4086/toc.2007.v003a006.
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact