Abstract
Most of the streets of Moscow are either radii emanating from the Kremlin, or pieces of circles around it. We show that Voronoi diagrams for n points based on this metric can be computed in optimal O(n log n) time and linear space. To this end, we prove a general theorem stating that bisectors of suitably separated point sets do not contain loops if, beside other properties, there are no holes in the circles of the underlying metric. Then the Voronoi diagrams can be computed within O(n log n) steps, using a divide-and-conquer algorithm. Our theorem not only applies to the Moscow metric but to a large class of metrics including the symmetric convex distance functions and all composite metrics obtained by assigning the L 1 or the L 2 metric to the regions of a planar map.
This work was partially supported by the grant Ot64/4-2 from the Deutsche Forschungsgemeinschaft
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© 1989 Springer-Verlag Berlin Heidelberg
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Klein, R. (1989). Voronoi diagrams in the moscow metric. In: van Leeuwen, J. (eds) Graph-Theoretic Concepts in Computer Science. WG 1988. Lecture Notes in Computer Science, vol 344. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-50728-0_61
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DOI: https://doi.org/10.1007/3-540-50728-0_61
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