An Efficient Randomized Algorithm for Higher-Order Abstract Voronoi Diagrams

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)log2n+nlog3n) steps, where O(k(n-k)) is the number of faces in the worst case. This result applies to disjoint line segments in the Lp 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 L1 or Euclidean metric before.