scholarly journals Randomized incremental construction for the Hausdorff Voronoi diagram revisited and extended

2018 ◽  
Vol 37 (2) ◽  
pp. 579-600 ◽  
Author(s):  
Elena Arseneva ◽  
Evanthia Papadopoulou
Keyword(s):  
Voronoi Diagram ◽  
Randomized Incremental Construction ◽  
Hausdorff Voronoi Diagram ◽  
Incremental Construction

scholarly journals The Hausdorff Voronoi Diagram of Point Clusters in the Plane

Algorithmica ◽  
2004 ◽  
Vol 40 (2) ◽  
pp. 63-82 ◽  
Author(s):  
Evanthia Papadopoulou
Keyword(s):  
Voronoi Diagram ◽  
Hausdorff Voronoi Diagram

scholarly journals THE HAUSDORFF VORONOI DIAGRAM OF POLYGONAL OBJECTS: A DIVIDE AND CONQUER APPROACH

2004 ◽  
Vol 14 (06) ◽  
pp. 421-452 ◽  
Author(s):  
EVANTHIA PAPADOPOULOU ◽  
D. T. LEE
Keyword(s):  
Voronoi Diagram ◽  
Vlsi Design ◽  
Companion Paper ◽  
Direct Application ◽  
Divide And Conquer ◽  
Vlsi Layout ◽  
Manufacturing Defects ◽  
Combinatorial Properties ◽  
Divide And Conquer Algorithm ◽  
Hausdorff Voronoi Diagram

We study the Hausdorff Voronoi diagram of a set S of polygonal objects in the plane, a generalization of Voronoi diagrams based on the maximum distance of a point from a polygon, and show that it is equivalent to the Voronoi diagram of S under the Hausdorff distance function. We investigate the structural and combinatorial properties of the Hausdorff Voronoi diagram and give a divide and conquer algorithm for the construction of this diagram that improves upon previous results. As a byproduct we introduce the Hausdorff hull, a structure that relates to the Hausdorff Voronoi diagram in the same way as a convex hull relates to the ordinary Voronoi diagram. The Hausdorff Voronoi diagram finds direct application in the problem of computing the critical area of a VLSI Layout, a measure reflecting the sensitivity of a VLSI design to random manufacturing defects, described in a companion paper.13

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].

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