Abstract
Let P be a path between two points s and t in a polygonal subdivision \(\mathcal T\) with obstacles and weighted regions. Given a relative error tolerance ε ∈ (0,1), we present the first algorithm to compute a path between s and t that can be deformed to P without passing over any obstacle and the path cost is within a factor 1 + ε of the optimum. The running time is \(O(\frac{h^3}{\varepsilon^2}kn\,\mathrm{polylog}(k,n,\frac{1}{\varepsilon}))\), where k is the number of segments in P and h and n are the numbers of obstacles and vertices in \(\mathcal T\), respectively. The constant in the running time of our algorithm depends on some geometric parameters and the ratio of the maximum region weight to the minimum region weight.
The research of Cheng and Jin was supported by the Research Grant Council, Hong Kong, China (project no. 612107).
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References
Aleksandrov, L., Maheshwari, A., Sack, J.-R.: Determining approximate shortest paths on weighted polyhedral surfaces. J. ACM 52, 25–53 (2005)
Bespamyatnikh, S.: Computing homotopic shortest paths in the plane. J. Alg. 49, 284–303 (2003)
Cabello, S., Liu, Y., Mantler, A., Snoeyink, J.: Testing Homotopy for Paths in the Plane. Discr. Comput. Geom. 31, 61–81 (2004)
Efrat, A., Kobourov, S.G., Lubiw, A.: Computing homotopic shortest paths efficiently. Comput. Geom. Theory and Appl. 35, 162–172 (2006)
Farach, M.: Optimal suffix tree construction with large alphabets. In: Proc. 38th Annu. Sympos. Found. Comput. Sci., pp. 137–143 (1997)
Forbus, K.D., Uhser, J., Chapman, V.: Qualitative spatial reasoning about sketch maps. AI Magazine 24, 61–72 (2004)
Fredman, M.L., Tarjan, R.E.: Fibonacci heaps and their uses in improved network optimization algorithms. J. ACM 34, 596–615 (1987)
Gao, S., Jerrum, M., Kaufmann, M., Kehlhorn, K., Rülling, W., Storb, C.: On continuous homotopic one layer routing. In: Proc. 4th Annu. Sympos. Comput. Geom., pp. 392–402 (1998)
Hershberger, J., Snoeyink, J.: Computing minimum length paths of a given homotopy class. Comput. Geom. Theory and Appl. 4, 63–98 (1994)
Kaufmann, M., Mehlhorn, K.: On local routing of two-terminal nets. J. Comb. Theory, Ser. B 55, 33–72 (1992)
Leiserson, C.E., Maley, F.M.: Algorithms for routing and testing routability of planar VLSI layouts. In: Proc. 17th Annu. Sympos. Theory of Comput., pp. 69–78 (1985)
Mitchell, J., Papadimitriou, C.: The weighted region problem: Finding shortest paths through a weighted planar subdivision. J. ACM 38, 18–73 (1991)
Sun, Z., Reif, J.: On finding approximate optimal paths in weighted regions. J. Alg. 58, 1–32 (2006)
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Cheng, SW., Jin, J., Vigneron, A., Wang, Y. (2010). Approximate Shortest Homotopic Paths in Weighted Regions. In: Cheong, O., Chwa, KY., Park, K. (eds) Algorithms and Computation. ISAAC 2010. Lecture Notes in Computer Science, vol 6507. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17514-5_10
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DOI: https://doi.org/10.1007/978-3-642-17514-5_10
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