Abstract
We investigate higher-order Voronoi diagrams in the city metric. This metric is induced by quickest paths in the L 1 metric in the presence of an accelerating transportation network of axis-parallel line segments. For the structural complexity of k th-order city Voronoi diagrams of n point sites, we show an upper bound of O(k(n − k) + kc) and a lower bound of Ω(n + kc), where c is the complexity of the transportation network. This is quite different from the bound O(k(n − k)) in the Euclidean metric [12]. For the special case where k = n − 1 the complexity in the Euclidean metric is O(n), while that in the city metric is Θ(nc). Furthermore, we develop an O(k 2(n + c)log(n + c))-time iterative algorithm to compute the k th-order city Voronoi diagram and an O(nclog2(n + c)logn)-time divide-and-conquer algorithm to compute the farthest-site city Voronoi diagram.
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References
Aichholzer, O., Aurenhammer, F., Palop, B.: Quickest paths, straight skeletons, and the city Voronoi diagram. Discrete Comput. Geom. 31, 17–35 (2004)
Agarwal, P.K., de Berg, M., Matoušek, J., Schwarzkopf, O.: Constructing levels in arrangements and higher order Voronoi diagrams. SIAM J. Comput. 27(3), 654–667 (1998)
Aurenhammer, F., Schwarzkopf, O.: A simple on-line randomized incremental algorithm for computing higher order Voronoi diagrams. Internat. J. Comput. Geom. Appl. 2, 363–381 (1992)
Bae, S.W., Chwa, K.-Y.: Shortest Paths and Voronoi Diagrams with Transportation Networks Under General Distances. In: Deng, X., Du, D.-Z. (eds.) ISAAC 2005. LNCS, vol. 3827, pp. 1007–1018. Springer, Heidelberg (2005)
Boissonnat, J.D., Devillers, O., Teillaud, M.: A semidynamic construction of higher-order Voronoi diagrams and its randomized analysis. Algorithmica 9, 329–356 (1993)
Bae, S.W., Kim, J.-H., Chwa, K.-Y.: Optimal construction of the city Voronoi diagram. Internat. J. Comput. Geom. Appl. 19(2), 95–117 (2009)
Chazelle, B., Edelsbrunner, H.: An improved algorithm for constructing kth-order Voronoi diagrams. IEEE Transactions on Computers 36(11), 1349–1454 (1987)
Cheong, O., Everett, H., Glisse, M., Gudmundsson, J., Hornus, S., Lazard, S., Lee, M., Na, H.-S.: Farthest-polygon Voronoi diagrams. Comput. Geom. Theory Appl. 44, 234–247 (2011)
Edelsbrunner, H., Guibas, L.J., Stolfi, J.: Optimal point location in a monotone subdivision. SIAM J. Comput. 15(2), 317–340 (1986)
Gemsa, A., Lee, D.T., Liu, C.-H., Wagner, D.: Higher order city Voronoi diagrams. ArXiv e-prints, arXiv:1204.4374 (April 2012)
Görke, R., Shin, C.-S., Wolff, A.: Constructing the city Voronoi diagram faster. Internat. J. Comput. Geom. Appl. 18(4), 275–294 (2008)
Lee, D.T.: On k-nearest neighbor Voronoi diagrams in the plane. IEEE Trans. Comput. 31(6), 478–487 (1982)
Mulmuley, K.: A fast planar partition algorithm, I. J. Symbolic Comput. 10(3-4), 253–280 (1990)
Mulmuley, K.: On levels in arrangements and Voronoi diagrams. Discrete Comput. Geom. 6, 307–338 (1991)
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Gemsa, A., Lee, D.T., Liu, CH., Wagner, D. (2012). Higher Order City Voronoi Diagrams. In: Fomin, F.V., Kaski, P. (eds) Algorithm Theory – SWAT 2012. SWAT 2012. Lecture Notes in Computer Science, vol 7357. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31155-0_6
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