Document Open Access Logo

Computation of Cycle Bases in Surface Embedded Graphs

Authors Kyle Fox, Thomas Stanley

PDF

File

Thumbnail PDF
PDF
LIPIcs.ISAAC.2022.13.pdf
  • Filesize: 0.65 MB
  • 13 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Computational geometry
  • Mathematics of computing → Graphs and surfaces
Keywords
  • cycle basis
  • surface embedded graphs
  • homology

Metrics

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

Abstract

We present an O(n³ g²log g + m) + Õ(n^{ω + 1}) time deterministic algorithm to find the minimum cycle basis of a directed graph embedded on an orientable surface of genus g. This result improves upon the previous fastest known running time of O(m³n + m²n² log n) applicable to general directed graphs.
While an O(n^ω + 2^{2g}n² + m) time deterministic algorithm was known for undirected graphs, the use of the underlying field ℚ in the directed case (as opposed to ℤ₂ for the undirected case) presents extra challenges. It turns out that some of our new observations are useful for both variants of the problem, so we present an O(n^ω + n² g² log g + m) time deterministic algorithm for undirected graphs as well.

Cite As Get BibTex

Kyle Fox and Thomas Stanley. Computation of Cycle Bases in Surface Embedded Graphs. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 13:1-13:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ISAAC.2022.13

Author Details

Kyle Fox
  • University of Texas at Dallas, Richardson, TX, USA
Thomas Stanley
  • Unaffiliated, Dallas, TX, USA

Funding

  • Fox, Kyle: Research partially supported by NSF grant CCF-1942597.

Acknowledgements

Research by T. Stanley was partially performed while this author was a student at the University of Texas at Dallas.

References

  1. Edoardo Amaldi, Claudio Iuliano, Tomasz Jurkiewicz, Kurt Mehlhorn, and Romeo Rizzi. Breaking the o(m2n) barrier for minimum cycle bases. In Amos Fiat and Peter Sanders, editors, Algorithms - ESA 2009, pages 301-312, Berlin, Heidelberg, 2009. Springer Berlin Heidelberg. Google Scholar
  2. Glencora Borradaile, Erin Wolf Chambers, Kyle Fox, and Amir Nayyeri. Minimum cycle and homology bases of surface-embedded graphs. J. Comput. Geom., 8(2):58-79, 2017. URL: https://doi.org/10.20382/jocg.v8i2a4.
  3. Glencora Borradaile, Piotr Sankowski, and Christian Wulff-Nilsen. Min st-cut oracle for planar graphs with near-linear preprocessing time. ACM Trans. Algorithms, 11(3):16:1-16:29, 2015. URL: https://doi.org/10.1145/2684068.
  4. Sergio Cabello, Éric Colin de Verdière, and Francis Lazarus. Finding shortest non-trivial cycles in directed graphs on surfaces. In Proceedings of the Twenty-Sixth Annual Symposium on Computational Geometry, SoCG '10, pages 156-165, New York, NY, USA, 2010. Association for Computing Machinery. URL: https://doi.org/10.1145/1810959.1810988.
  5. A. C. Cassell, J. C. De C. Henderson, K. Ramachandran, and Alec Westley Skempton. Cycle bases of minimal measure for the structural analysis of skeletal structures by the flexibility method. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 350(1660):61-70, 1976. URL: https://doi.org/10.1098/rspa.1976.0095.
  6. L. Chua and Li-Kuan Chen. On optimally sparse cycle and coboundary basis for a linear graph. IEEE Transactions on Circuit Theory, 20(5):495-503, 1973. URL: https://doi.org/10.1109/TCT.1973.1083752.
  7. J. Coelho de Pina. Applications of shortest path methods. PhD thesis, University of Amsterdam, 1995. Google Scholar
  8. David Eppstein. Dynamic generators of topologically embedded graphs. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '03, pages 599-608, USA, 2003. Society for Industrial and Applied Mathematics. Google Scholar
  9. Jeff Erickson and Amir Nayyeri. Minimum cuts and shortest non-separating cycles via homology covers. In Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '11, pages 1166-1176, USA, 2011. Society for Industrial and Applied Mathematics. Google Scholar
  10. Petra Gleiss, Josef Leydold, and Peter Stadler. Circuit bases of strongly connected digraphs. Santa Fe Institute, Working Papers, 23, January 2001. URL: https://doi.org/10.7151/dmgt.1200.
  11. Joshua Evan Greene. On loops intersecting at most once. Geometric and Functional Analysis, 29(6):1828-1843, December 2019. URL: https://doi.org/10.1007/s00039-019-00517-0.
  12. Ramesh Hariharan, Telikepalli Kavitha, and Kurt Mehlhorn. Faster algorithms for minimum cycle basis in directed graphs. SIAM Journal on Computing, 38(4):1430-1447, 2008. URL: https://doi.org/10.1137/060670730.
  13. David Hartvigsen and Russell Mardon. The all-pairs min cut problem and the minimum cycle basis problem on planar graphs. SIAM J. Discret. Math., 7(3):403-418, August 1994. URL: https://doi.org/10.1137/S0895480190177042.
  14. Joseph Horton. A polynomial-time algorithm to find the shortest cycle basis of a graph. SIAM J. Comput., 16:358-366, April 1987. URL: https://doi.org/10.1137/0216026.
  15. Telikepalli Kavitha and Kurt Mehlhorn. Algorithms to compute minimum cycle basis in directed graphs. Theory of Computing Systems, 40:485-505, June 2007. URL: https://doi.org/10.1007/s00224-006-1319-6.
  16. Donald E. Knuth. The Art of Computer Programming, Volume 1 (3rd Ed.): Fundamental Algorithms. Addison Wesley Longman Publishing Co., Inc., USA, 1997. Google Scholar
  17. Kurt Mehlhorn and Dimitrios Michail. Minimum cycle bases: Faster and simpler. ACM Trans. Algorithms, 6(1), December 2010. URL: https://doi.org/10.1145/1644015.1644023.
  18. Abhishek Rathod. Fast algorithms for minimum cycle basis and minimum homology basis. In Sergio Cabello and Danny Z. Chen, editors, 36th International Symposium on Computational Geometry, SoCG 2020, June 23-26, 2020, Zürich, Switzerland, volume 164 of LIPIcs, pages 64:1-64:11. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.SoCG.2020.64.
  19. Geetika Tewari, Craig Gotsman, and Steven Gortler. Meshing genus-1 point clouds using discrete one-forms. Computers & Graphics, 30:917-926, December 2006. URL: https://doi.org/10.1016/j.cag.2006.08.019.
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