On the complexity of higher order abstract Voronoi diagrams

Abstract Voronoi diagrams [15,16] are based on bisecting curves enjoying simple combinatorial properties, rather than on the geometric notions of sites and circles. They serve as a unifying concept. Once the bisector system of any concrete type of Voronoi diagram is shown to fulfill the AVD properties, structural results and efficient algorithms become available without further effort. For example, the first optimal algorithms for constructing nearest Voronoi diagrams of disjoint convex objects, or of line segments under the Hausdorff metric, have been obtained this way [20]. In a concrete order-k Voronoi diagram, all points are placed into the same region that have the same k nearest neighbors among the given sites. This paper is the first to study abstract Voronoi diagrams of arbitrary order k. We prove that their complexity is upper bounded by 2k(n-k). So far, an O(k (n-k)) bound has been shown only for point sites in the Euclidean and L_p plane [18,19], and, very recently, for line segments [23]. These proofs made extensive use of the geometry of the sites. Our result on AVDs implies a 2k (n-k) upper bound for a wide range of cases for which only trivial upper complexity bounds were previously known, and a slightly sharper bound for the known cases. Also, our proof shows that the reasons for this bound are combinatorial properties of certain permutation sequences.