Shortest Rectilinear Path Queries to Rectangles in a Rectangular Domain (original) (raw)

Abstract

Given a set of open axis-aligned disjoint rectangles in the plane, each of which behaves as both an obstacle and a target, we seek to find shortest obstacle-avoiding rectilinear paths from a query to the nearest target and the farthest target. In our problem, the distance to a target is determined by the point on the target achieving the minimum or maximum geodesic distance among all points on the boundary of the target. This problem arises in facility location and robot motion planning problems. We show how to construct a data structure for such shortest path queries to the nearest and farthest neighbors efficiently.

M. Kim and H.-K. Ahn were supported by the Institute of Information & communications Technology Planning & Evaluation(IITP) grant funded by the Korea government(MSIT) (No. 2017-0-00905, Software Star Lab (Optimal Data Structure and Algorithmic Applications in Dynamic Geometric Environment)) and (No. 2019-0-01906, Artificial Intelligence Graduate School Progra (POSTECH)). S.D. Yoon was supported by “Cooperative Research Program for Agriculture Science & Technology Development (Project No. PJ01526903)” Rural Development Administration, Republic of Korea.

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References

  1. Ben-Moshe, B., Bhattacharya, B., Shi, Q.: Farthest neighbor Voronoi diagram in the presence of rectangular obstacles. In: Proceedings of the 13th Canadian Conference on Computational Geometry (CCCG), pp. 243–246 (2005)
    Google Scholar
  2. Ben-Moshe, B., Katz, M., Mitchell, J.: Farthest neighbors and center points in the presence of rectangular obstacles. In: Proceedings of the 17th Annual Symposium on Computational Geometry (SoCG), pp. 164–171 (2001)
    Google Scholar
  3. Chen, D., Inkulu, R., Wang, H.: Two-point \(L_1\) shortest path queries in the plane. In: Proceedings of the 30th Annual Symposium on Computational Geometry (SoCG), pp. 406–415 (2014)
    Google Scholar
  4. Chiang, Y.J., Mitchell, J.: Two-point Euclidean shortest path queries in the plane. In: Proceedings of the 10th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 215–224 (1999)
    Google Scholar
  5. De Berg, M., Cheong, O., Van Kreveld, M., Overmars, M.: Computational Geometry: Algorithms and Applications, 3rd edn. Springer, Santa Clara (2008). https://doi.org/10.1007/978-3-540-77974-2
    Book MATH Google Scholar
  6. De Rezende, P., Lee, D.T., Wu, Y.F.: Rectilinear shortest paths with rectangular barriers. In: Proceedings of the 1st Annual Symposium on Computational Geometry (SoCG), pp. 204–213 (1985)
    Google Scholar
  7. Elgindy, H., Mitra, P.: Orthogonal shortest route queries among axes parallel rectangular obstacles. Int. J. Comput. Geom. Appl. 4(1), 3–24 (1994)
    Article MathSciNet Google Scholar
  8. Gester, M., Müller, D., Nieberg, T., Panten, C., Schulte, C., Vygen, J.: BonnRoute: Algorithms and data structures for fast and good VLSI routing. ACM Trans. Des. Autom. Electron. Syst. 18(2), 32:1–32:24 (2013)
    Article Google Scholar
  9. Giora, Y., Kaplan, H.: Optimal dynamic vertical ray shooting in rectilinear planar subdivisions. ACM Trans. Algorithms 5(3), 28:1–28:51 (2009)
    Article MathSciNet Google Scholar
  10. Guo, H., Maheshwari, A., Sack, J.-R.: Shortest path queries in polygonal domains. In: Fleischer, R., Xu, J. (eds.) AAIM 2008. LNCS, vol. 5034, pp. 200–211. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-68880-8_20
    Chapter MATH Google Scholar
  11. Hershberger, J.: Finding the upper envelope of \(n\) line segments in \({O}(n \log n)\) time. Inf. Process. Lett. 33(4), 169–174 (1989)
    Article MathSciNet Google Scholar
  12. Hershberger, J., Suri, S.: An optimal algorithm for Euclidean shortest paths in the plane. SIAM J. Comput. 28(6), 2215–2256 (1999)
    Article MathSciNet Google Scholar
  13. Liu, C.H., Lee, D.: Higher-order geodesic Voronoi diagrams in a polygonal domain with holes. In: Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1633–1645 (2013)
    Google Scholar
  14. Mitchell, J.: \(L_1\) shortest paths among polygonal obstacles in the plane. Algorithmica 8(1–6), 55–88 (1992). https://doi.org/10.1007/BF01758836
    Article MathSciNet MATH Google Scholar
  15. Papadopoulou, E., Lee, D.: The \(L_\infty \) Voronoi diagram of segments and VLSI applications. Int. J. Comput. Geom. Appl. 11(05), 503–528 (2001)
    Article MathSciNet Google Scholar
  16. Wang, H.: A divide-and-conquer algorithm for two-point \(L_1\) shortest path queries in polygonal domains. In: Proceedings of the 35th International Symposium on Computational Geometry (SoCG), pp. 59:1–59:14 (2019)
    Google Scholar

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Authors and Affiliations

  1. Department of Computer Science and Engineering, Graduate School of Artificial Intelligence, Pohang University of Science and Technology, Pohang, Korea
    Mincheol Kim & Hee-Kap Ahn
  2. Department of Service and Design Engineering, SungShin Women’s University, Seoul, Korea
    Sang Duk Yoon

Authors

  1. Mincheol Kim
  2. Sang Duk Yoon
  3. Hee-Kap Ahn

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Correspondence toHee-Kap Ahn .

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  1. University of São Paulo, São Paulo, Brazil
    Yoshiharu Kohayakawa
  2. University of Campinas, Campinas, Brazil
    Flávio Keidi Miyazawa

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Kim, M., Yoon, S.D., Ahn, HK. (2020). Shortest Rectilinear Path Queries to Rectangles in a Rectangular Domain. In: Kohayakawa, Y., Miyazawa, F.K. (eds) LATIN 2020: Theoretical Informatics. LATIN 2021. Lecture Notes in Computer Science(), vol 12118. Springer, Cham. https://doi.org/10.1007/978-3-030-61792-9\_22

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