Randomized incremental construction for the hausdorff voronoi diagram revisited and extended

The Hausdorff Voronoi diagram of clusters of points in the plane is a generalization of Voronoi diagrams based on the Hausdorff distance function. Its combinatorial complexity is O(n+m), where n is the total number of points and m is the number of crossings between the input clusters (formula presented) the number of clusters is k. We present efficient algorithms to construct this diagram via the randomized incremental construction (RIC) framework For non-crossing clusters (m=0), our algorithm runs in expected (formula presented) time and deterministic O(n) space. For arbitrary clusters the algorithm runs in expected (formula presented) space. The two algorithms can be combined in a crossing-oblivious scheme within the same bounds. We show how to apply the RIC framework efficiently to handle non-standard characteristics of generalized Voronoi diagrams, including sites (and bisectors) of non-constant complexity, sites that are not enclosed in their Voronoi regions, empty Voronoi regions, and finally, disconnected bisectors and Voronoi regions. The diagram finds direct applications.SCOPUS: cp.kinfo:eu-repo/semantics/publishe

Randomized incremental construction for the hausdorff voronoi diagram revisited and extended

We study the amortized number of combinatorial changes (edge insertions and removals) needed to update the graph structure of the Voronoi diagram V(S) (and several variants thereof) of a set S of n sites in the plane as sites are added.We define a general update operation for planar graphs modeling the incremental construction of several variants of Voronoi diagrams as well as the incremental construction of an intersection of halfspaces in R3. We show that the amortized number of edge insertions and removals needed to add a new site is O(n−−√). A matching Ω(n−−√) combinatorial lower bound is shown, even in the case where the graph of the diagram is a tree. This contrasts with the O(logn) upper bound of Aronov et al. (2006) for farthest-point Voronoi diagrams when the points are inserted in order along their convex hull.We present a semi-dynamic data structure that maintains the Voronoi diagram of a set S of n sites in convex position. This structure supports the insertion of a new site p and finds the asymptotically minimal number K of edge insertions and removals needed to obtain the diagram of S∪{p} from the diagram of S, in time O(Kpolylog n) worst case, which is O(n−−√polylog n) amortized by the aforementioned result.The most distinctive feature of this data structure is that the graph of the Voronoi diagram is maintained at all times and can be traversed in the natural way; this contrasts with other known data structures supporting nearest neighbor queries. Our data structure supports general search operations on the current Voronoi diagram, which can, for example, be used to perform point location queries in the cells of the current Voronoi diagram in O(logn) time, or to determine whether two given sites are neighbors in the Delaunay triangulation.info:eu-repo/semantics/publishe