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An \(\varOmega (n^3)\) Lower Bound on the Number of Cell Crossings for Weighted Shortest Paths in 3-Dimensional Polyhedral Structures

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LATIN 2020: Theoretical Informatics (LATIN 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12118))

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Abstract

A new lower bound of \(\varOmega (n^3)\) on the maximum number of cell crossings for weighted shortest paths in 3-dimensional polyhedral structures consisting of a linear number of \(\mathcal {O}(n)\) polyhedral cells and cell faces is derived. This is a generalization and sharpening of the formerly known \(\varOmega (n^2)\) lower bound on the maximum number of cell crossings for weighted shortest path in 2-dimensional polyhedral structures and has been a long-standing open problem for the 3-dimensional case.

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References

  1. Aleksandrov, L., Djidjev, H.N., Guo, H., Maheshwari, A., Nussbaum, D., Sack, J.-R.: Algorithms for approximate shortest path queries on weighted polyhedral surfaces. Discrete Comput. Geom. 44(4), 762–801 (2009). https://doi.org/10.1007/s00454-009-9204-0

    Article  MathSciNet  MATH  Google Scholar 

  2. Aleksandrov, L., Djidjev, H., Maheshwari, A., Sack, J.-R.: An approximation algorithm for computing shortest paths in weighted 3-D domains. Discrete Comput. Geom. 50(1), 124–184 (2013). https://doi.org/10.1007/s00454-013-9486-0

    Article  MathSciNet  MATH  Google Scholar 

  3. Aleksandrov, L., Maheshwari, A., Sack, J.: Determining approximate shortest paths on weighted polyhedral surfaces. J. ACM 52(1), 25–53 (2005). https://doi.org/10.1145/1044731.1044733

    Article  MathSciNet  MATH  Google Scholar 

  4. The Boost C++ Library. http://www.boost.org/

  5. Bose, P., Maheshwari, A., Shu, C., Wuhrer, S.: A survey of geodesic paths on 3D surfaces. Comput. Geom. 44(9), 486–498 (2011). https://doi.org/10.1016/j.comgeo.2011.05.006

    Article  MathSciNet  MATH  Google Scholar 

  6. Canny, J.F., Reif, J.H.: New lower bound techniques for robot motion planning problems. In: Proceedings of the 28th Annual Symposium on Foundations of Computer Science, Los Angeles, California, USA, 27–29 October 1987, pp. 49–60. IEEE Computer Society (1987). https://doi.org/10.1109/SFCS.1987.42

  7. Carufel, J.D., Grimm, C., Maheshwari, A., Owen, M., Smid, M.H.M.: A note on the unsolvability of the weighted region shortest path problem. Comput. Geom. 47(7), 724–727 (2014). https://doi.org/10.1016/j.comgeo.2014.02.004

    Article  MathSciNet  MATH  Google Scholar 

  8. Dijkstra, E.: A note on two problems in connexion with graphs. Numerische Mathematik 1(1), 269–271 (1959). https://doi.org/10.1007/BF01386390

    Article  MathSciNet  MATH  Google Scholar 

  9. Leonhardt, U., Philbin, T.: Geometry and Light: The Science of Invisibility. Courier Dover Publications (2012)

    Google Scholar 

  10. Mitchell, J.S.B., Papadimitriou, C.H.: The weighted region problem: finding shortest paths through a weighted planar subdivision. J. ACM 38(1), 18–73 (1991). https://doi.org/10.1145/102782.102784

    Article  MathSciNet  MATH  Google Scholar 

  11. Mitchell, J.S.B., Sharir, M.: New results on shortest paths in three dimensions. In: Proceedings of the 20th Annual Symposium on Computational Geometry, SCG 2004, pp. 124–133. ACM, New York (2004). https://doi.org/10.1145/997817.997839

  12. OpenVolumeMesh - A Generic and Versatile Index-Based Data Structure for Polytopal Meshes. https://www.openvolumemesh.org/

  13. P. E. Hart, N.J.N., Raphael, B.: A formal basis for the heuristic determination of minimum cost paths. IEEE Trans. Syst. Sci. Cybern. SSC 4(2), 100–107 (1968)

    Google Scholar 

  14. ParaView - An Open-Source, Multi-Platform Data Analysis and Visualization Application. http://www.paraview.org/

  15. Si, H.: TetGen - A Quality Tetrahedral Mesh Generator and a 3D Delaunay Triangulator. http://www.tetgen.org/

  16. Si, H.: TetGen - A Quality Tetrahedral Mesh Generator and a 3D Delaunay Triangulator. Technical report, Karl-Weierstraß-Institut für Angewandte Analysis und Stochastik, Berlin, Germany (2013)

    Google Scholar 

  17. Sun, Z., Reif, J.H.: On finding approximate optimal paths in weighted regions. J. Algorithms 58(1), 1–32 (2006). https://doi.org/10.1016/j.jalgor.2004.07.004

    Article  MathSciNet  MATH  Google Scholar 

  18. Visualization Toolkit Website. http://www.vtk.org/

  19. wsp3dovm - Weighted Shortest Paths with OpenVolumeMesh. https://github.com/FrankBau/wsp3dovm/

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Acknowledgement

The authors would like to thank the referees for valuable comments made. The third author would like to thank Erik van Leeuwen, Ioana Bercea, Karl Bringmann, and Michael Sagraloff for fruitful initial discussions which took place while that author was visiting Max-Planck Institute, Algorithms, Saarbrücken.

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Authors and Affiliations

  1. Hochschule für Technik und Wirtschaft, Computer Engineering, Berlin, Germany

    Frank Bauernöppel

  2. Carleton University, School of Computer Science, Ottawa, Canada

    Anil Maheshwari & Jörg-Rüdiger Sack

Authors
  1. Frank Bauernöppel
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  2. Anil Maheshwari
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  3. Jörg-Rüdiger Sack
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Corresponding author

Correspondence to Frank Bauernöppel .

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Editors and Affiliations

  1. University of São Paulo, São Paulo, Brazil

    Yoshiharu Kohayakawa

  2. University of Campinas, Campinas, Brazil

    Flávio Keidi Miyazawa

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Bauernöppel, F., Maheshwari, A., Sack, JR. (2020). An \(\varOmega (n^3)\) Lower Bound on the Number of Cell Crossings for Weighted Shortest Paths in 3-Dimensional Polyhedral Structures. 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_19

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  • DOI: https://doi.org/10.1007/978-3-030-61792-9_19

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  • Publisher Name: Springer, Cham

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  • Online ISBN: 978-3-030-61792-9

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