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Fully-Adaptive Dynamic Connectivity of Square Intersection Graphs

Authors Ivor van der Hoog , André Nusser, Eva Rotenberg , Frank Staals

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Author Details

Ivor van der Hoog
  • Technical University of Denmark, Lyngby, Denmark
André Nusser
  • CNRS, Inria, I3S, Université Côte d'Azur, France
Eva Rotenberg
  • Technical University of Denmark, Lyngby, Denmark
Frank Staals
  • Utrecht University, The Netherlands

Funding

  • van der Hoog, Ivor: Received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 899987.
  • Nusser, André: Supported by the French government through the France 2030 investment plan managed by the National Research Agency (ANR), as part of the Initiative of Excellence of Université Côte d’Azur under reference number ANR-15-IDEX-01.
  • Rotenberg, Eva: Supported by the Independent Research Fund Denmark grant 2020-2023 (9131-00044B) "Dynamic Network Analysis" and the Carlsberg Foundation Young Researcher Fellowship CF21-0302 "Graph Algorithms with Geometric Applications".

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Ivor van der Hoog, André Nusser, Eva Rotenberg, and Frank Staals. Fully-Adaptive Dynamic Connectivity of Square Intersection Graphs. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 63:1-63:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.MFCS.2024.63

Abstract

A classical problem in computational geometry and graph algorithms is: given a dynamic set 𝒮 of geometric shapes in the plane, efficiently maintain the connectivity of the intersection graph of 𝒮. Previous papers studied the setting where, before the updates, the data structure receives some parameter P. Then, updates could insert and delete disks as long as at all times the disks have a diameter that lies in a fixed range [1/P, 1]. As a consequence of that prerequisite, the aspect ratio ψ (i.e. the ratio between the largest and smallest diameter) of the disks would at all times satisfy ψ ≤ P. The state-of-the-art for storing disks in a dynamic connectivity data structure is a data structure that uses O(Pn) space and that has amortized O(P log⁴ n) expected amortized update time. Connectivity queries between disks are supported in O(log n / log log n) time. 
In the dynamic setting, one wishes for a more flexible data structure in which disks of any diameter may arrive and leave, independent of their diameter, changing the aspect ratio freely. Ideally, the aspect ratio should merely be part of the analysis. We restrict our attention to axis-aligned squares, and study fully-dynamic square intersection graph connectivity. Our result is fully-adaptive to the aspect ratio, spending time proportional to the current aspect ratio ψ, as opposed to some previously given maximum P. Our focus on squares allows us to simplify and streamline the connectivity pipeline from previous work. When n is the number of squares and ψ is the aspect ratio after insertion (or before deletion), our data structure answers connectivity queries in O(log n / log log n) time. We can update connectivity information in O(ψ log⁴ n + log⁶ n) amortized time. We also improve space usage from O(P ⋅ n log n) to O(n log³ n log ψ) - while generalizing to a fully-adaptive aspect ratio - which yields a space usage that is near-linear in n for any polynomially bounded ψ.

Subject Classification

ACM Subject Classification
  • Theory of computation → Dynamic graph algorithms
  • Theory of computation → Computational geometry
Keywords
  • Computational geometry
  • planar geometry
  • data structures
  • geometric intersection graphs
  • fully-dynamic algorithms

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References

  1. Anders Aamand, Adam Karczmarz, Jakub Lacki, Nikos Parotsidis, Peter M. R. Rasmussen, and Mikkel Thorup. Optimal decremental connectivity in non-sparse graphs. CoRR, abs/2111.09376, 2021. URL: https://arxiv.org/abs/2111.09376.
  2. Stephen Alstrup, Thore Husfeldt, and Theis Rauhe. Marked ancestor problems. In Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. no. 98CB36280), pages 534-543. IEEE, 1998. Google Scholar
  3. Stephen Alstrup, Jens P. Secher, and Maz Spork. Optimal on-line decremental connectivity in trees. Inf. Process. Lett., 64(4):161-164, 1997. URL: https://doi.org/10.1016/S0020-0190(97)00170-1.
  4. Arne Andersson. General balanced trees. J. Algorithms, 30(1):1-18, 1999. URL: https://doi.org/10.1006/jagm.1998.0967.
  5. Kevin Buchin, Maarten Löffler, Pat Morin, and Wolfgang Mulzer. Preprocessing imprecise points for delaunay triangulation: Simplified and extended. Algorithmica, 61(3):674-693, 2011. Google Scholar
  6. Timothy M Chan and Zhengcheng Huang. Dynamic geometric connectivity in the plane with constant query time. arXiv preprint arXiv:2402.05357, 2024. Google Scholar
  7. Timothy M. Chan, Mihai Pătraşcu, and Liam Roditty. Dynamic connectivity: Connecting to networks and geometry. SIAM J. Comput., 40(2):333-349, 2011. URL: https://doi.org/10.1137/090751670.
  8. Mark de Berg, Otfried Cheong, Marc J. van Kreveld, and Mark H. Overmars. Computational geometry: Algorithms and applications, 3rd Edition. Springer, 2008. URL: https://www.worldcat.org/oclc/227584184.
  9. David Eppstein. Dynamic generators of topologically embedded graphs. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, January 12-14, 2003, Baltimore, Maryland, USA, pages 599-608. ACM/SIAM, 2003. URL: http://dl.acm.org/citation.cfm?id=644108.644208.
  10. David Eppstein, Zvi Galil, Giuseppe F Italiano, and Thomas H Spencer. Separator based sparsification for dynamic planar graph algorithms. In Proceedings of the twenty-fifth annual ACM symposium on Theory of Computing, pages 208-217, 1993. Google Scholar
  11. Greg N. Frederickson. Data structures for on-line updating of minimum spanning trees, with applications. SIAM J. Comput., 14(4):781-798, 1985. URL: https://doi.org/10.1137/0214055.
  12. Sariel Har-Peled. Quadtrees-hierarchical grids. Lecture notes, 2010. Google Scholar
  13. Monika Rauch Henzinger and Valerie King. Randomized fully dynamic graph algorithms with polylogarithmic time per operation. J. ACM, 46(4):502-516, 1999. URL: https://doi.org/10.1145/320211.320215.
  14. Jacob Holm, Kristian De Lichtenberg, and Mikkel Thorup. Poly-logarithmic deterministic fully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity. Journal of the ACM (JACM), 48(4):723-760, 2001. Google Scholar
  15. Jacob Holm and Eva Rotenberg. Good r-divisions imply optimal amortized decremental biconnectivity. In Markus Bläser and Benjamin Monmege, editors, 38th International Symposium on Theoretical Aspects of Computer Science, STACS 2021, March 16-19, 2021, Saarbrücken, Germany (Virtual Conference), volume 187 of LIPIcs, pages 42:1-42:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.STACS.2021.42.
  16. Haim Kaplan, Alexander Kauer, Katharina Klost, Kristin Knorr, Wolfgang Mulzer, Liam Roditty, and Paul Seiferth. Dynamic connectivity in disk graphs. In 38th International Symposium on Computational Geometry (SoCG 2022). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. Google Scholar
  17. 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(3):838-904, 2020. Google Scholar
  18. Bruce M. Kapron, Valerie King, and Ben Mountjoy. Dynamic graph connectivity in polylogarithmic worst case time. In Sanjeev Khanna, editor, Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013, New Orleans, Louisiana, USA, January 6-8, 2013, pages 1131-1142. SIAM, 2013. URL: https://doi.org/10.1137/1.9781611973105.81.
  19. Katharina Klost. An algorithmic framework for the single source shortest path problem with applications to disk graphs. Computational Geometry, 111:101979, 2023. Google Scholar
  20. Fabian Kuhn, Rogert Wattenhofer, and Aaron Zollinger. Ad-hoc networks beyond unit disk graphs. In Proceedings of the 2003 Joint Workshop on Foundations of Mobile Computing, DIALM-POMC '03, pages 69-78, New York, NY, USA, 2003. Association for Computing Machinery. URL: https://doi.org/10.1145/941079.941089.
  21. Jakub Lacki and Piotr Sankowski. Optimal decremental connectivity in planar graphs. In Ernst W. Mayr and Nicolas Ollinger, editors, 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015, March 4-7, 2015, Garching, Germany, volume 30 of LIPIcs, pages 608-621. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015. URL: https://doi.org/10.4230/LIPIcs.STACS.2015.608.
  22. Chih-Hung Liu. Nearly optimal planar k nearest neighbors queries under general distance functions. SIAM Journal on Computing, 51(3):723-765, 2022. Google Scholar
  23. Maarten Löffler, Joseph A Simons, and Darren Strash. Dynamic planar point location with sub-logarithmic local updates. In Algorithms and Data Structures: 13th International Symposium, WADS 2013, London, ON, Canada, August 12-14, 2013. Proceedings 13, pages 499-511. Springer, 2013. Google Scholar
  24. Danupon Nanongkai, Thatchaphol Saranurak, and Christian Wulff-Nilsen. Dynamic minimum spanning forest with subpolynomial worst-case update time. In Chris Umans, editor, 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 950-961. IEEE Computer Society, 2017. URL: https://doi.org/10.1109/FOCS.2017.92.
  25. Daniel Dominic Sleator and Robert Endre Tarjan. A data structure for dynamic trees. J. Comput. Syst. Sci., 26(3):362-391, 1983. URL: https://doi.org/10.1016/0022-0000(83)90006-5.
  26. Robert Endre Tarjan. A class of algorithms which require nonlinear time to maintain disjoint sets. J. Comput. Syst. Sci., 18(2):110-127, 1979. URL: https://doi.org/10.1016/0022-0000(79)90042-4.
  27. Robert Endre Tarjan and Jan van Leeuwen. Worst-case analysis of set union algorithms. J. ACM, 31(2):245-281, 1984. URL: https://doi.org/10.1145/62.2160.
  28. Mikkel Thorup. Decremental dynamic connectivity. J. Algorithms, 33(2):229-243, 1999. URL: https://doi.org/10.1006/jagm.1999.1033.
  29. Mikkel Thorup. Near-optimal fully-dynamic graph connectivity. In F. Frances Yao and Eugene M. Luks, editors, Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, May 21-23, 2000, Portland, OR, USA, pages 343-350. ACM, 2000. URL: https://doi.org/10.1145/335305.335345.
  30. Dan E Willard and George S Lueker. Adding range restriction capability to dynamic data structures. Journal of the ACM (JACM), 32(3):597-617, 1985. Google Scholar
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