Given a set of sites in the plane, their order-k Voronoi diagram partitions the plane into regions such that all points within one region have the same k nearest sites. The order-k abstract Voronoi diagram is defined in terms of bisecting curves satisfying some simple combinatorial properties, rather than the geometric notions of sites and distance, and it represents a wide class of order-k concrete Voronoi diagrams. In this paper we develop a randomized divide-andconquer algorithm to compute the order-k abstract Voronoi diagram in expected O(kn1+ε) operations. For solving small sub-instances in the divide-and-conquer process, we also give two sub-algorithms with expected O(k2n log n) and O(n22α(n) log n) time, respectively. This directly implies an O(kn1+ε)-time algorithm for several concrete order-k instances such as points in any convex distance, disjoint line segments and convex polygons of constant size in the Lpnorm, and others.
CITATION STYLE
Bohler, C., Liu, C. H., Papadopoulou, E., & Zavershynskyi, M. (2014). A randomized divide and conquer algorithm for higher-order abstract Voronoi diagrams. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 8889, pp. 27–37). Springer Verlag. https://doi.org/10.1007/978-3-319-13075-0_3
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