Abstract
In this paper we study a facility location problem in the plane in which a single point (facility) and a rapid transit line (highway) are simultaneously located in order to minimize the total travel time of the clients to the facility, using the L1 or Manhattan metric. The rapid transit line is represented by a line segment with fixed length and arbitrary orientation. The highway is an alternative transportation system that can be used by the clients to reduce their travel time to the facility. This problem was introduced by Espejo and Rodríguez-Chía in [8]. They gave both a characterization of the optimal solutions and an algorithm running in O(n3logn) time, where n represents the number of clients. In this paper we show that the Espejo and Rodríguez-Chía’s algorithm does not always work correctly. At the same time, we provide a proper characterization of the solutions with a simpler proof and give an algorithm solving the problem in O(n3) time.
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Abellanas, M., Hurtado, F., Icking, C., Klein, R., Langetepe, E., Ma, L., Palop, B., Sacristán, V.: Voronoi diagram for services neighboring a highway. Information Processing Letters 86, 283–288 (2003)
Ahn, H.-K., Alt, H., Asano, T., Bae, S.W., Brass, P., Cheong, O., Knauer, C., Na, H.-S., Shin, C.-S., Wolff, A.: Constructing optimal highways. In: Proceedings of the 13th Computing: The Australasian Theory Symposium (CATS 2007), pp. 7–14 (2007)
Aichholzer, O., Aurenhammer, F., Palop, B.: Quickest paths, straight skeletons, and the city Voronoi diagram. Discrete & Computational Geometry 31, 17–35 (2004)
Aloupis, G., Cardinal, J., Collette, S., Hurtado, F., Langerman, S., O’Rourke, J., Palop, B.: Highway hull revisited. Computational Geometry: Theory and Applications 43, 115–130 (2010)
Bae, S.W., Korman, M., Tokuyama, T.: All Farthest Neighbors in the Presence of Highways and Obstacles. In: Das, S., Uehara, R. (eds.) WALCOM 2009. LNCS, vol. 5431, pp. 71–82. Springer, Heidelberg (2009)
Cardinal, J., Collette, S., Hurtado, F., Langerman, S., Palop, B.: Optimal location of transportation devices. Computational Geometry: Theory and Applications 41, 219–229 (2008)
Díaz-Báñez, J.-M., Korman, M., Pérez-Lantero, P., Ventura, I.: Locating a service facility and a rapid transit line. CoRR, abs/1104.0753 (2011)
Espejo, I., Rodríguez-Chía, A.M.: Simultaneous location of a service facility and a rapid transit line. Computers and Operations Research 38, 525–538 (2011)
Korman, M., Tokuyama, T.: Optimal Insertion of a Segment Highway in a City Metric. In: Hu, X., Wang, J. (eds.) COCOON 2008. LNCS, vol. 5092, pp. 611–620. Springer, Heidelberg (2008)
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Díaz-Báñez, J.M., Korman, M., Pérez-Lantero, P., Ventura, I. (2012). Locating a Service Facility and a Rapid Transit Line. In: Márquez, A., Ramos, P., Urrutia, J. (eds) Computational Geometry. EGC 2011. Lecture Notes in Computer Science, vol 7579. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34191-5_12
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DOI: https://doi.org/10.1007/978-3-642-34191-5_12
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