Document Open Access Logo

A Clique-Based Separator for Intersection Graphs of Geodesic Disks in ℝ²

Authors Boris Aronov , Mark de Berg , Leonidas Theocharous

PDF
Thumbnail PDF

File

LIPIcs.SoCG.2024.9.pdf
  • Filesize: 0.87 MB
  • 15 pages

Document Identifiers

Author Details

Boris Aronov
  • Department of Computer Science and Engineering, Tandon School of Engineering, New York University, Brooklyn, NY, USA
Mark de Berg
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
Leonidas Theocharous
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands

Funding

  • Aronov, Boris: Work has been supported by NSF grant CCF 20-08551.
  • de Berg, Mark: MdB is supported by the Dutch Research Council (NWO) through Gravitation-grant NETWORKS-024.002.003.
  • Theocharous, Leonidas: LT is supported by the Dutch Research Council (NWO) through Gravitation-grant NETWORKS-024.002.003.

Cite AsGet BibTex

Boris Aronov, Mark de Berg, and Leonidas Theocharous. A Clique-Based Separator for Intersection Graphs of Geodesic Disks in ℝ². In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 9:1-9:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SoCG.2024.9

Abstract

Let d be a (well-behaved) shortest-path metric defined on a path-connected subset of ℝ² and let 𝒟 = {D_1,…,D_n} be a set of geodesic disks with respect to the metric d. We prove that 𝒢^×(𝒟), the intersection graph of the disks in 𝒟, has a clique-based separator consisting of O(n^{3/4+ε}) cliques. This significantly extends the class of objects whose intersection graphs have small clique-based separators. Our clique-based separator yields an algorithm for q-Coloring that runs in time 2^O(n^{3/4+ε}), assuming the boundaries of the disks D_i can be computed in polynomial time. We also use our clique-based separator to obtain a simple, efficient, and almost exact distance oracle for intersection graphs of geodesic disks. Our distance oracle uses O(n^{7/4+ε}) storage and can report the hop distance between any two nodes in 𝒢^×(𝒟) in O(n^{3/4+ε}) time, up to an additive error of one. So far, distance oracles with an additive error of one that use subquadratic storage and sublinear query time were not known for such general graph classes.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Computational geometry
  • intersection graphs
  • separator theorems

Metrics

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

Related Versions

References

  1. Ittai Abraham and Cyril Gavoille. On approximate distance labels and routing schemes with affine stretch. In Proc. 25^th International Symposium on Distributed Computing (DISC 2011), volume 6950 of Lecture Notes in Computer Science (ARCoSS), pages 404-415, 2011. URL: https://doi.org/10.1007/978-3-642-24100-0_39.
  2. M. Ajtai, V. Chvátal, M.M. Newborn, and E. Szemerédi. Crossing-free subgraphs. In Peter L. Hammer, Alexander Rosa, Gert Sabidussi, and Jean Turgeon, editors, Theory and Practice of Combinatorics, volume 60 of North-Holland Mathematics Studies, pages 9-12. North-Holland, 1982. URL: https://doi.org/doi.org/10.1016/S0304-0208(08)73484-4.
  3. Srinivasa Rao Arikati, Danny Z. Chen, L. Paul Chew, Gautam Das, Michiel H. M. Smid, and Christos D. Zaroliagis. Planar spanners and approximate shortest path queries among obstacles in the plane. In Proc. 4th Annual European Symposium on Algorithms (ESA 1996), volume 1136 of Lecture Notes in Computer Science, pages 514-528, 1996. URL: https://doi.org/10.1007/3-540-61680-2_79.
  4. Boris Aronov, Mark de Berg, and Leonidas Theocharous. A clique-based separator for intersection graphs of geodesic disks in ℝ². arXiv:2403.04905, 2024. URL: https://doi.org/10.48550/arxiv.2403.04905.
  5. Édouard Bonnet and Pawel Rzazewski. Optimality program in segment and string graphs. Algorithmica, 81(7):3047-3073, 2019. URL: https://doi.org/10.1007/s00453-019-00568-7.
  6. Timothy M. Chan and Dimitrios Skrepetos. Approximate Shortest Paths and Distance Oracles in Weighted Unit-Disk Graphs. In Proc. 34th International Symposium on Computational Geometry (SoCG 2018), volume 99, pages 24:1-24:13, 2018. URL: https://doi.org/10.4230/LIPIcs.SoCG.2018.24.
  7. Panagiotis Charalampopoulos, Pawel Gawrychowski, Yaowei Long, Shay Mozes, Seth Pettie, Oren Weimann, and Christian Wulff-Nilsen. Almost optimal exact distance oracles for planar graphs. J. ACM, 70(2):12:1-12:50, 2023. URL: https://doi.org/10.1145/3580474.
  8. Shiri Chechik. Approximate distance oracles with constant query time. In Proc. 46th Symposium on Theory of Computing (STOC 2014), pages 654-663, 2014. URL: https://doi.org/10.1145/2591796.2591801.
  9. Mark de Berg. A note on reachability and distance oracles for transmission graphs. Computing in Geometry and Topology, 2(1):4:1-4:15, 2023. URL: https://doi.org/doi.org/10.57717/cgt.v2i1.25.
  10. Mark de Berg, Hans L. Bodlaender, Sándor Kisfaludi-Bak, Dániel Marx, and Tom C. van der Zanden. A framework for Exponential-Time-Hypothesis-tight algorithms and lower bounds in geometric intersection graphs. SIAM J. Comput., 49:1291-1331, 2020. URL: https://doi.org/10.1137/20M1320870.
  11. Mark de Berg, Sándor Kisfaludi-Bak, Morteza Monemizadeh, and Leonidas Theocharous. Clique-based separators for geometric intersection graphs. Algorithmica, 85(6):1652-1678, 2023. URL: https://doi.org/10.1007/S00453-022-01041-8.
  12. Hristo Djidjev and Shankar M. Venkatesan. Reduced constants for simple cycle graph separation. Acta Informatica, 34(3):231-243, 1997. URL: https://doi.org/10.1007/s002360050082.
  13. Jacob Fox and János Pach. Separator theorems and Turán-type results for planar intersection graphs. Advances in Mathematics, 219(3):1070-1080, 2008. URL: https://doi.org/doi.org/10.1016/j.aim.2008.06.002.
  14. Bin Fu. Theory and application of width bounded geometric separators. Journal of Computer and System Sciences, 77(2):379-392, 2011. URL: https://doi.org/10.1016/j.jcss.2010.05.003.
  15. Jie Gao and Li Zhang. Well-separated pair decomposition for the unit-disk graph metric and its applications. SIAM Journal on Computing, 35(1):151-169, 2005. URL: https://doi.org/10.1137/S0097539703436357.
  16. Sariel Har-Peled and Kent Quanrud. Approximation algorithms for polynomial-expansion and low-density graphs. SIAM J. Comput., 46(6):1712-1744, 2017. URL: https://doi.org/10.1137/16M1079336.
  17. Hung Le and Christian Wulff-Nilsen. Optimal approximate distance oracle for planar graphs. In Proc. 62nd IEEE Annual Symposium on Foundations of Computer Science (FOCS 2021), pages 363-374. IEEE, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00044.
  18. James R. Lee. Separators in region intersection graphs. In Proc. 8th Innovations in Theoretical Computer Science Conference (ITCS 2017), volume 67 of LIPIcs, pages 1:1-1:8, 2017. URL: https://doi.org/10.4230/LIPIcs.ITCS.2017.1.
  19. Thomas Leighton. Complexity Issues in VLSI. Foundations of Computing Series. MIT Press, 2003. Google Scholar
  20. Richard J. Lipton and Robert Endre Tarjan. A separator theorem for planar graphs. SIAM J. Appl. Math, 36(2):177-189, 1977. URL: https://doi.org/doi/10.1137/0136016.
  21. Joseph S. B. Mitchell and Christos H. Papadimitriou. The weighted region problem: Finding shortest paths through a weighted planar subdivision. J. ACM, 38(1):18-73, January 1991. URL: https://doi.org/10.1145/102782.102784.
  22. Mihai Patrascu and Liam Roditty. Distance oracles beyond the Thorup-Zwick bound. In Proc. 51st Annual Symposium on Foundations of Computer Science (FOCS 2010), pages 815-823, 2010. URL: https://doi.org/10.1109/FOCS.2010.83.
  23. Christian Sommer. Shortest-path queries in static networks. ACM Comput. Surv., 46(4):45:1-45:31, 2014. URL: https://doi.org/10.1145/2530531.
  24. Mikkel Thorup and Uri Zwick. Approximate distance oracles. J. ACM, 52(1):1-24, 2005. URL: https://doi.org/10.1145/1044731.1044732.
  25. William Thurston. The Geometry and Topology of 3-Manifolds. Princeton Lecture Notes, 1978-1981. 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
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact