scholarly journals Randomized Incremental Construction of Abstract Voronoi Diagrams

1992 ◽  
pp. 283-308 ◽  
Author(s):  
Rolf Klein ◽  
Kurt Mehlhorn ◽  
Stefan Meiser
Keyword(s):  
Voronoi Diagrams ◽  
Randomized Incremental Construction ◽  
Incremental Construction

ABSTRACT VORONOI DIAGRAMS WITH DISCONNECTED REGIONS

2014 ◽  
Vol 24 (04) ◽  
pp. 347-372 ◽  
Author(s):  
CECILIA BOHLER ◽  
ROLF KLEIN
Keyword(s):  
Hausdorff Metric ◽  
Voronoi Diagrams ◽  
Construction Technique ◽  
Voronoi Region ◽  
Expected Time ◽  
Randomized Incremental Construction ◽  
Concrete Type ◽  
Trapezoidal Decomposition ◽  
Number Of Faces ◽  
Incremental Construction

Abstract Voronoi diagrams, AVDs for short, are based on bisecting curves enjoying simple combinatorial properties, rather than on the geometric notions of sites and distance. They serve as a unifying concept. Once the bisector system of any concrete type of Voronoi diagram is shown to fulfill the AVD axioms, structural results and efficient algorithms become available without further effort; for example, the first optimal algorithms for constructing nearest Voronoi diagrams of disjoint convex objects, or of line segments under the Hausdorff metric, have been obtained this way. One of these axioms stated that all Voronoi regions must be pathwise connected, a property quite useful in divide&conquer and randomized incremental construction algorithms. Yet, there are concrete Voronoi diagrams where this axiom fails to hold. In this paper we consider, for the first time, abstract Voronoi diagrams with disconnected regions. By combining a randomized incremental construction technique with trapezoidal decomposition we obtain an algorithm that runs in expected time [Formula: see text], where s is the maximum number of faces a Voronoi region in a subdiagram of three sites can have, and where mj denotes the average number of faces per region in any subdiagram of j sites. In the connected case, where s = 1 = mj , this results in the known optimal bound [Formula: see text].

scholarly journals Randomized incremental construction of abstract Voronoi diagrams

1993 ◽  
Vol 3 (3) ◽  
pp. 157-184 ◽  
Author(s):  
Rolf Klein ◽  
Kurt Mehlhorn ◽  
Stefan Meiser
Keyword(s):  
Voronoi Diagrams ◽  
Randomized Incremental Construction ◽  
Incremental Construction

Randomized incremental construction of Delaunay and Voronoi diagrams

Algorithmica ◽  
1992 ◽  
Vol 7 (1-6) ◽  
pp. 381-413 ◽  
Author(s):  
Leonidas J. Guibas ◽  
Donald E. Knuth ◽  
Micha Sharir
Keyword(s):  
Voronoi Diagrams ◽  
Randomized Incremental Construction ◽  
Incremental Construction

scholarly journals Incremental Construction of Generalized Voronoi Diagrams on Pointerless Quadtrees

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Quanjun Yin ◽  
Long Qin ◽  
Xiaocheng Liu ◽  
Yabing Zha
Keyword(s):  
Voronoi Diagrams ◽  
Surrounding Area ◽  
Repair Mechanism ◽  
Local Repair ◽  
Order Of Magnitude ◽  
Generalized Voronoi Diagrams ◽  
Incremental Construction ◽  
Coarse To Fine ◽  
Multiple Granularities ◽  
Magnitude Improvement

In robotics, Generalized Voronoi Diagrams (GVDs) are widely used by mobile robots to represent the spatial topologies of their surrounding area. In this paper we consider the problem of constructing GVDs on discrete environments. Several algorithms that solve this problem exist in the literature, notably the Brushfire algorithm and its improved versions which possess local repair mechanism. However, when the area to be processed is very large or is of high resolution, the size of the metric matrices used by these algorithms to compute GVDs can be prohibitive. To address this issue, we propose an improvement on the current algorithms, using pointerless quadtrees in place of metric matrices to compute and maintain GVDs. Beyond the construction and reconstruction of a GVD, our algorithm further provides a method to approximate roadmaps in multiple granularities from the quadtree based GVD. Simulation tests in representative scenarios demonstrate that, compared with the current algorithms, our algorithm generally makes an order of magnitude improvement regarding memory cost when the area is larger than210×210. We also demonstrate the usefulness of the approximated roadmaps for coarse-to-fine pathfinding tasks.

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