Abstract
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-and-conquer algorithm to compute the order-\(k\) abstract Voronoi diagram in expected \(O(kn^{1+\varepsilon })\) operations. For solving small sub-instances in the divide-and-conquer process, we also give two sub-algorithms with expected \(O(k^2n\log n)\) and \(O(n^22^{\alpha (n)}\log n)\) time, respectively. This directly implies an \(O(kn^{1+\varepsilon })\)-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 \(L_p\) norm, and others.
This work was supported by the European Science Foundation (ESF) in the EUROCORES collaborative research project EuroGIGA/VORONOI, projects DFG Kl 655/17-1 and SNF 20GG21-134355. The work of the last two authors was also supported by the Swiss National Science Foundation, project 200020-149658.
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Bohler, C., Liu, CH., Papadopoulou, E., Zavershynskyi, M. (2014). A Randomized Divide and Conquer Algorithm for Higher-Order Abstract Voronoi Diagrams. In: Ahn, HK., Shin, CS. (eds) Algorithms and Computation. ISAAC 2014. Lecture Notes in Computer Science(), vol 8889. Springer, Cham. https://doi.org/10.1007/978-3-319-13075-0_3
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