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Computing Diameter+2 in Truly-Subquadratic Time for Unit-Disk Graphs

Authors Hsien-Chih Chang , Jie Gao , Hung Le

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Author Details

Hsien-Chih Chang
  • Department of Computer Science, Dartmouth College, Hanover, NH, USA
Jie Gao
  • Department of Computer Science, Rutgers University, Piscataway, NJ, USA
Hung Le
  • Manning CICS, University of Massachusetts Amherst, MA, USA

Funding

  • Gao, Jie: Supported partially by NSF DMS-2220271, IIS-2229876, CCF-2208663, CCF-2118953.
  • Le, Hung: Supported by the NSF CAREER Award No. CCF-2237288 and an NSF Grant No. CCF-2121952.

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Hsien-Chih Chang, Jie Gao, and Hung Le. Computing Diameter+2 in Truly-Subquadratic Time for Unit-Disk Graphs. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 38:1-38:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SoCG.2024.38

Abstract

Finding the diameter of a graph in general cannot be done in truly subquadratic assuming the Strong Exponential Time Hypothesis (SETH), even when the underlying graph is unweighted and sparse. When restricting to concrete classes of graphs and assuming SETH, planar graphs and minor-free graphs admit truly subquadratic algorithms, while geometric intersection graphs of unit balls, congruent equilateral triangles, and unit segments do not. Unit-disk graphs is one of the major open cases where the complexity of diameter computation remains unknown. More generally, it is conjectured that a truly subquadratic time algorithm exists for pseudo-disk graphs where each pair of objects has at most two intersections on the boundary. In this paper, we show a truly-subquadratic algorithm of running time O^~(n^{2-1/18}), for finding the diameter in a unit-disk graph, whose output differs from the optimal solution by at most 2. This is the first algorithm that provides an additive guarantee in distortion, independent of the size or the diameter of the graph. Our algorithm requires two important technical elements. First, we show that for the intersection graph of pseudo-disks, the graph VC-dimension - either of k-hop balls or the distance encoding vectors - is 4. This contrasts to the VC dimension of the pseudo-disks themselves as geometric ranges (which is known to be 3). Second, we introduce a clique-based r-clustering for geometric intersection graphs, which is an analog of the r-division construction for planar graphs. We also showcase the new techniques by establishing new results for distance oracles for unit-disk graphs with subquadratic storage and O(1) query time. The results naturally extend to unit L₁ or L_∞-disks and fat pseudo-disks of similar size. Last, if the pseudo-disks additionally have bounded ply, we have a truly subquadratic algorithm to find the exact diameter.

Subject Classification

ACM Subject Classification
  • Theory of computation → Shortest paths
  • Theory of computation → Computational geometry
Keywords
  • Unit-disk graph
  • pseudo-disks
  • r-division
  • VC-dimension
  • distance oracle
  • clique-based separator

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References

  1. A. Karim Abu-Affash, Paz Carmi, Anil Maheshwari, Pat Morin, Michiel Smid, and Shakhar Smorodinsky. Approximating maximum diameter-bounded subgraph in unit disk graphs. Discrete Comput. Geom., 66(4):1401-1414, December 2021. Google Scholar
  2. Noga Alon, Graham Brightwell, H. A. Kierstead, A. V . Kostochka, and Peter Winkler. Dominating sets in k-majority tournaments. J. Combin. Theory Ser. B, 96(3):374-387, May 2006. Google Scholar
  3. Mark de Berg. A note on reachability and distance oracles for transmission graphs. Computing in Geometry and Topology, page Vol. 2 No. 1 (2023), 2023. URL: https://doi.org/10.57717/CGT.V2I1.25.
  4. 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 Journal on Computing, 49(6):1291-1331, 2020. URL: https://doi.org/10.1137/20m1320870.
  5. Mark de Berg, Sándor Kisfaludi-Bak, Morteza Monemizadeh, and Leonidas Theocharous. Clique-based separators for geometric intersection graphs. Algorithmica, 85(6):1652-1678, 2023. Google Scholar
  6. Nicolas Bousquet, Aurélie Lagoutte, Zhentao Li, Aline Parreau, and Stéphan Thomassé. Identifying codes in hereditary classes of graphs and VC-dimension. SIAM Journal on Discrete Mathematics, 29(4):2047-2064, 2015. URL: https://doi.org/10.1137/14097879X.
  7. Nicolas Bousquet and Stéphan Thomassé. VC-dimension and Erdős-Pósa property. Discrete Math., 338(12):2302-2317, December 2015. Google Scholar
  8. Heinz Breu. Algorithmic Aspects of Constrained Unit Disk Graphs. PhD thesis, University of British Columbia, April 1996. Google Scholar
  9. Karl Bringmann, Sándor Kisfaludi‑Bak, Marvin Künnemann, André Nusser, and Zahra Parsaeian. Towards sub-quadratic diameter computation in geometric intersection graphs. In Xavier Goaoc and Michael Kerber, editors, 38th International Symposium on Computational Geometry (SoCG 2022), pages 21:1-21:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, June 2022. Google Scholar
  10. Sarit Buzaglo, Rom Pinchasi, and Günter Rote. Topological hypergraphs. Thirty Essays on Geometric Graph Theory, pages 71-81, 2013. Google Scholar
  11. Sergio Cabello and Miha Jejčič. Shortest paths in intersection graphs of unit disks. Comput. Geom., 48(4):360-367, May 2015. Google Scholar
  12. Timothy M. Chan and Dimitrios Skrepetos. All-pairs shortest paths in unit-disk graphs in slightly subquadratic time. In LIPIcs-Leibniz International Proceedings in Informatics, volume 64, 2016. Google Scholar
  13. Timothy M. Chan and Dimitrios Skrepetos. All-Pairs shortest paths in geometric intersection graphs. In Algorithms and Data Structures, pages 253-264. Springer International Publishing, 2017. Google Scholar
  14. Timothy M. Chan and Dimitrios Skrepetos. Approximate shortest paths and distance oracles in weighted Unit-Disk graphs. J. Comput. Geom., 10(2), 2019. Google Scholar
  15. Bernard Chazelle and Emo Welzl. Quasi-optimal range searching in spaces of finite VC-dimension. Discrete & Computational Geometry, 4(5):467-489, October 1989. URL: https://doi.org/10.1007/BF02187743.
  16. Victor Chepoi, Bertrand Estellon, and Yann Vaxès. Covering planar graphs with a fixed number of balls. Discrete Comput. Geom., 37(2):237-244, February 2007. Google Scholar
  17. Guillaume Ducoffe, Michel Habib, and Laurent Viennot. Diameter, eccentricities and distance oracle computations on h-minor free graphs and graphs of bounded (distance) vapnik-chervonenkis dimension. SIAM Journal on Computing, 51(5):1506-1534, 2022. Google Scholar
  18. Lech Duraj, Filip Konieczny, and Krzysztof Potępa. Better diameter algorithms for bounded VC-dimension graphs and geometric intersection graphs, 2023. URL: https://arxiv.org/abs/2307.08162.
  19. Alon Efrat, Alon Itai, and Matthew J Katz. Geometry helps in bottleneck matching and related problems. Algorithmica, 31(1):1-28, September 2001. Google Scholar
  20. Greg N. Frederickson. Fast algorithms for shortest paths in planar graphs, with applications. SIAM Journal on Computing, 16(6):1004-1022, 1987. URL: https://doi.org/10.1137/0216064.
  21. Viktor Fredslund-Hansen, Shay Mozes, and Christian Wulff-Nilsen. Truly subquadratic exact distance oracles with constant query time for planar graphs. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021), Leibniz International Proceedings in Informatics (LIPIcs), pages 25:1-25:12, 2021. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2021.25.
  22. 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.
  23. Sariel Har-Peled and Kent Quanrud. Approximation algorithms for polynomial-expansion and low-density graphs. SIAM Journal on Computing, 46(6):1712-1744, 2017. URL: https://doi.org/10.1137/16m1079336.
  24. David Haussler and Emo Welzl. Epsilon-nets and simplex range queries. In Proceedings of the second annual symposium on Computational geometry, SCG '86, pages 61-71, New York, NY, USA, August 1986. Association for Computing Machinery. Google Scholar
  25. Harry B. Hunt III, Madhav V. Marathe, Venkatesh Radhakrishnan, Shankar S. Ravi, Daniel J. Rosenkrantz, and Richard E. Stearns. NC-Approximation schemes for NP- and PSPACE-Hard problems for geometric graphs. J. Algorithm. Comput. Technol., 26(2):238-274, February 1998. Google Scholar
  26. Haim Kaplan, Matthew J. Katz, Rachel Saban, and Micha Sharir. The Unweighted and Weighted Reverse Shortest Path Problem for Disk Graphs. In 31st Annual European Symposium on Algorithms (ESA 2023), pages 67:1-67:14, 2023. URL: https://doi.org/10.4230/LIPIcs.ESA.2023.67.
  27. Katharina Klost. An algorithmic framework for the single source shortest path problem with applications to disk graphs. Comput. Geom., 111:101979, April 2023. Google Scholar
  28. Hung Le and Christian Wulff-Nilsen. VC set systems in minor-free (di)graphs and applications. arXiv 2304.01790, 2023. Google Scholar
  29. Jason Li and Merav Parter. Planar diameter via metric compression. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, pages 152-163, New York, NY, USA, June 2019. Association for Computing Machinery. Google Scholar
  30. Jiří Matoušek, Raimund Seidel, and Emo Welzl. How to net a lot with little: small ε-nets for disks and halfspaces. In Proceedings of the sixth annual symposium on Computational geometry, SCG '90, pages 16-22, New York, NY, USA, May 1990. Association for Computing Machinery. Google Scholar
  31. Tim Nieberg, Johann Hurink, and Walter Kern. A robust PTAS for maximum weight independent sets in unit disk graphs. In Graph-Theoretic Concepts in Computer Science, pages 214-221. Springer Berlin Heidelberg, 2005. Google Scholar
  32. Ryan Williams. A new algorithm for optimal 2-constraint satisfaction and its implications. Theoretical Computer Science, 348(2-3):357-365, 2005. Google Scholar
  33. Christian Wulff-Nilsen. Separator theorems for minor-free and shallow minor-free graphs with applications. In 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, pages 37-46, 2011. URL: https://doi.org/10.1109/FOCS.2011.15.
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