To fill a theory gap of Voronoi diagrams that there have been no reports of extended diagrams in spatial density so far. A new concept of spatial density Voronoi diagram was proposed. An important property was presented and proven. And a construction algorithm was presented. Spatial density can be used to indicate factors related to density such as conveyance and the traffic conditions. Some pr...

Objective Voronoi diagrams are important in many fields in a series of sciences. Network Voronoi diagrams are useful to investigate dominance regions in a grid street system or a radial-circular street system. However, all generators may have different effect. To deal with a network Voronoi diagram with varied functions of generators, it must be worth formulating a power network Voronoi diagram...

We present linear-time algorithms to construct tree-structured Voronoi diagrams, after the sequence of their regions at infinity or along a given boundary is known. We focus on Voronoi diagrams of line segments, including the farthest-segment Voronoi diagram, the order-(k+1) subdivision within a given order-k Voronoi region, and deleting a segment from a nearest-neighbor diagram. Although tree-...

The Voronoi diagram of n distinct generating points divides the plane into cells, each of which consists of points most close to one particular generator. After introducing ‘limit Voronoi diagrams’ by analyzing diagrams of moving and coinciding points, we define compactifications of the configuration space of n distinct, labeled points. On elements of these compactifications we define Voronoi d...

Abstract Voronoi diagrams were introduced by R . Klein [Kle89b, Kle88a, Kle88b] as an axiomatic basis ofVoronoi diagrams. We show how to construct abstract Voronoi diagrams in time O(nlogn) by a randomized algorithm, which is based on Clarkson and Shor's randomized incremental construction technique [CS89]. The new algorithm has the following advantages over previous algorithms: • It can handle...

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 various variants of Voronoi diagrams depending on the class of objects, the distance function and the embedding space. In this paper, we investigate a framewo...

Voronoi diagrams are a fundamental structure used in many areas of science. For a given set of objects, called sites, the Voronoi diagram separates the plane into regions, such that points belonging to the same region have got the same nearest site. This definition clearly depends on the type of given objects, they may be points, line segments, polygons, etc. and the distance measure used. To f...