Abstract
Given a terrain defined as a piecewise-linear function with n triangles, and m point sites on it, we would like to identify the location on the terrain that minimizes the maximum distance to the sites. The distance is measured as the length of the Euclidean shortest path along the terrain. To simplify the problem somewhat, we extend the terrain to (the surface of) a polyhedron. To compute the optimum placement, we compute the furthest-site Voronoi diagram of the sites on the polyhedron. The diagram has maximum combinatorial complexity Θ(mn) 2, and the algorithm runs in O(mn 2 log2 m(logm + logn) time.
B.A. has been partially supported by a Sloan Research Fellowship. M.v.K. and R.v.O. have been partially supported by the ESPRIT IV LTR Project No. 21957 (CGAL). K.V. has been supported by National Science Foundation Grant CCR-93-01259, by an Army Research O.ce MURI grant DAAH04-96-1-0013, by a Sloan fellowship, by an NYI award, by matching funds from Xerox Corporation, and by a grant from the U.S.-Israeli Binational Science Foundation. Part of the work was carried out while B.A. was visiting Utrecht University.
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Aronov, B., van Kreveld, M., van Oostrum, R., Varadarajan, K. (1998). Facility Location on Terrains. In: Chwa, KY., Ibarra, O.H. (eds) Algorithms and Computation. ISAAC 1998. Lecture Notes in Computer Science, vol 1533. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-49381-6_4
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DOI: https://doi.org/10.1007/3-540-49381-6_4
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