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A WSPD, Separator and Small Tree Cover for c-Packed Graphs

Authors Lindsey Deryckere , Joachim Gudmundsson , André van Renssen , Yuan Sha , Sampson Wong

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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Well-separated pair decomposition
  • separator
  • tree cover
  • distance oracles
  • realistic graphs

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Abstract

The c-packedness property, proposed in 2010, is a geometric property that captures the spatial distribution of a set of edges. Despite the recent interest in c-packedness, its utility has so far been limited to Fréchet distance problems. An open problem is whether a wider variety of algorithmic and data structure problems can be solved efficiently under the c-packedness assumption, and more specifically, on c-packed graphs.
In this paper, we prove two fundamental properties of c-packed graphs: that there exists a linear-size well-separated pair decomposition under the graph metric, and there exists a constant size balanced separator. We then apply these fundamental properties to obtain a small tree cover for the metric space and distance oracles under the shortest path metric. In particular, we obtain a tree cover of constant size, an exact distance oracle of near-linear size and an approximate distance oracle of linear size.

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Lindsey Deryckere, Joachim Gudmundsson, André van Renssen, Yuan Sha, and Sampson Wong. A WSPD, Separator and Small Tree Cover for c-Packed Graphs. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 21:1-21:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.WADS.2025.21

Author Details

Lindsey Deryckere
  • The University of Sydney, Australia
Joachim Gudmundsson
  • The University of Sydney, Australia
André van Renssen
  • The University of Sydney, Australia
Yuan Sha
  • The University of Sydney, Australia
Sampson Wong
  • The University of Copenhagen, Denmark

Funding

This research was partially funded by the Australian Government through the Australian Research Council (project number DP240101353).
  • Gudmundsson, Joachim: This research was partially funded by the Australian Government through the Australian Research Council (project number DP240101353).
  • van Renssen, André: This research was partially funded by the Australian Government through the Australian Research Council (project number DP240101353).
  • Wong, Sampson: This research was partially funded by the European Union through the Marie Skłodowska-Curie Actions Postdoctoral Fellowship (project number 101146276).

References

  1. N. Alon, P. D. Seymour, and R. Thomas. A separator theorem for graphs with an excluded minor and its applications. In Proceedings of the 22nd Annual ACM Symposium on Theory of Computing (STOC), pages 293-299, 1990. Google Scholar
  2. S. Arya, G. Das, D. M. Mount, J. S. Salowe, and M. H. M. Smid. Euclidean spanners: short, thin, and lanky. In Proceedings of the 27th Annual ACM Symposium on Theory of Computing (STOC), pages 489-498, 1995. Google Scholar
  3. Y. Bartal, O. N. Fandina, and O. Neiman. Covering metric spaces by few trees. J. Comput. Syst. Sci., 130:26-42, 2022. URL: https://doi.org/10.1016/J.JCSS.2022.06.001.
  4. U. Bertele and F. Brioschi. Nonserial dynamic programming. Academic Press, Inc., 1972. Google Scholar
  5. F. Brüning, J. Conradi, and A. Driemel. Faster approximate covering of subcurves under the Fréchet distance. In Proceedings of the 30th Annual European Symposium on Algorithms, (ESA), volume 244 of LIPIcs, pages 28:1-28:16, 2022. URL: https://doi.org/10.4230/LIPICS.ESA.2022.28.
  6. K. Buchin, M. Buchin, J. Gudmundsson, A. Popov, and S. Wong. Map matching queries under Fréchet distance on low-density spanners. In Proceedings of the 40th Annual Symposium on Computational Geometry (SoCG), 2024. Google Scholar
  7. P. B. Callahan and S. R. Kosaraju. A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields. J. ACM, 42(1):67-90, 1995. URL: https://doi.org/10.1145/200836.200853.
  8. H. Chang, J. Conroy, H. Le, L. Milenkovic, and C. Than. Covering planar metrics (and beyond): O(1) trees suffice. In Proceedings of the 64th IEEE Annual Symposium on Foundations of Computer Science (FOCS), pages 2231-2261, 2023. Google Scholar
  9. S. Chaudhuri and C. D. Zaroliagis. Shortest paths in digraphs of small treewidth. part I: sequential algorithms. Algorithmica, 27(3):212-226, 2000. URL: https://doi.org/10.1007/S004530010016.
  10. B. Chazelle. Computing on a free tree via complexity-preserving mappings. Algorithmica, 2:337-361, 1987. URL: https://doi.org/10.1007/BF01840366.
  11. D. Chen, A. Driemel, L. J. Guibas, A. Nguyen, and C. Wenk. Approximate map matching with respect to the Fréchet distance. In Proceedings of the 13th Workshop on Algorithm Engineering and Experiments (ALENEX), pages 75-83, 2011. Google Scholar
  12. J. Conradi, A. Driemel, and B. Kolbe. (1+ε)-ANN data structure for curves via subspaces of bounded doubling dimension. Comput. Geom. Topol., 3(2):6:1-6:22, 2024. URL: https://www.cgt-journal.org/index.php/cgt/article/view/45.
  13. A. Driemel and S. Har-Peled. Jaywalking your dog: Computing the Fréchet distance with shortcuts. SIAM Journal on Computing, 42(5):1830-1866, 2013. URL: https://doi.org/10.1137/120865112.
  14. A. Driemel, S. Har-Peled, and C. Wenk. Approximating the Fréchet distance for realistic curves in near linear time. Discret. Comput. Geom., 48(1):94-127, 2012. URL: https://doi.org/10.1007/S00454-012-9402-Z.
  15. Z. Dvorák and S. Norin. Treewidth of graphs with balanced separations. J. Comb. Theory B, 137:137-144, 2019. URL: https://doi.org/10.1016/J.JCTB.2018.12.007.
  16. J. Gao and L. Zhang. Well-separated pair decomposition for the unit-disk graph metric and its applications. SIAM J. Comput., 35(1):151-169, 2005. URL: https://doi.org/10.1137/S0097539703436357.
  17. J. R. Gilbert, J. P. Hutchinson, and R. E. Tarjan. A separator theorem for graphs of bounded genus. J. Algorithms, 5(3):391-407, 1984. URL: https://doi.org/10.1016/0196-6774(84)90019-1.
  18. J. Gudmundsson, Z. Huang, A. van Renssen, and S. Wong. Computing a subtrajectory cluster from c-packed trajectories. In Proceedings of the 34th International Symposium on Algorithms and Computation (ISAAC), volume 283 of LIPIcs, pages 34:1-34:15, 2023. URL: https://doi.org/10.4230/LIPICS.ISAAC.2023.34.
  19. J. Gudmundsson, C. Levcopoulos, G. Narasimhan, and M. H. M. Smid. Approximate distance oracles for geometric spanners. ACM Trans. Algorithms, 4(1):10:1-10:34, 2008. URL: https://doi.org/10.1145/1328911.1328921.
  20. J. Gudmundsson, M. P. Seybold, and S. Wong. Map matching queries on realistic input graphs under the Fréchet distance. In Proceedings of the 34th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1464-1492, 2023. Google Scholar
  21. J. Gudmundsson and M. H. M. Smid. Fast algorithms for approximate Fréchet matching queries in geometric trees. Comput. Geom., 48(6):479-494, 2015. URL: https://doi.org/10.1016/J.COMGEO.2015.02.003.
  22. J. Gudmundsson and S. Wong. A well-separated pair decomposition for low density graphs. CoRR, abs/2411.08204, 2024. URL: https://doi.org/10.48550/arXiv.2411.08204.
  23. A. Gupta, R. Krauthgamer, and J. R. Lee. Bounded geometries, fractals, and low-distortion embeddings. In Proceedings of the 44th Annual Symposium on Foundations of Computer Science (FOCS), pages 534-543, 2003. Google Scholar
  24. S. Har-Peled. Geometric Approximation Algorithms. American Mathematical Society, 2011. Google Scholar
  25. S. Har-Peled and M. Mendel. Fast construction of nets in low-dimensional metrics and their applications. SIAM J. Comput., 35(5):1148-1184, 2006. URL: https://doi.org/10.1137/S0097539704446281.
  26. S. Har-Peled and K. 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.
  27. S. Har-Peled and Benjamin Raichel. The Fréchet distance revisited and extended. ACM Trans. Algorithms, 10(1):3:1-3:22, 2014. URL: https://doi.org/10.1145/2532646.
  28. K. Kawarabayashi, P. N. Klein, and C. Sommer. Linear-space approximate distance oracles for planar, bounded-genus and minor-free graphs. In Proceedings of the 38th International Colloquium on Automata, Languages and Programming (ICALP), volume 6755 of Lecture Notes in Computer Science, pages 135-146, 2011. URL: https://doi.org/10.1007/978-3-642-22006-7_12.
  29. K. Kawarabayashi, C. Sommer, and M. Thorup. More compact oracles for approximate distances in undirected planar graphs. In Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 550-563, 2013. Google Scholar
  30. P. N. Klein. Preprocessing an undirected planar network to enable fast approximate distance queries. In Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 820-827, 2002. Google Scholar
  31. Jon M. Kleinberg and Éva Tardos. Algorithm design. Addison-Wesley, 2006. Google Scholar
  32. R. Kyng, R. Peng, R. Schwieterman, and P. Zhang. Incomplete nested dissection. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing (STOC), pages 404-417, 2018. Google Scholar
  33. H. Le and C. Wulff-Nilsen. Optimal approximate distance oracle for planar graphs. In Proceedings of the 62nd IEEE Annual Symposium on Foundations of Computer Science (FOCS), pages 363-374, 2021. Google Scholar
  34. R. J. Lipton and R. E. Tarjan. A separator theorem for planar graphs. SIAM J. Appl. Math, 36(2):177-189, 1979. Google Scholar
  35. R. J. Lipton and R. E. Tarjan. Applications of a planar separator theorem. SIAM J. Comput., 9(3):615-627, 1980. URL: https://doi.org/10.1137/0209046.
  36. G. L. Miller and W. P. Thurston. Separators in two and three dimensions. In Proceedings of the 22nd Annual ACM Symposium on Theory of Computing (STOC), pages 300-309, 1990. Google Scholar
  37. G. Narasimhan and M. H. M. Smid. Geometric spanner networks. Cambridge University Press, 2007. Google Scholar
  38. K. Talwar. Bypassing the embedding: algorithms for low dimensional metrics. In László Babai, editor, Proceedings of the 36th Annual ACM Symposium on Theory of Computing (STOC), pages 281-290, 2004. Google Scholar
  39. M. Thorup. Compact oracles for reachability and approximate distances in planar digraphs. J. ACM, 51(6):993-1024, 2004. URL: https://doi.org/10.1145/1039488.1039493.
  40. M. Thorup and U. Zwick. Approximate distance oracles. J. ACM, 52(1):1-24, 2005. URL: https://doi.org/10.1145/1044731.1044732.
  41. C. Wulff-Nilsen. Approximate distance oracles for planar graphs with improved query time-space tradeoff. In Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 351-362, 2016. Google Scholar
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