Abstract
We study the amortized number of combinatorial changes (edge insertions and removals) needed to update the graph structure of the Voronoi diagram (and several variants thereof) of a set S of n sites in the plane as sites are added to the set. To that effect, we define a general update operation for planar graphs that can be used to model 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 to the Voronoi diagram is \(O(\sqrt{n})\). A matching \(\Omega (\sqrt{n})\) combinatorial lower bound is shown, even in the case where the graph representing the Voronoi diagram is a tree. This contrasts with the \(O(\log {n})\) upper bound of Aronov et al. (LATIN 2006: Theoretical Informatics. Lecture Notes in Computer Science, Springer, Berlin, 2006) for farthest-point Voronoi diagrams in the special case where the points are inserted in clockwise order along their convex hull. We then present a semi-dynamic data structure that maintains the Voronoi diagram of a set S of n sites in convex position. This data structure supports the insertion of a new site p (and hence the addition of its Voronoi cell) 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 combinatorial result. The most distinctive feature of this data structure is that the graph of the Voronoi diagram is maintained explicitly at all times and can be retrieved and 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.
Highlights
Let S be a set of n sites in the plane
We present a semi-dynamic data structure that maintains the Voronoi diagram of a set S of n sites in convex position
This might come as a surprise in light of the fact that the number of combinatorial changes to the Delaunay triangulation of S upon the insertion of a point can be Ω(n) with each insertion, even when the sites are in convex position and are added in clockwise order. (Note that in that case the Voronoi diagram of S is a tree and the standard flip operation is a rotation in the tree.)
Summary
Let S be a set of n sites in the plane. The graph structures of the Voronoi diagram VD(S) and its dual the Delaunay triangulation DT(S) capture much of the proximity information of that set. The goal of this paper is to show that despite this worst-case behavior, the amortized number of structural changes to the graph of the Voronoi diagram of S, i.e., the minimum number of edge insertions and deletions needed to update VD(S) throughout a sequence of site insertions to S, is much smaller This might come as a surprise in light of the fact that the number of combinatorial changes (usually modeled as flips) to the Delaunay triangulation of S upon the insertion of a point can be Ω(n) with each insertion, even when the sites are in convex position and are added in clockwise order. It can be used to represent the changes to the 1-skeleton of a polyhedron in R3 after it is intersected with a halfspace
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