Abstract
Consider a polyhedral surface consisting of n triangular faces where each face has an associated positive weight. The cost of travel through each face is the Euclidean distance traveled multiplied by the weight of the face. We present an approximation algorithm for computing a path such that the ratio of the cost of the computed path with respect to the cost of a shortest path is bounded by (1+ε), for a given 0 < ε<1. The algorithm is based on a novel way of discretizing the polyhedral surface. We employ a generic greedy approach for solving shortest path problems in geometric graphs produced by such discretization. We improve upon existing approximation algorithms for computing shortest paths on polyhedral surfaces [1,4,5,10,12,15].
Research supported in part by NSERC
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References
Varadarajan, K.R., Agarwal, P.K.: Approximating Shortest Paths on Nonconvex Polyhedron. SIAM Jl. Comput. 30(4), 1321–1340 (2000)
Agarwal, P.K., Har-Peled, S., Karia, M.: Computing approximate shortest paths on convex polytopes. Algorithmica 33, 227–242 (2002)
Agarwal, P.K., et al.: Approximating Shortest Paths on a Convex Polytope in Three Dimensions. Jl. ACM 44, 567–584 (1997)
Aleksandrov, L., Lanthier, M., Maheshwari, A., Sack, J.-R.: An ε-approximation algorithm for weighted shortest paths. In: Arnborg, S. (ed.) SWAT 1998. LNCS, vol. 1432, pp. 11–22. Springer, Heidelberg (1998)
Aleksandrov, L., Maheshwari, A., Sack, J.-R.: Approximation Algorithms for Geometric Shortest Path Problems. In: 32nd STOC, pp. 286–295 (2000)
Canny, J., Reif, J.H.: New Lower Bound Techniques for Robot Motion Planning Problems. In: 28th FOCS, pp. 49–60 (1987)
Chen, J., Han, Y.: Shortest Paths on a Polyhedron. In: 6th SoACM-CG, pp. 360–369 (1990); Appeared in Internat. J. Comput. Geom. Appl., 6, 127–144 (1996)
Hershberger, J., Suri, S.: Practical Methods for Approximating Shortest Paths on a Convex Polytope in \(\mathcal{R}^3\). In: 6SODA, pp. 447–456 (1995)
Kapoor, S.: Efficient Computation of Geodesic Shortest Paths. In: 31st STOC (1999)
Lanthier, M., Maheshwari, A., Sack, J.-R.: Approximating Weighted Shortest Paths on Polyhedral Surfaces. Algorithmica 30(4), 527–562 (2001)
Mitchell, J.S.B., Mount, D.M., Papadimitriou, C.H.: The Discrete Geodesic Problem. SIAM Jl. Computing 16, 647–668 (1987)
Mitchell, J.S.B., Papadimitriou, C.H.: The Weighted Region Problem: Finding Shortest Paths Through a Weighted Planar Subdivision. JACM 38, 18–73 (1991)
Papadimitriou, C.H.: An Algorithm for Shortest Path Motion in Three Dimensions. IPL 20, 259–263 (1985)
Sharir, M., Schorr, A.: On Shortest Paths in Polyhedral Spaces. SIAM J. of Comp. 15, 193–215 (1986)
Sun, Z., Reif, J.: BUSHWACK: An approximation algorithm for minimal paths through pseudo-Euclidean spaces. In: Eades, P., Takaoka, T. (eds.) ISAAC 2001. LNCS, vol. 2223, pp. 160–171. Springer, Heidelberg (2001)
Ziegelmann, M.: Constrained Shortest Paths and Related Problems Ph.D. thesis, Universität des Saarlandes (Max-Planck Institut für Informatik)(2001)
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Aleksandrov, L., Maheshwari, A., Sack, JR. (2003). An Improved Approximation Algorithm for Computing Geometric Shortest Paths. In: Lingas, A., Nilsson, B.J. (eds) Fundamentals of Computation Theory. FCT 2003. Lecture Notes in Computer Science, vol 2751. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45077-1_23
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DOI: https://doi.org/10.1007/978-3-540-45077-1_23
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