Document Open Access Logo

Rectilinear Link Diameter and Radius in a Rectilinear Polygonal Domain

Authors Elena Arseneva , Man-Kwun Chiu, Matias Korman, Aleksandar Markovic, Yoshio Okamoto , Aurélien Ooms , André van Renssen , Marcel Roeloffzen

PDF

File

Thumbnail PDF
PDF
LIPIcs.ISAAC.2018.58.pdf
  • Filesize: 433 kB
  • 13 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Rectilinear link distance
  • polygonal domain
  • diameter
  • radius

Metrics

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

Abstract

We study the computation of the diameter and radius under the rectilinear link distance within a rectilinear polygonal domain of n vertices and h holes. We introduce a graph of oriented distances to encode the distance between pairs of points of the domain. This helps us transform the problem so that we can search through the candidates more efficiently. Our algorithm computes both the diameter and the radius in O(min(n^omega, n^2 + nh log h + chi^2)) time, where omega<2.373 denotes the matrix multiplication exponent and chi in Omega(n) cap O(n^2) is the number of edges of the graph of oriented distances. We also provide an alternative algorithm for computing the diameter that runs in O(n^2 log n) time.

Cite As Get BibTex

Elena Arseneva, Man-Kwun Chiu, Matias Korman, Aleksandar Markovic, Yoshio Okamoto, Aurélien Ooms, André van Renssen, and Marcel Roeloffzen. Rectilinear Link Diameter and Radius in a Rectilinear Polygonal Domain. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 58:1-58:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.ISAAC.2018.58

Author Details

Elena Arseneva
  • St. Petersburg State University, St. Petersburg, Russia
Man-Kwun Chiu
  • Institut für Informatik, Freie Universität Berlin, Berlin, Germany
Matias Korman
  • Tufts University, Boston, USA
Aleksandar Markovic
  • TU Eindhoven, Eindhoven, The Netherlands
Yoshio Okamoto
  • University of Electro-Communications, Tokyo, Japan, RIKEN Center for Advanced Intelligent Project, Tokyo, Japan
Aurélien Ooms
  • Université libre de Bruxelles (ULB), Brussels, Belgium
André van Renssen
  • University of Sydney, Sydney, Australia
Marcel Roeloffzen
  • TU Eindhoven, Eindhoven, The Netherlands

Funding

  • Arseneva, Elena: Partially supported by the SNF Early Postdoc Mobility grant P2TIP2-168563, Switzerland, and F.R.S.-FNRS, Belgium.
  • Chiu, Man-Kwun: Supported in part by ERC StG 757609.
  • Korman, Matias: Supported in part by KAKENHI No. 17K12635, Japan and NSF award CCF-1422311.
  • Markovic, Aleksandar: Supported by the Netherlands' Organisation for Scientific Research (NWO) under project no. 024.002.003.
  • Okamoto, Yoshio: Partially supported by JSPS KAKENHI Grant Number 15K00009 and JST CREST Grant Number JPMJCR1402, and Kayamori Foundation of Informational Science Advancement.
  • Ooms, Aurélien: Supported by the Fund for Research Training in Industry and Agriculture (FRIA).
  • van Renssen, André: Supported by JST ERATO Grant Number JPMJER1201, Japan.

References

  1. Hee-Kap Ahn, Luis Barba, Prosenjit Bose, Jean-Lou De Carufel, Matias Korman, and Eunjin Oh. A Linear-Time Algorithm for the Geodesic Center of a Simple Polygon. Discrete & Computational Geometry, 56(4):836-859, 2016. URL: http://dx.doi.org/10.1007/s00454-016-9796-0.
  2. 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. Discrete & Computational Geometry, 57(3):674-701, 2017. URL: http://dx.doi.org/10.1007/s00454-016-9841-z.
  3. Sang Won Bae, Matias Korman, and Yoshio Okamoto. The Geodesic Diameter of Polygonal Domains. Discrete & Computational Geometry, 50(2):306-329, 2013. URL: http://dx.doi.org/10.1007/s00454-013-9527-8.
  4. Sang Won Bae, Matias Korman, Yoshio Okamoto, and Haitao Wang. Computing the L₁ geodesic diameter and center of a simple polygon in linear time. Computational Geometry: Theory and Applications, 48(6):495-505, 2015. URL: http://dx.doi.org/10.1016/j.comgeo.2015.02.005.
  5. Timothy M. Chan and Dimitrios Skrepetos. All-Pairs Shortest Paths in Geometric Intersection Graphs. In Proc. 16th Algorithms and Data Structures Symposium, pages 253-264, 2017. Google Scholar
  6. Hristo Djidjev, Andrzej Lingas, and Jörg-Rüdiger Sack. An O(n log n) Algorithm for Computing the Link Center of a Simple Polygon. Discrete & Computational Geometry, 8:131-152, 1992. URL: http://dx.doi.org/10.1007/BF02293040.
  7. Yoav Giyora and Haim Kaplan. Optimal dynamic vertical ray shooting in rectilinear planar subdivisions. ACM Transactions on Algorithms, 5(3):28:1-28:51, 2009. URL: http://dx.doi.org/10.1145/1541885.1541889.
  8. John Hershberger and Subhash Suri. Matrix Searching with the Shortest-Path Metric. SIAM Journal on Computing, 26(6):1612-1634, 1997. URL: http://dx.doi.org/10.1137/S0097539793253577.
  9. John Hershberger and Subhash Suri. An Optimal Algorithm for Euclidean Shortest Paths in the Plane. SIAM Journal on Computing, 28(6):2215-2256, 1999. URL: http://dx.doi.org/10.1137/S0097539795289604.
  10. François Le Gall. Powers of tensors and fast matrix multiplication. In Katsusuke Nabeshima, Kosaku Nagasaka, Franz Winkler, and Ágnes Szántó, editors, Proc. 25th International Symposium on Symbolic and Algebraic Computation, ISSAC, pages 296-303. ACM, 2014. URL: http://dx.doi.org/10.1145/2608628.2608664.
  11. Joseph S. B. Mitchell. An optimal algorithm for shortest rectilinear paths among obstacles. In Proc. 1st Canadian Conference on Computational Geometry, CCCG, 1989. Google Scholar
  12. Joseph S. B. Mitchell. L₁ shortest paths among polygonal obstacles in the plane. Algorithmica, 8(1):55-88, 1992. Google Scholar
  13. Joseph S. B. Mitchell, Valentin Polishchuk, and Mikko Sysikaski. Minimum-link paths revisited. Computational Geometry: Theory and Applications, 47(6):651-667, 2014. URL: http://dx.doi.org/10.1016/j.comgeo.2013.12.005.
  14. Joseph S. B. Mitchell, Valentin Polishchuk, Mikko Sysikaski, and Haitao Wang. An Optimal Algorithm for Minimum-Link Rectilinear Paths in Triangulated Rectilinear Domains. In Magnús M. Halldórsson, Kazuo Iwama, Naoki Kobayashi, and Bettina Speckmann, editors, Proc. Automata, Languages, and Programming - 42nd International Colloquium, ICALP, volume 9134 of Lecture Notes in Computer Science, pages 947-959. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-662-47672-7_77.
  15. Bengt J. Nilsson and Sven Schuierer. Computing the Rectilinear Link Diameter of a Polygon. In Proc. Computational Geometry - Methods, Algorithms and Applications, International Workshop on Computational Geometry CG'91, pages 203-215, 1991. URL: http://dx.doi.org/10.1007/3-540-54891-2_15.
  16. Bengt J. Nilsson and Sven Schuierer. An Optimal Algorithm for the Rectilinear Link Center of a Rectilinear Polygon. Computational Geometry: Theory and Applications, 6:169-194, 1996. URL: http://dx.doi.org/10.1016/0925-7721(95)00026-7.
  17. Subhash Suri. On some link distance problems in a simple polygon. IEEE Transactions on Robotics and Automation, 6(1):108-113, 1990. URL: http://dx.doi.org/10.1109/70.88124.
  18. Haitao Wang. On the geodesic centers of polygonal domains. Journal of Computational Geometry, 9(1):131-190, 2018. 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