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Oriented Spanners

Authors Kevin Buchin , Joachim Gudmundsson , Antonia Kalb , Aleksandr Popov , Carolin Rehs , André van Renssen , Sampson Wong

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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • computational geometry
  • spanner
  • oriented graph
  • greedy triangulation

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Abstract

Given a point set P in the Euclidean plane and a parameter t, we define an oriented t-spanner as an oriented subgraph of the complete bi-directed graph such that for every pair of points, the shortest cycle in G through those points is at most a factor t longer than the shortest oriented cycle in the complete bi-directed graph. We investigate the problem of computing sparse graphs with small oriented dilation.
As we can show that minimising oriented dilation for a given number of edges is NP-hard in the plane, we first consider one-dimensional point sets. While obtaining a 1-spanner in this setting is straightforward, already for five points such a spanner has no plane embedding with the leftmost and rightmost point on the outer face. This leads to restricting to oriented graphs with a one-page book embedding on the one-dimensional point set. For this case we present a dynamic program to compute the graph of minimum oriented dilation that runs in 𝒪(n⁸) time for n points, and a greedy algorithm that computes a 5-spanner in 𝒪(nlog n) time.
Expanding these results finally gives us a result for two-dimensional point sets: we prove that for convex point sets the greedy triangulation results in an oriented 𝒪(1)-spanner.

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Kevin Buchin, Joachim Gudmundsson, Antonia Kalb, Aleksandr Popov, Carolin Rehs, André van Renssen, and Sampson Wong. Oriented Spanners. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 26:1-26:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ESA.2023.26

Author Details

Kevin Buchin
  • Technical University of Dortmund, Germany
Joachim Gudmundsson
  • University of Sydney, Australia
Antonia Kalb
  • Technical University of Dortmund, Germany
Aleksandr Popov
  • Technical University of Eindhoven, The Netherlands
Carolin Rehs
  • Technical University of Dortmund, Germany
André van Renssen
  • University of Sydney, Australia
Sampson Wong
  • BARC, University of Copenhagen, Denmark

Funding

  • Popov, Aleksandr: Supported by the Dutch Research Council (NWO) under the project number 612.001.801.

References

  1. Hugo A. Akitaya, Ahmad Biniaz, and Prosenjit Bose. On the spanning and routing ratios of the directed Θ_6-graph. Comput. Geom., 105-106:101881, 2022. URL: https://doi.org/10.1016/j.comgeo.2022.101881.
  2. Boris Aronov, Kevin Buchin, Maike Buchin, Bart Jansen, Tom De Jong, Marc van Kreveld, Maarten Löffler, Jun Luo, Bettina Speckmann, and Rodrigo Ignacio Silveira. Connect the dot: Computing feed-links for network extension. J. Spatial Information Science, 3:3-31, 2011. URL: https://doi.org/10.5311/JOSIS.2011.3.47.
  3. Michael A. Bekos, Martin Gronemann, and Chrysanthi N. Raftopoulou. Two-page book embeddings of 4-planar graphs. Algorithmica, 75(1):158-185, 2016. URL: https://doi.org/10.1007/s00453-015-0016-8.
  4. Prosenjit Bose, Aaron Lee, and Michiel H. M. Smid. On generalized diamond spanners. In Proc. 10th Internat. Sympos. Algorithms and Data Struct., volume 4619 of Lecture Notes in Computer Science, pages 325-336. Springer-Verlag, 2007. URL: https://doi.org/10.1007/978-3-540-73951-7_29.
  5. Prosenjit Bose and Michiel Smid. On plane geometric spanners: A survey and open problems. Comput. Geom. Theory Appl., 46(7):818-830, 2013. URL: https://doi.org/10.1016/j.comgeo.2013.04.002.
  6. Aléx F. Brandt, Miguel M. Gaiowski, Cid C. de Souza, and Pedro J. de Rezende. Minimum dilation triangulation: Reaching optimality efficiently. In Proc. 26th Canad. Conf. Comput. Geom. (CCCG), 2014. Google Scholar
  7. Martin Burkhart, Pascal Von Rickenbach, Rogert Wattenhofer, and Aaron Zollinger. Does topology control reduce interference? In Proc. 5th ACM Internat. Sympos. Mobile Ad Hoc Networking and Computing, pages 9-19, 2004. URL: https://doi.org/10.1145/989459.989462.
  8. Fan R. K. Chung, Frank Thomson Leighton, and Arnold L. Rosenberg. Embedding graphs in books: A layout problem with applications to vlsi design. SIAM J. Algebraic Discrete Methods, 8(1):33-58, 1987. URL: https://doi.org/10.1137/0608002.
  9. Lenore J. Cowen and Christopher G. Wagner. Compact roundtrip routing for digraphs. In Proc. 10th Annu. ACM-SIAM Sympos. Discrete Algorithms (SODA), pages 885-886. SIAM, 1999. URL: https://doi.org/10.5555/314500.315068.
  10. Lenore J. Cowen and Christopher G. Wagner. Compact roundtrip routing in directed networks. J. Algorithms, 50(1):79-95, 2004. URL: https://doi.org/10.1016/j.jalgor.2003.08.001.
  11. Gautam Das and Deborah Joseph. Which triangulations approximate the complete graph? In Optimal Algorithms, volume 401 of Lecture Notes in Computer Science, pages 168-192. Springer-Verlag, 1989. URL: https://doi.org/10.1007/3-540-51859-2_15.
  12. Andrew Dobson and Kostas E. Bekris. Sparse roadmap spanners for asymptotically near-optimal motion planning. Internat. J. Robotics Research, 33(1):18-47, 2014. URL: https://doi.org/10.1177/0278364913498.
  13. Vida Dujmovic and David R. Wood. On linear layouts of graphs. Discret. Math. Theor. Comput. Sci., 6(2):339-358, 2004. URL: https://doi.org/10.46298/dmtcs.317.
  14. David Eppstein. Spanning trees and spanners. In Handbook of Computational Geometry, pages 425-461. Elsevier, 2000. URL: https://doi.org/h10.1016/B978-044482537-7/50010-3.
  15. Panos Giannopoulos, Rolf Klein, Christian Knauer, Martin Kutz, and Dániel Marx. Computing geometric minimum-dilation graphs is NP-hard. Internat. J. Comput. Geom. Appl., 20(2):147-173, 2010. URL: https://doi.org/10.1142/S0218195910003244.
  16. Robert L. (Scot) Drysdale III, Scott A. McElfresh, and Jack Snoeyink. On exclusion regions for optimal triangulations. Discret. Appl. Math., 109(1-2):49-65, 2001. URL: https://doi.org/10.1016/S0166-218X(00)00236-5.
  17. J. Mark Keil and Carl A. Gutwin. Classes of graphs which approximate the complete euclidean graph. Discrete Comput. Geom., 7:13-28, 1992. URL: https://doi.org/10.1007/BF02187821.
  18. Christos Levcopoulos and Drago Krznaric. The greedy triangulation can be computed from the Delaunay triangulation in linear time. Comput. Geom. Theory Appl., 14(4):197-220, 1999. URL: https://doi.org/10.1016/S0925-7721(99)00037-1.
  19. Christos Levcopoulos and Andrzej Lingas. Fast algorithms for greedy triangulation. BIT, 32(2):280-296, 1992. URL: https://doi.org/10.1007/BF01994882.
  20. Giri Narasimhan and Michiel Smid. Geometric Spanner Networks. Cambridge University Press, 2007. URL: https://doi.org/10.1017/CBO9780511546884.
  21. Christian Schindelhauer, Klaus Volbert, and Martin Ziegler. Geometric spanners with applications in wireless networks. Comput. Geom. Theory Appl., 36(3):197-214, 2007. URL: https://doi.org/10.1016/j.comgeo.2006.02.001.
  22. Mihalis Yannakakis. Embedding planar graphs in four pages. J. Comput. Syst. Sci., 38(1):36-67, 1989. URL: https://doi.org/10.1016/0022-0000(89)90032-9.
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