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Voronoi Diagrams--A Survey of a Fundamental Geometric Data Structure Voronoi Diagrams — A Survey of a Fundamental Geometric Data Structure FRANZ AURENHAMMER Institute fur Informationsverarbeitung Technische Universitat Graz, Sch iet!stattgasse 4a, Austria This paper presents a survey of the Voronoi diagram, one of the most fundamental data structures in computational geometry. It demonstrates the importance and usefulness of the Voronoi diagram in a wide variety of fields inside and outside computer science and surveys the history of its development. The paper puts particular emphasis on the unified exposition of its mathematical and algorithmic properties. Finally, the paper provides the first comprehensive bibliography on Voronoi diagrams and related structures. Categories and Subject Descriptors: F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problems–geometrical problems and computations; G. 2.1 [Discrete Mathematics]: Combinatorics— combinatorial algorithms; I. 3.5 [Computer Graphics]: Computational Geometry and Object Modeling—geometric algorithms, languages, and systems General Terms: Algorithms, Theory Additional Key Words and Phrases: Cell complex, clustering, combinatorial complexity, convex hull, crystal structure, divide-and-conquer, geometric data structure, growth model, higher dimensional embedding, hyperplane arrangement, k-set, motion planning, neighbor searching, object modeling, plane-sweep, proximity, randomized insertion, spanning tree, triangulation INTRODUCTION [19841 and to the textbooks by Preparata and Shames [1985] and Edelsbrunner Computational geometry is concerned [1987bl.) with the design and analysis of algo- Readers familiar with the literature of rithms for geometrical problems. In add- computational geometry will have no- ition, other more practically oriented, ticed, especially in the last few years, an areas of computer science— such as com- increasing interest in a geometrical con- puter graphics, computer-aided design, struct called the Voronoi diagram. This robotics, pattern recognition, and opera- trend can also be observed in combinato- tions research—give rise to problems that rial geometry and in a considerable num- inherently are geometrical. This is one ber of articles in natural science journals reason computational geometry has at- that address the Voronoi diagram under tracted enormous research interest in the different names specific to the respective past decade and is a well-established area area. Given some number of points in the today. (For standard sources, we refer to plane, their Voronoi diagram divides the the survey article by Lee and Preparata plane according to the nearest-neighbor Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. @ 1991 ACM 0360-0300/91/0900-0345 $01.50 ACM Computing Surveys, Vol. 23, No. 3, September 1991 346 ● Franz A urenhammer CONTENTS developed for the computer construction and representation of Voronoi diagrams. The intention of this survey is three- fold: First, motivated by the fact that INTRODUCTION Voronoi diagrams have been (reinvented 1. HISTORICAL PERSPECTIVE and studied fairly independently in the 11 The Natural Scientist’s Viewpoint applied natural sciences, in mathemat - 1,2 The Mathematician’s Viewpoint 13 The Computer Scientist’s Viewpoint its, and in computer science, it presents 2. ALGORITHMIC APPLICATIONS sketches of their historical development 2 1 Closest-Site Problems in these three areas. Second, it surveys 2,2 Placement and Motion Planning the literature on Voronoi diagrams and 23 Triangulating Sites related structures, with particular em- 2,4 Connectivity Graphs for Sites 2,5 Clustering Point Sites phasis on the unified exposition of their 3, SELECTED TOPICS mathematical and computational proper- 3.1 The Geometry of Voronol Diagrams, Them ties and their applications in computer Relatlon to Higher Dimensional ObJects. science. Finally, it provides the first 3.2 The Topology of Planar Diagrams: Divide-and-Conquer ConstructIon comprehensive bibliography on Voronoi and its Variants. diagrams. 33 A Deformation of the Voronoi Diagram: The Plane-Sweep Technique Basic Properties of the Voronoi Diagram REFERENCES We begin with a description of elemen- tary, though important, properties of the rule: Each point is associated with Voronoi diagram that will suggest some the region o~ the plane closest to it; feeling for this structure. We also intro- (Figure 1). duce notation used throughout this pa- Why do Voronoi diagrams receive so per. See also Preparata and Shames much attention? What is special about [1985] or Edelsbrunner [1987] for sources this easily defined and visualized con- on this material. struct? It seems three main reasons are We first give a usual generic definition responsible. First, Voronoi diagrams of the Voronoi diagram. Let S denote a arise in nature in various situations. In- set of n points (called sites) in the plane. deed, several natural processes can be For two distinct sites p, q e S, the domi- used to define particular classes of nance of p over q is defined as the sub- Voronoi diagrams. Human intuition is set of the plane being at least as close to often guided by visual perception. If one p as to q. Formally, sees an underlying structure, the whole situation may be understood at a higher dom(p, q) = {x~l?zl~(x,p) <b(x, q)}, level. Second, Voronoi diagrams have in- teresting and surprising mathematical for 6 denoting the euclidean distance properties; for instance, they are related function. Clearly, dom( p, q) is a closed to many well-known geometrical struc- half plane bounded by the perpendicuhm tures. Thifi has led several authors to bisector of’ p and q. This bisector sepa- believe that the Voronoi diagram is one rates all points of the plane closer to p of the most fundamental constructs de- from those closer to q and will be termed fined by a discrete set of points. Finally, the separator of p and q. The region of a Voronoi diagrams have proved to be a site p e S is the portion of the plane ly- powerful tool in solving seemingly un- ing in all of the dominances of p over the related computational problems and remaining sites in S. Formally therefore have increasingly attracted the attention of computer scientists in the last few years. Efficient and reg(p) = ,e~~{P1 dom(p, q). reasonably simple techniques have been ACM Computmg Surveys, Vol. 23, No. 3, September 1991 Voronoi Diagrams ● 347 should be observed that each vertex is the center of a circle that passes through at least three sites but encloses no site. Although n sites give rise to ();= 0( n2) separators, only linearly many separators contribute an edge to V(S). This can be seen by viewing a Voronoi diagram as a planar graph with n re- gions and minimum vertex degree 3. Each of the e edges has two vertices, and each of the u vertices belongs to at least three edges. Hence, 2 e >3 U. Euler’s re- lation n + v – e ? 2 now implies e < 3n – 6 and u s 2 n – 4, Thus, for example, Figure 1. Voronoi diagram for eight sites in the the average number of edges of a region plane. does not achieve six; there are less than 3 n edges, and each of them belongs to exactly two of the n regions. Since the regions are coming from in- The linear behavior of the size of the tersecting n – 1 half planes, they are Voronoi diagram in the plane means that, convex polygons. Thus the boundary of a roughly speaking, this structure is not region consists of at most n – 1 edges much more complex than the underlying (maximal open straight-line segments) configuration of sites. This is one of the and vertices (their endpoints). Each point main reasons for the frequent use of on an edge is equidistant from exactly Voronoi diagrams. A second reason is two sites, and each vertex is equidistant that V(S) comprises the entire proximity from at least three. As a consequence, information about S in an explicit and the regions are edge to edge and vertex computationally useful manner. For ex- to vertex, that is to say, they form a ample, its applicability to the important polygonal partition of the plane. This post-office problem (see below) is based partition is called the Voronoi diagram, on the trivial observation that a point x V(S), of the finite point-set S (Figure 1). falls into the region of a site p if and Note that a region, say reg( p), cannot only if p is closest to x among all sites in be empty since it contains all points of S. Moreover, if site p is closest to site q, the plane at least as close to p as to any then reg( p) and reg( q) share a common other sites in S. In particular, p e reg( p). edge. This particularly implies that the It follows that V(S) contains exactly n closest pair of sites in S gives rise to regions. Some of them are necessarily some edge of V(S). unbounded. They are defined by sites ly. ing on the boundary of the convex hull of Applications in Computer Science S because just for those sites there exist To substantiate the usefulness of the points arbitrarily far away but still clos- Voronoi diagram in computer science, we est.1 No vertices occur if and only if all briefly describe four situations where this sites in S lie on a single straight line. structure is used. The practicality y and Such degenerate configurations also im- diversity of these applications will ply the existence of regions with only one impart the appeal of Voronoi diagrams.
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