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We study the amortized number of combinatorial changes (edge insertions and removals) needed to update the graph structure of the Voronoi diagram $mathcal{V}(S)$ (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 $mathbb{R}^3$. We show that the amortized number of edge insertions and removals needed to add a new site is $O(sqrt{n})$. A matching $Omega(sqrt{n})$ combinatorial lower bound is shown, even in the case where the graph of the diagram is a tree. This contrasts with the $O(log{n})$ 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 $S cup {p}$ from the diagram of $S$, in time $O(K,mathrm{polylog} n)$ worst case, which is $O(sqrt{n};mathrm{polylog} n)$ 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 $O(log n)$ time, or to determine whether two given sites are neighbors in the Delaunay triangulation.
A data structure is presented that explicitly maintains the graph of a Voronoi diagram of $N$ point sites in the plane or the dual graph of a convex hull of points in three dimensions while allowing insertions of new sites/points. Our structure suppo
We describe a new data structure for dynamic nearest neighbor queries in the plane with respect to a general family of distance functions. These include $L_p$-norms and additively weighted Euclidean distances. Our data structure supports general (con
Given a set $S$ of $m$ point sites in a simple polygon $P$ of $n$ vertices, we consider the problem of computing the geodesic farthest-point Voronoi diagram for $S$ in $P$. It is known that the problem has an $Omega(n+mlog m)$ time lower bound. Previ
Given a tesselation of the plane, defined by a planar straight-line graph $G$, we want to find a minimal set $S$ of points in the plane, such that the Voronoi diagram associated with $S$ fits $G$. This is the Generalized Inverse Voronoi Problem (GIV
The Voronoi diagram of a finite set of objects is a fundamental geometric structure that subdivides the embedding space into regions, each region consisting of the points that are closer to a given object than to the others. We may define many varian