Average case complexity of Voronoi diagrams of n sites from the unit cube
نویسندگان
چکیده
We consider the expected number of Voronoi vertices (or number of Delaunay cells for the dual structure) for a set of n i.i.d. random point sites chosen uniformly from the unit d-hypercube [0, 1]. We show an upper bound for this number which is linear in n, the number of random point sites, where d is assumed to be a constant. This result matches the trivial lower bound of n. This is an open problem since several years. In 1991, Dwyer [2] showed that for a uniform distribution from the unit d-ball the average number of Voronoi vertices is linear in n and it is commonly assumed that this holds for any reasonable probability distribution.
منابع مشابه
Higher-order Voronoi diagrams on triangulated surfaces
We study the complexity of higher-order Voronoi diagrams on triangulated surfaces under the geodesic distance, when the sites may be polygonal domains of constant complexity. More precisely, we show that on a surface defined by n triangles the sum of the combinatorial complexities of the order-j Voronoi diagrams of m sites, for j = 1, . . . , k, is O(kn + km+ knm), which is asymptotically tight...
متن کاملNew Results on Abstract Voronoi Diagrams
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...
متن کاملOn the Hausdorff and Other Cluster Voronoi Diagrams
The Voronoi diagram is a fundamental geometric structure that encodes proximity information. Given a set of geometric objects, called sites, their Voronoi diagram is a subdivision of the underlying space into maximal regions, such that all points within one region have the same nearest site. Problems in diverse application domains (such as VLSI CAD, robotics, facility location, etc.) demand var...
متن کاملOn the Average Complexity of 3D-Voronoi Diagrams of Random Points on Convex Polytopes
It is well known that the complexity, i.e. the number of vertices, edges and faces, of the 3-dimensional Voronoi diagram of n points can be as bad as (n2). It is also known that if the points are chosen Independently Identically Distributed uniformly from a 3-dimensional region such as a cube or sphere, then the expected complexity falls to O(n). In this paper we introduce the problem of analyz...
متن کاملThe Complexity of Bisectors and Voronoi Diagrams on Realistic Terrains
We prove tight bounds on the complexity of bisectors and Voronoi diagrams on so-called realistic terrains, under the geodesic distance. In particular, if n denotes the number of triangles in the terrain, we show the following two results. (i) If the triangles of the terrain have bounded slope and the projection of the set of triangles onto the xy-plane has low density, then the worst-case compl...
متن کامل