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Quickest Paths in Anisotropic Media

  • Radwa El Shawi
  • Joachim Gudmundsson
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6831)

Abstract

In this paper we study the quickest path problem where speed is direction-dependent (anisotropic). The problem arises in sailing, robotics, aircraft navigation, and routing of autonomous vehicles, where the speed is affected by the direction of waves, winds or slope of the terrain. We present an approximation algorithm to find a quickest path for a point robot moving in planar subdivision, where each face is assigned a translational flow that reflects the cost of travelling within this face.

Our main contribution is a data structure that given a subdivision with translational flows returns a (1 + ε)-approximate quickest path in the subdivision between any two query points in the plane.

Keywords

Optimal Path Anisotropic Medium Query Point Boundary Edge Steiner Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Aleksandrov, L., Lanthier, M., Maheshwari, A., Sack, J.-R.: An epsilon-Approximation for Weighted Shortest Paths on Polyhedral Surfaces. In: Proceedings of the 6th Scandinavian Workshop on Algorithm Theory (1998)Google Scholar
  2. 2.
    de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry: Algorithms and Applications, 3rd edn. Springer, Heidelberg (2008)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bose, P., Gudmundsson, J., Morin, P.: Ordered theta graphs. Computational geometry – Theory & Applications 28(1), 11–18 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Callahan, P.B., Kosaraju, S.R.: A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields. Journal of the ACM 42, 67–90 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cheng, S.-W., Na, H.-S., Vigneron, A., Wang, Y.: Approximate Shortest Paths in Anisotropic Regions. SIAM Journal on Computing 38(3), 802–824 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dijkstra, E.W.: A note on two problems in connexion with graphs. Numerische Mathematik 1, 269–271 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Johnson, D.B.: Efficient algorithms for shortest paths in sparse networks. Journal of the ACM 24(1), 1–13 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Lanthier, M., Maheshwari, A., Sack, J.-R.: Approximating Weighted Shortest Paths on Polyhedral Surfaces. In: Proceedings of the 13th Symposium on Computational Geometry, pp. 274–283 (1997)Google Scholar
  9. 9.
    Mata, C., Mitchell, J.: A New Algorithm for Computing Shortest Paths in Weighted Planar Subdivisions. In: Proceedings of the 13th Symposium on Computational Geometry, pp. 264–273 (1997)Google Scholar
  10. 10.
    Mitchell, J.: Geometric shortest paths and network optimization. Handbook of Computational Geometry, 633–701 (2000)Google Scholar
  11. 11.
    Mitchell, J., Papadimitriou, C.: The weighted region problem: Finding shortest paths through a weighted planar subdivision. Journal of the ACM 38(1), 18–73 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Narasimhan, G., Smid, M.: Geometric Spanner Networks. Cambridge University Press, Cambridge (2007)CrossRefzbMATHGoogle Scholar
  13. 13.
    Papadakis, N., Perakis, A.: Deterministic Minimal Time Vessel Routing. Journal of Operations Research 38(3), 426–438 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Reif, J., Sun, Z.: Movement Planning in the Presence of Flows. Algorithmica 39(2), 127–153 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Rowe, N.: Obtaining Optimal Mobile-Robot Paths with Nonsmooth Anisotropic Cost Functions Using Qualitative-State Reasoning. International Journal of Robotics Research 16(3), 375–399 (1997)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Radwa El Shawi
    • 1
    • 2
  • Joachim Gudmundsson
    • 1
    • 2
  1. 1.School of Information TechnologyUniversity of SydneyAustralia
  2. 2.NICTASydneyAustralia

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