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Algorithms for Distance Problems in Continuous Graphs

Authors Sergio Cabello , Delia Garijo , Antonia Kalb , Fabian Klute , Irene Parada , Rodrigo I. Silveira

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ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • diameter
  • mean distance
  • continuous graph
  • treewidth
  • planar graph

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Abstract

We study the problem of computing the diameter and the mean distance of a continuous graph, i.e., a connected graph where all points along the edges, instead of only the vertices, must be taken into account. It is known that for continuous graphs with m edges these values can be computed in roughly O(m²) time. In this paper, we use geometric techniques to obtain subquadratic time algorithms to compute the diameter and the mean distance of a continuous graph for two well-established classes of sparse graphs. We show that the diameter and the mean distance of a continuous graph of treewidth at most k can be computed in O(n log^O(k) n) time, where n is the number of vertices in the graph. We also show that computing the diameter and mean distance of a continuous planar graph with n vertices and F faces takes O(n F log n) time.

Cite As Get BibTex

Sergio Cabello, Delia Garijo, Antonia Kalb, Fabian Klute, Irene Parada, and Rodrigo I. Silveira. Algorithms for Distance Problems in Continuous Graphs. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 13:1-13:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.WADS.2025.13

Author Details

Sergio Cabello
  • Faculty of Mathematics and Physics, University of Ljubljana, Slovenia
  • Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia
Delia Garijo
  • University of Seville, Spain
Antonia Kalb
  • Technical University of Dortmund, Germany
Fabian Klute
  • Universitat Politècnica de Catalunya, Barcelona, Spain
Irene Parada
  • Universitat Politècnica de Catalunya, Barcelona, Spain
Rodrigo I. Silveira
  • Universitat Politècnica de Catalunya, Barcelona, Spain

Funding

  • Cabello, Sergio: Funded in part by the Slovenian Research and Innovation Agency (P1-0297, N1-0218, N1-0285). Funded in part by the European Union (ERC, KARST, project number 101071836). Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.
  • Garijo, Delia: Partially supported by grants PID2019-104129GB-I00 and PID2023-150725NB-I00 funded by MICIU/AEI/10.13039/501100011033.
  • Klute, Fabian: Partially supported by grants PID2019-104129GB-I00 and PID2023-150725NB-I00 funded by MICIU/AEI/10.13039/501100011033.
  • Parada, Irene: Serra Húnter Fellow. Partially supported by grants PID2019-104129GB-I00 and PID2023-150725NB-I00 funded by MICIU/AEI/10.13039/501100011033.
  • Silveira, Rodrigo I.: Partially supported by grants PID2019-104129GB-I00 and PID2023-150725NB-I00 funded by MICIU/AEI/ 10.13039/501100011033.

References

  1. A. Abboud, V. Vassilevska Williams, and J. Wang. Approximation and fixed parameter subquadratic algorithms for radius and diameter in sparse graphs. In Proc. 27th Annu. ACM-SIAM Sympos. Discrete Algorithms (SODA), pages 377-391. SIAM, 2016. URL: https://doi.org/10.1137/1.9781611974331.ch28.
  2. S. Won Bae, M. De Berg, O. Cheong, J. Gudmundsson, and C. Levcopoulos. Shortcuts for the circle. Comput. Geom. Theory Appl., 79:37-54, 2019. URL: https://doi.org/10.1016/J.COMGEO.2019.01.006.
  3. L. N. Baptista, J. B. Kennedy, and D. Mugnolo. Mean distance on metric graphs. The Journal of Geometric Analysis, 34(137):1-25, 2024. URL: https://doi.org/10.1007/s12220-024-01574-0.
  4. P. Bergé, G. Ducoffe, and M. Habib. Subquadratic-time algorithm for the diameter and all eccentricities on median graphs. Theory of Computing Systems, 68(1):144-193, 2024. URL: https://doi.org/10.1007/S00224-023-10153-9.
  5. P. Bergé, G. Ducoffe, and M. Habib. Quasilinear-time eccentricities computation, and more, on median graphs. In Proc. 36th Annu. ACM-SIAM Sympos. Discrete Algorithms (SODA), pages 1679-1704. SIAM, 2025. URL: https://doi.org/10.1137/1.9781611978322.52.
  6. H. L. Bodlaender. A partial k-arboretum of graphs with bounded treewidth. Theor. Comput. Sci., 209(1-2):1-45, 1998. URL: https://doi.org/10.1016/S0304-3975(97)00228-4.
  7. H. L. Bodlaender, P. Grønås Drange, M. S. Dregi, F. V. Fomin, D. Lokshtanov, and M. Pilipczuk. A c^k n 5-approximation algorithm for treewidth. SIAM J. Comput., 45(2):317-378, 2016. URL: https://doi.org/10.1137/130947374.
  8. K. Bringmann, T. Husfeldt, and M. Magnusson. Multivariate analysis of orthogonal range searching and graph distances. Algorithmica, 82:2292-2315, 2020. URL: https://doi.org/10.1007/S00453-020-00680-Z.
  9. S. Cabello. Subquadratic algorithms for the diameter and the sum of pairwise distances in planar graphs. ACM Trans. Algorithms, 15(2):21:1-21:38, 2019. URL: https://doi.org/10.1145/3218821.
  10. S. Cabello. Computing the inverse geodesic length in planar graphs and graphs of bounded treewidth. ACM Trans. Algorithms, 18(2):14:1-14:26, 2022. URL: https://doi.org/10.1145/3501303.
  11. S. Cabello, D. Garijo, A. Kalb, F. Klute, I. Parada, and R. I. Silveira. Algorithms for distance problems in continuous graphs, 2025. URL: https://arxiv.org/abs/2503.07769.
  12. S. Cabello and C. Knauer. Algorithms for graphs of bounded treewidth via orthogonal range searching. Comput. Geom. Theory Appl., 42(9):815-824, 2009. URL: https://doi.org/10.1016/j.comgeo.2009.02.001.
  13. J. Cáceres, D. Garijo, A. González, A. Márquez, M. L. Puertas, and P. Ribeiro. Shortcut sets for the locus of plane Euclidean networks. Applied Mathematics and Computation, 334:192-205, 2018. URL: https://doi.org/10.1016/j.amc.2018.04.010.
  14. J.-L. De Carufel, C. Grimm, A. Maheshwari, S. Schirra, and M. H. M. Smid. Minimizing the continuous diameter when augmenting a geometric tree with a shortcut. Comput. Geom. Theory Appl., 89:101631, 2020. URL: https://doi.org/10.1016/J.COMGEO.2020.101631.
  15. J.-L. De Carufel, C. Grimm, A. Maheshwari, and M. H. M. Smid. Minimizing the continuous diameter when augmenting paths and cycles with shortcuts. In Proc. 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT'16), volume 53 of LIPIcs, pages 27:1-27:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPICS. SWAT.2016.27.
  16. C. E. Chen and R. S. Garfinkel. The generalized diameter of a graph. Networks, 12(3):335-340, 1982. URL: https://doi.org/10.1002/net.3230120310.
  17. G. Ducoffe. Eccentricity queries and beyond using hub labels. Theoretical Computer Science, 930(21):128-141, 2022. URL: https://doi.org/10.1016/j.tcs.2022.07.017.
  18. G. Ducoffe, M. Habib, and L. 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. URL: https://doi.org/10.1137/20m136551x.
  19. L. Friedlander. Genericity of simple eigenvalues for a metric graph. Israel Journal of Mathematics, 146:149-156, 2005. URL: https://doi.org/10.1007/BF02773531.
  20. D. Garijo, A. Márquez, N. Rodríguez, and R. I. Silveira. Computing optimal shortcuts for networks. Eur. J. Oper. Res., 279(1):26-37, 2019. URL: https://doi.org/10.1016/j.ejor.2019.05.018.
  21. D. Garijo, A. Márquez, and R. I. Silveira. Continuous mean distance of a weighted graph. Results in Mathematics, 78(139):1-36, 2023. URL: https://doi.org/10.1007/s00025-023-01902-w.
  22. P. Gawrychowski, H. Kaplan, S. Mozes, M. Sharir, and O. Weimann. Voronoi diagrams on planar graphs, and computing the diameter in deterministic O(n^5/3) time. SIAM J. Comput., 50(2):509-554, 2021. URL: https://doi.org/10.1137/18M1193402.
  23. J. Gudmundsson and Y. Sha. Augmenting graphs to minimize the radius. Comput. Geom. Theory Appl., 114(101996):1-14, 2023. URL: https://doi.org/10.1016/j.comgeo.2023.101996.
  24. J. Gudmundsson and S. Wong. Improving the dilation of a metric graph by adding edges. ACM Trans. Algorithms, 18(3-Article 20):1-14, 2022. URL: https://doi.org/10.1145/3517807.
  25. S. L. Hakimi. Optimum locations of switching centers and the absolute centers and medians of a graph. Operations Research, 12(3):450-459, 1964. Google Scholar
  26. H. Le and C. Wulff-Nilsen. Vc set systems in minor-free (di)graphs and applications. In Proc. 35th Annu. ACM-SIAM Sympos. Discrete Algorithms (SODA), pages 5332-5360. SIAM, 2024. URL: https://doi.org/10.1137/1.9781611977912.192.
  27. G. Lumer. Connecting of local operators and evolution equations on networks. In Potential Theory Copenhagen 1979, pages 219-234. Springer-Verlag, 1980. URL: https://doi.org/10.1007/BFb0086338.
  28. L. Monier. Combinatorial solutions of multidimensional divide-and-conquer recurrences. Algorithms, 1(1):60-74, 1980. URL: https://doi.org/10.1016/0196-6774(80)90005-X.
  29. L. Roditty and V. Vassilevska Williams. Fast approximation algorithms for the diameter and radius of sparse graphs. In Proc. 45th Annu. ACM Sympos. Theory Comput. (STOC), pages 515-524. ACM, 2013. URL: https://doi.org/10.1145/2488608.2488673.
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