Document Open Access Logo

Computing the L1 Geodesic Diameter and Center of a Polygonal Domain

Authors Sang Won Bae, Matias Korman, Joseph S. B. Mitchell, Yoshio Okamoto, Valentin Polishchuk, Haitao Wang

PDF

File

Thumbnail PDF
PDF
LIPIcs.STACS.2016.14.pdf
  • Filesize: 0.66 MB
  • 14 pages

Document Identifiers

Subject Classification

Keywords
  • geodesic diameter
  • geodesic center
  • shortest paths
  • polygonal domains
  • L1 metric

Metrics

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

Abstract

For a polygonal domain with h holes and a total of n vertices, we present algorithms that compute the L_1 geodesic diameter in O(n^2+h^4) time and the L_1 geodesic center in O((n^4+n^2 h^4)*alpha(n)) time, where alpha(.) denotes the inverse Ackermann function. No algorithms were known for these problems before. For the Euclidean counterpart, the best algorithms compute the geodesic diameter in O(n^{7.73}) or O(n^7(h+log(n))) time, and compute the geodesic center in O(n^{12+epsilon}) time. Therefore, our algorithms are much faster than the algorithms for the Euclidean problems. Our algorithms are based on several interesting observations on L_1 shortest paths in polygonal domains.

Cite As Get BibTex

Sang Won Bae, Matias Korman, Joseph S. B. Mitchell, Yoshio Okamoto, Valentin Polishchuk, and Haitao Wang. Computing the L1 Geodesic Diameter and Center of a Polygonal Domain. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 14:1-14:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.STACS.2016.14

Author Details

Sang Won Bae
Matias Korman
Joseph S. B. Mitchell
Yoshio Okamoto
Valentin Polishchuk
Haitao Wang

References

  1. H.-K. Ahn, L. Barba, P. Bose, J.-L. De Carufel, M. Korman, and E. Oh. A linear-time algorithm for the geodesic center of a simple polygon. In Proc. of the 31st Symposium on Computational Geometry (SoCG), pages 209-223, 2015. Google Scholar
  2. T. Asano and G. Toussaint. Computing the geodesic center of a simple polygon. Technical Report SOCS-85.32, McGill University, Montreal, Canada, 1985. Google Scholar
  3. S.W. Bae, M. Korman, J.S.B. Mitchell, Y. Okamoto, V. Polishchuk, and H. Wang. Computing the L₁ geodesic diameter and center of a polygonal domain. arXiv:1512.07160, 2015. Google Scholar
  4. S.W. Bae, M. Korman, and Y. Okamoto. The geodesic diameter of polygonal domains. Discrete and Computational Geometry, 50:306-329, 2013. Google Scholar
  5. S.W. Bae, M. Korman, and Y. Okamoto. Computing the geodesic centers of a polygonal domain. In Proc. of the 26th Canadian Conference on Computational Geometry, 2014. Google Scholar
  6. S.W. Bae, M. Korman, Y. Okamoto, and H. Wang. Computing the L₁ geodesic diameter and center of a simple polygon in linear time. Computational Geometry: Theory and Applications, 48:495-505, 2015. Google Scholar
  7. R. Bar-Yehuda and B. Chazelle. Triangulating disjoint Jordan chains. International Journal of Computational Geometry and Applications, 4(4):475-481, 1994. Google Scholar
  8. B. Chazelle. A theorem on polygon cutting with applications. In Proc. of the 23rd Annual Symposium on Foundations of Computer Science, pages 339-349, 1982. Google Scholar
  9. D.Z. Chen and H. Wang. A nearly optimal algorithm for finding L₁ shortest paths among polygonal obstacles in the plane. In Proc. of the 19th European Symposium on Algorithms, pages 481-492, 2011. Google Scholar
  10. D.Z. Chen and H. Wang. Computing the visibility polygon of an island in a polygonal domain. In Proc. of the 39th International Colloquium on Automata, Languages and Programming, pages 218-229, 2012. Journal version published online in Algorithmica, 2015. Google Scholar
  11. D.Z. Chen and H. Wang. L₁ shortest path queries among polygonal obstacles in the plane. In Proc. of the 30th Symposium on Theoretical Aspects of Computer Science, pages 293-304, 2013. Google Scholar
  12. D.Z. Chen and H. Wang. Visibility and ray shooting queries in polygonal domains. Computational Geometry: Theory and Applications, 48:31-41, 2015. Google Scholar
  13. H. Edelsbrunner, L.J. Guibas, and M. Sharir. The upper envelope of piecewise linear functions: Algorithms and applications. Discrete and Computational Geometry, 4:311-336, 1989. Google Scholar
  14. L.J. Guibas, J. Hershberger, D. Leven, M. Sharir, and R.E. Tarjan. Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons. Algorithmica, 2(1-4):209-233, 1987. Google Scholar
  15. J. Hershberger and J. Snoeyink. Computing minimum length paths of a given homotopy class. Computational Geometry: Theory and Applications, 4(2):63-97, 1994. Google Scholar
  16. J. Hershberger and S. Suri. Matrix searching with the shortest-path metric. SIAM Journal on Computing, 26(6):1612-1634, 1997. Google Scholar
  17. R. Inkulu and S. Kapoor. Planar rectilinear shortest path computation using corridors. Computational Geometry: Theory and Applications, 42(9):873-884, 2009. Google Scholar
  18. S. Kapoor, S.N. Maheshwari, and J.S.B. Mitchell. An efficient algorithm for Euclidean shortest paths among polygonal obstacles in the plane. Discrete and Computational Geometry, 18(4):377-383, 1997. Google Scholar
  19. J.S.B. Mitchell. An optimal algorithm for shortest rectilinear paths among obstacles. In the 1st Canadian Conference on Computational Geometry, 1989. Google Scholar
  20. J.S.B. Mitchell. L₁ shortest paths among polygonal obstacles in the plane. Algorithmica, 8(1):55-88, 1992. Google Scholar
  21. R. Pollack, M. Sharir, and G. Rote. Computing the geodesic center of a simple polygon. Discrete and Computational Geometry, 4(1):611-626, 1989. Google Scholar
  22. S. Schuierer. Computing the L₁-diameter and center of a simple rectilinear polygon. In Proc. of the International Conference on Computing and Information, pages 214-229, 1994. Google Scholar
  23. S. Suri. Computing geodesic furthest neighbors in simple polygons. Journal of Computer and System Sciences, 39:220-235, 1989. 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