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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
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)
de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry: Algorithms and Applications, 3rd edn. Springer, Heidelberg (2008)
Bose, P., Gudmundsson, J., Morin, P.: Ordered theta graphs. Computational geometry – Theory & Applications 28(1), 11–18 (2004)
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)
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)
Dijkstra, E.W.: A note on two problems in connexion with graphs. Numerische Mathematik 1, 269–271 (1959)
Johnson, D.B.: Efficient algorithms for shortest paths in sparse networks. Journal of the ACM 24(1), 1–13 (1977)
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)
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)
Mitchell, J.: Geometric shortest paths and network optimization. Handbook of Computational Geometry, 633–701 (2000)
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)
Narasimhan, G., Smid, M.: Geometric Spanner Networks. Cambridge University Press, Cambridge (2007)
Papadakis, N., Perakis, A.: Deterministic Minimal Time Vessel Routing. Journal of Operations Research 38(3), 426–438 (1990)
Reif, J., Sun, Z.: Movement Planning in the Presence of Flows. Algorithmica 39(2), 127–153 (2004)
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)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
El Shawi, R., Gudmundsson, J. (2011). Quickest Paths in Anisotropic Media. In: Wang, W., Zhu, X., Du, DZ. (eds) Combinatorial Optimization and Applications. COCOA 2011. Lecture Notes in Computer Science, vol 6831. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22616-8_20
Download citation
DOI: https://doi.org/10.1007/978-3-642-22616-8_20
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22615-1
Online ISBN: 978-3-642-22616-8
eBook Packages: Computer ScienceComputer Science (R0)