, Volume 81, Issue 6, pp 2317–2345 | Cite as

An Efficient Randomized Algorithm for Higher-Order Abstract Voronoi Diagrams

  • Cecilia Bohler
  • Rolf Klein
  • Chih-Hung LiuEmail author


Given a set of n sites in the plane, the order-k Voronoi diagram is a planar subdivision such that all points in a region share the same k nearest sites. The order-k Voronoi diagram arises for the k-nearest-neighbor problem, and there has been a lot of work for point sites in the Euclidean metric. In this paper, we study order-k Voronoi diagrams defined by an abstract bisecting curve system that satisfies several practical axioms, and thus our study covers many concrete order-k Voronoi diagrams. We propose a randomized incremental construction algorithm that runs in \(O(k(n-k)\log ^2 n +n\log ^3 n)\) steps, where \(O(k(n-k))\) is the number of faces in the worst case. This result applies to disjoint line segments in the \(L_p\) norm, convex polygons of constant size, points in the Karlsruhe metric, and so on. In fact, a running time with a polylog factor to the number of faces was only achieved for point sites in the \(L_1\) or Euclidean metric before.


Order-k Voronoi diagrams Abstract Voronoi diagrams Randomized geometric algorithms Computational geometry 



We deeply appreciate all valuable comments from the anonymous reviewers for SoCG 2016 and Algorithmica.


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Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of BonnBonnGermany
  2. 2.Department of Computer ScienceETH ZürichZurichSwitzerland

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