Selected topics from computational geometry, data structures and motion planning

Abstract

The Voronoi diagram of a set of sites in the plane partitions the plane into regions, called Voronoi regions, one to a site. The Voronoi region of a site s is the set of points in the plane for which 8 is the closest site among all the sites. The Voronoi diagram has many applications in diverse fields, cf. Leven and Sharir [LS86] or Aurenhammer [Aur90] for a list of applications and a history of Voronoi diagrams. Different types of diagrams result from considering different notions of distance, e.g. Euclidean or Lp-norm or convex distance functions, and different sorts of sites, e.g. points, line segments, or circles. For many types of diagrams efficient construction algorithms have been found; these are either based on the divide-and-conquer technique due to Shamos and Hoey [SH75], the sweepline technique due to Fortune [For87], or geometric transforms due to Brown [Bro79] and Edelsbrunner and Seidel [ES86]. A unifying approach to Voronoi diagrams was proposed by Klein [Kle88a, Kle88b, Kle89a, KleS9b], cf. [ES86] for a related approach. Klein does not use the concept of distance as the basic notion but rather the concept of bisecting curve, i.e. he assumes for each pair {p, q} of sites the existence of a bisector J(p, q) which is homeomorphic to a line and divides the plane into a p-region and a q-region. The intersection of all p-regions for different q's is then the Voronoi-region of site p. He also postulates that Voronoi-regions are simply-connected and partition the plane. He shows that these so-called abstract Voronoi diagrams have already many of the properties of concrete Voronoi diagrams. In [MMO91] and the refinement [KMM91] we present a randomized incremental algorithm that can handle abstract Voronoi diagrams in (almost) their full generality. When n denotes the number of sites, the algorithm runs in O(nlog n) expected time, the average being taken over all permutations of the input. The algorithm is simple enough to be of great practical importance. It is uniform in the sense that only a single operation, namely the construction of a Voronoi diagram for 5 sites, depends on the specific type of Voronoi diagram and has to be newly programmed in order to adapt the algorithm to the type of the diagram. Moreover, this operation is the only geometric operation in our algorithm, and using this operation, abstract Voronoi diagrams can be constructed in a