Document Open Access Logo

Isometric Embeddings in Trees and Their Use in Distance Problems

Author Guillaume Ducoffe

PDF

File

Thumbnail PDF
PDF
LIPIcs.MFCS.2021.43.pdf
  • Filesize: 0.81 MB
  • 16 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Graph algorithms analysis
Keywords
  • Tree embeddings
  • Range queries
  • Centroid decomposition
  • Heavy-path decomposition
  • Diameter
  • Radius and all Eccentricities computations

Metrics

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

Abstract

We present powerful techniques for computing the diameter, all the eccentricities, and other related distance problems on some geometric graph classes, by exploiting their "tree-likeness" properties. We illustrate the usefulness of our approach as follows:  
- We propose a subquadratic-time algorithm for computing all eccentricities on partial cubes of bounded lattice dimension and isometric dimension O(n^{0.5-ε}). This is one of the first positive results achieved for the diameter problem on a subclass of partial cubes beyond median graphs. 
- Then, we obtain almost linear-time algorithms for computing all eccentricities in some classes of face-regular plane graphs, including benzenoid systems, with applications to chemistry. Previously, only a linear-time algorithm for computing the diameter and the center was known (and an Õ(n^{5/3})-time algorithm for computing all the eccentricities). 
- We also present an almost linear-time algorithm for computing the eccentricities in a polygon graph with an additive one-sided error of at most 2. 
- Finally, on any cube-free median graph, we can compute its absolute center in almost linear time. Independently from this work, Bergé and Habib have recently presented a linear-time algorithm for computing all eccentricities in this graph class (LAGOS'21), which also implies a linear-time algorithm for the absolute center problem.  Our strategy here consists in exploiting the existence of some embeddings of these graphs in either a system or a product of trees, or in a single tree but where each vertex of the graph is embedded in a subset of nodes. While this may look like a natural idea, the way it can be done efficiently, which is our main technical contribution in the paper, is surprisingly intricate.

Cite As Get BibTex

Guillaume Ducoffe. Isometric Embeddings in Trees and Their Use in Distance Problems. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 43:1-43:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.MFCS.2021.43

Author Details

Guillaume Ducoffe
  • National Institute of Research and Development in Informatics, Bucharest, Romania
  • University of Bucharest, Romania

Funding

  • Ducoffe, Guillaume: This work was supported by project PN-19-37-04-01 “New solutions for complex problems in current ICT research fields based on modelling and optimization”, funded by the Romanian Core Program of the Ministry of Research and Innovation (MCI) 2019-2022.

References

  1. A. Abboud, V. Vassilevska Williams, and J. Wang. Approximation and fixed parameter subquadratic algorithms for radius and diameter in sparse graphs. In Proceedings of the twenty-seventh annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 377-391. SIAM, 2016. Google Scholar
  2. Muad Abu-Ata and Feodor F. Dragan. Metric tree-like structures in real-world networks: an empirical study. Networks, 67(1):49-68, 2016. URL: https://doi.org/10.1002/net.21631.
  3. H. Bandelt and V. Chepoi. Metric graph theory and geometry: a survey. Contemporary Mathematics, 453:49-86, 2008. Google Scholar
  4. H. Bandelt and H. Mulder. Distance-hereditary graphs. Journal of Combinatorial Theory, Series B, 41(2):182-208, 1986. Google Scholar
  5. H.-J. Bandelt. Recognition of tree metrics. SIAM Journal on Discrete Mathematics, 3(1):1-6, 1990. Google Scholar
  6. H.-J. Bandelt, V. Chepoi, and D. Eppstein. Combinatorics and geometry of finite and infinite squaregraphs. SIAM Journal on Discrete Mathematics, 24(4):1399-1440, 2010. Google Scholar
  7. H.-J. Bandelt, V. Chepoi, and D. Eppstein. Ramified rectilinear polygons: coordinatization by dendrons. Discrete & Computational Geometry, 54(4):771-797, 2015. Google Scholar
  8. H.-J. Bandelt and M. van De Vel. Embedding topological median algebras in products of dendrons. Proceedings of the London Mathematical Society, 3(3):439-453, 1989. Google Scholar
  9. V. Batagelj, T. Pisanski, and J. Simoes-Pereira. An algorithm for tree-realizability of distance matrices. International Journal of Computer Mathematics, 34(3-4):171-176, 1990. Google Scholar
  10. L. Bénéteau, J. Chalopin, V. Chepoi, and Y. Vaxès. Medians in median graphs and their cube complexes in linear time. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2020. Google Scholar
  11. P. Bergé and M. Habib. Diameter in linear time for constant-dimension median graphs. In XI Latin and American Algorithms, Graphs and Optimization Symposium (LAGOS 2021), 2021. To appear. Google Scholar
  12. J. A. Bondy and U. S. R. Murty. Graph theory. Springer, 2008. Google Scholar
  13. M. Borassi, P. Crescenzi, and M. Habib. Into the square: On the complexity of some quadratic-time solvable problems. Electronic Notes in Theoretical Computer Science, 322:51-67, 2016. Google Scholar
  14. M. Borassi, P. Crescenzi, and L. Trevisan. An axiomatic and an average-case analysis of algorithms and heuristics for metric properties of graphs. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 920-939. SIAM, 2017. Google Scholar
  15. A. Brandstädt, V. Chepoi, and F. Dragan. The algorithmic use of hypertree structure and maximum neighbourhood orderings. Discrete Applied Mathematics, 82(1-3):43-77, 1998. Google Scholar
  16. B. Bresar, S. Klavzar, and R. Skrekovski. On cube-free median graphs. Discrete Mathematics, 307(3):345-351, 2007. Google Scholar
  17. K. Bringmann, T. Husfeldt, and M. Magnusson. Multivariate Analysis of Orthogonal Range Searching and Graph Distances. Algorithmica, pages 1-24, 2020. Google Scholar
  18. S. Cabello. Subquadratic algorithms for the diameter and the sum of pairwise distances in planar graphs. ACM Transactions on Algorithms (TALG), 15(2):1-38, 2018. Google Scholar
  19. S. Cabello and C. Knauer. Algorithms for graphs of bounded treewidth via orthogonal range searching. Computational Geometry, 42(9):815-824, 2009. Google Scholar
  20. C. Cheng. A poset-based approach to embedding median graphs in hypercubes and lattices. Order, 29(1):147-163, 2012. Google Scholar
  21. V. Chepoi. On distances in benzenoid systems. Journal of chemical information and computer sciences, 36(6):1169-1172, 1996. Google Scholar
  22. V. Chepoi, F. Dragan, M. Habib, Y. Vaxès, and H. Alrasheed. Fast approximation of eccentricities and distances in hyperbolic graphs. Journal of Graph Algorithms and Applications, 23(2):393-433, 2019. Google Scholar
  23. V. Chepoi, F. Dragan, and Y. Vaxès. Center and diameter problems in plane triangulations and quadrangulations. In Symposium on Discrete Algorithms (SODA'02), pages 346-355, 2002. Google Scholar
  24. V Chepoi, F Dragan, and Y Vaxes. Distance and routing problems in plane graphs of non-positive curvature. J. Algorithms, 61:1-30, 2006. Google Scholar
  25. V. Chepoi and M. Hagen. On embeddings of CAT(0) cube complexes into products of trees via colouring their hyperplanes. Journal of Combinatorial Theory, Series B, 103(4):428-467, 2013. Google Scholar
  26. V. Chepoi, A. Labourel, and S. Ratel. Distance and routing labeling schemes for cube-free median graphs. Algorithmica, pages 1-45, 2020. Google Scholar
  27. V. Chepoi and D. Maftuleac. Shortest path problem in rectangular complexes of global nonpositive curvature. Computational Geometry, 46(1):51-64, 2013. Google Scholar
  28. Victor Chepoi, Arnaud Labourel, and Sébastien Ratel. Distance labeling schemes for K₄-free bridged graphs. In International Colloquium on Structural Information and Communication Complexity, pages 310-327. Springer, 2020. Google Scholar
  29. Victor Chepoi, Y Vaxes, and FR Dragan. Distance-based location update and routing in irregular cellular networks. In Sixth International Conference on Software Engineering, Artificial Intelligence, Networking and Parallel/Distributed Computing and First ACIS International Workshop on Self-Assembling Wireless Network, pages 380-387. IEEE, 2005. Google Scholar
  30. D. Coudert, G. Ducoffe, and A. Popa. Fully polynomial FPT algorithms for some classes of bounded clique-width graphs. ACM Transactions on Algorithms (TALG), 15(3):1-57, 2019. Google Scholar
  31. P. Damaschke. Computing giant graph diameters. In International Workshop on Combinatorial Algorithms (IWOCA), pages 373-384. Springer, 2016. Google Scholar
  32. D. Della Giustina, Ni. Prezza, and R. Venturini. A new linear-time algorithm for centroid decomposition. In String Processing and Information Retrieval, pages 274-282. Springer International Publishing, 2019. Google Scholar
  33. R. Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012. Google Scholar
  34. D Ž Djoković. Distance-preserving subgraphs of hypercubes. Journal of Combinatorial Theory, Series B, 14(3):263-267, 1973. Google Scholar
  35. F. Dragan and M. Abu-Ata. Collective additive tree spanners of bounded tree-breadth graphs with generalizations and consequences. Theoretical Computer Science, 547:1-17, 2014. Google Scholar
  36. F. Dragan, D. Corneil, E. Köhler, and Y. Xiang. Collective additive tree spanners for circle graphs and polygonal graphs. Discrete Applied Mathematics, 160(12):1717-1729, 2012. Google Scholar
  37. F. Dragan, Y. Xiang, and C. Yan. Collective Tree Spanners for Unit Disk Graphs with Applications. Electronic Notes in Discrete Mathematics, 32:117-124, 2009. Google Scholar
  38. F. Dragan and C. Yan. Collective tree spanners in graphs with bounded parameters. Algorithmica, 57(1):22-43, 2010. Google Scholar
  39. F. Dragan, C. Yan, and D. Corneil. Collective Tree Spanners and Routing in AT-free Related Graphs. Journal of Graph Algorithms and Applications, 10(2):97-122, 2006. Google Scholar
  40. F. Dragan, C. Yan, and I. Lomonosov. Collective tree spanners of graphs. SIAM Journal on Discrete Mathematics, 20(1):240-260, 2006. Google Scholar
  41. F. Dragan, C. Yan, and Y. Xiang. Collective additive tree spanners of homogeneously orderable graphs. In Latin American Symposium on Theoretical Informatics, pages 555-567. Springer, 2008. Google Scholar
  42. G. Ducoffe. A New Application of Orthogonal Range Searching for Computing Giant Graph Diameters. In Symposium on Simplicity in Algorithms (SOSA), 2019. Google Scholar
  43. G. Ducoffe. Easy computation of eccentricity approximating trees. Discrete Applied Mathematics, 260:267-271, 2019. Google Scholar
  44. G. Ducoffe. Optimal diameter computation within bounded clique-width graphs. Technical report, arXiv, 2020. URL: http://arxiv.org/abs/2011.08448.
  45. G. Ducoffe and F.F. Dragan. A story of diameter, radius and (almost) helly property. Networks, 2021. To appear. Google Scholar
  46. G. Ducoffe, M. Habib, and L. Viennot. Fast diameter computation within split graphs. In International Conference on Combinatorial Optimization and Applications, pages 155-167. Springer, 2019. Google Scholar
  47. G. Ducoffe, M. Habib, and L. Viennot. Diameter computation on H-minor free graphs and graphs of bounded (distance) VC-dimension. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1905-1922. SIAM, 2020. Google Scholar
  48. D. Eppstein. The lattice dimension of a graph. European Journal of Combinatorics, 26(5):585-592, 2005. Google Scholar
  49. D. Eppstein. Recognizing Partial Cubes in Quadratic Time. Journal of Graph Algorithms and Applications, 5(2):269–293, 2011. Google Scholar
  50. S. Even and A. Litman. Layered cross product—A technique to construct interconnection networks. In Proceedings of the fourth annual ACM symposium on Parallel algorithms and architectures, pages 60-69, 1992. Google Scholar
  51. A. Farley and A. Proskurowski. Computation of the center and diameter of outerplanar graphs. Discrete Applied Mathematics, 2(3):185-191, 1980. Google Scholar
  52. A. Filtser and H. Le. Clan embeddings into trees, and low treewidth graphs. In 53rd Annual ACM Symposium on Theory of Computing (STOC 2021). ACM, 2021. To appear. Google Scholar
  53. H. Gabow, J. Bentley, and R. Tarjan. Scaling and related techniques for geometry problems. In Proceedings of the sixteenth annual ACM symposium on Theory of computing, pages 135-143, 1984. Google Scholar
  54. C. Gavoille, D. Peleg, S. Pérennes, and R. Raz. Distance labeling in graphs. Journal of Algorithms, 53(1):85-112, 2004. Google Scholar
  55. P. Gawrychowski, H. Kaplan, S. Mozes, M. Sharir, and O. Weimann. Voronoi diagrams on planar graphs, and computing the diameter in deterministic Õ(n^5/3) time. In Symposium on Discrete Algorithms (SODA), pages 495-514. SIAM, 2018. Google Scholar
  56. A. Goldman. Optimal center location in simple networks. Transportation science, 5(2):212-221, 1971. Google Scholar
  57. M. Golumbic. Algorithmic graph theory and perfect graphs, volume 57. Elsevier, 2004. Google Scholar
  58. Mikhaïl Gromov. Hyperbolic groups. In Essays in group theory, pages 75-263. Springer, 1987. URL: https://doi.org/10.1007/978-1-4613-9586-7_3.
  59. 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
  60. G.Y. Handler. Minimax location of a facility in an undirected tree graph. Transportation Science, 7(3):287-293, 1973. Google Scholar
  61. E. Howorka. On metric properties of certain clique graphs. Journal of Combinatorial Theory, Series B, 27(1):67-74, 1979. Google Scholar
  62. C. Jordan. Sur les assemblages de lignes. J. Reine Angew. Math, 70(185):81, 1869. Google Scholar
  63. O. Kariv and S.L. Hakimi. An algorithmic approach to network location problems. I: The p-centers. SIAM Journal on Applied Mathematics, 37(3):513-538, 1979. Google Scholar
  64. S. Khuller and Y. Sussmann. The capacitated k-center problem. SIAM Journal on Discrete Mathematics, 13(3):403-418, 2000. Google Scholar
  65. Y.-F. Lan, Y.-L. Wang, and H. Suzuki. A linear-time algorithm for solving the center problem on weighted cactus graphs. Information Processing Letters, 71(5-6):205-212, 1999. Google Scholar
  66. H. M. Mulder and A. Schrijver. Median graphs and Helly hypergraphs. Discrete Mathematics, 25(1):41-50, 1979. Google Scholar
  67. M. Nielsen, G. Plotkin, and G. Winskel. Petri nets, event structures and domains, part I. Theoretical Computer Science, 13(1):85-108, 1981. Google Scholar
  68. R. Nowakowski and I. Rival. The smallest graph variety containing all paths. Discrete Mathematics, 43(2-3):223-234, 1983. Google Scholar
  69. S. Olariu. A simple linear-time algorithm for computing the center of an interval graph. International Journal of Computer Mathematics, 34(3-4):121-128, 1990. Google Scholar
  70. L. Roditty and V. Vassilevska Williams. Fast approximation algorithms for the diameter and radius of sparse graphs. In Proceedings of the forty-fifth annual ACM symposium on Theory of computing (STOC), pages 515-524, 2013. Google Scholar
  71. D. Sleator and R. Tarjan. A data structure for dynamic trees. Journal of Computer and System Sciences, 26(3):362-391, 1983. Google Scholar
  72. L. Stewart and R. Valenzano. On polygon numbers of circle graphs and distance hereditary graphs. Discrete Applied Mathematics, 248:3-17, 2018. Google Scholar
  73. S. Ting. A linear-time algorithm for maxisum facility location on tree networks. Transportation Science, 18(1):76-84, 1984. Google Scholar
  74. D. Willard. New data structures for orthogonal range queries. SIAM Journal on Computing, 14(1):232–253, 1985. Google Scholar
  75. P.M. Winkler. Isometric embedding in products of complete graphs. Discrete Applied Mathematics, 7(2):221-225, 1984. 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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact