voronoi diagrams--a survey of a fundamental geometric data...

Voronoi Diagrams — A Survey of a Fundamental GeometricData 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

Computational geometry is concernedwith the design and analysis of algo-rithms for geometrical problems. In add-ition, other more practically oriented,areas of computer science— such as com-puter graphics, computer-aided design,robotics, pattern recognition, and opera-tions research—give rise to problems thatinherently are geometrical. This is onereason computational geometry has at-tracted enormous research interest in thepast decade and is a well-established areatoday. (For standard sources, we refer tothe survey article by Lee and Preparata

[19841 and to the textbooks by Preparataand Shames [1985] and Edelsbrunner[1987bl.)

Readers familiar with the literature ofcomputational geometry will have no-ticed, especially in the last few years, anincreasing interest in a geometrical con-struct called the Voronoi diagram. Thistrend can also be observed in combinato-rial geometry and in a considerable num-ber of articles in natural science journalsthat address the Voronoi diagram underdifferent names specific to the respectivearea. Given some number of points in theplane, their Voronoi diagram divides theplane according to the nearest-neighbor

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ACM Computing Surveys, Vol. 23, No. 3, September 1991

346 ● Franz A urenhammer

CONTENTS

INTRODUCTION1. HISTORICAL PERSPECTIVE

11 The Natural Scientist’s Viewpoint1,2 The Mathematician’s Viewpoint

13 The Computer Scientist’s Viewpoint2. ALGORITHMIC APPLICATIONS

2 1 Closest-Site Problems

2,2 Placement and Motion Planning23 Triangulating Sites2,4 Connectivity Graphs for Sites

2,5 Clustering Point Sites3, SELECTED TOPICS

3.1 The Geometry of Voronol Diagrams, ThemRelatlon to Higher Dimensional ObJects.

3.2 The Topology of Planar Diagrams:Divide-and-Conquer ConstructIon

and its Variants.

33 A Deformation of the Voronoi Diagram:The Plane-Sweep Technique

REFERENCES

rule: Each point is associated withthe region o~ the plane closest to it;(Figure 1).

Why do Voronoi diagrams receive somuch attention? What is special aboutthis easily defined and visualized con-struct? It seems three main reasons areresponsible. First, Voronoi diagramsarise in nature in various situations. In-deed, several natural processes can beused to define particular classes ofVoronoi diagrams. Human intuition isoften guided by visual perception. If onesees an underlying structure, the wholesituation may be understood at a higherlevel. Second, Voronoi diagrams have in-teresting and surprising mathematicalproperties; for instance, they are relatedto many well-known geometrical struc-tures. Thifi has led several authors to

believe that the Voronoi diagram is oneof the most fundamental constructs de-fined by a discrete set of points. Finally,Voronoi diagrams have proved to be apowerful tool in solving seemingly un-related computational problems andtherefore have increasingly attractedthe attention of computer scientistsin the last few years. Efficient andreasonably simple techniques have been

developed for the computer constructionand representation of Voronoi diagrams.

The intention of this survey is three-fold: First, motivated by the fact thatVoronoi diagrams have been (reinventedand studied fairly independently in theapplied natural sciences, in mathemat -its, and in computer science, it presentssketches of their historical developmentin these three areas. Second, it surveysthe literature on Voronoi diagrams andrelated structures, with particular em-phasis on the unified exposition of theirmathematical and computational proper-ties and their applications in computerscience. Finally, it provides the firstcomprehensive bibliography on Voronoidiagrams.

Basic Properties of the Voronoi Diagram

We begin with a description of elemen-tary, though important, properties of theVoronoi diagram that will suggest somefeeling for this structure. We also intro-duce notation used throughout this pa-per. See also Preparata and Shames[1985] or Edelsbrunner [1987] for sourceson this material.

We first give a usual generic definitionof the Voronoi diagram. Let S denote aset of n points (called sites) in the plane.For two distinct sites p, q e S, the domi-nance of p over q is defined as the sub-set of the plane being at least as close top as to q. Formally,

dom(p, q) = {x~l?zl~(x,p) <b(x, q)},

for 6 denoting the euclidean distancefunction. Clearly, dom( p, q) is a closedhalf plane bounded by the perpendicuhmbisector of’ p and q. This bisector sepa-rates all points of the plane closer to pfrom those closer to q and will be termedthe separator of p and q. The region of asite p e S is the portion of the plane ly-ing in all of the dominances of p over theremaining sites in S. Formally

reg(p) = ,e~~{P1 dom(p, q).

ACM Computmg Surveys, Vol. 23, No. 3, September 1991

Voronoi Diagrams ● 347

Figure 1. Voronoi diagram for eight sites in the

plane.

Since the regions are coming from in-tersecting n – 1 half planes, they areconvex polygons. Thus the boundary of aregion consists of at most n – 1 edges(maximal open straight-line segments)and vertices (their endpoints). Each pointon an edge is equidistant from exactlytwo sites, and each vertex is equidistantfrom at least three. As a consequence,the regions are edge to edge and vertexto vertex, that is to say, they form apolygonal partition of the plane. Thispartition is called the Voronoi diagram,V(S), of the finite point-set S (Figure 1).

Note that a region, say reg( p), cannotbe empty since it contains all points ofthe plane at least as close to p as to any

other sites in S. In particular, p e reg( p).It follows that V(S) contains exactly nregions. Some of them are necessarilyunbounded. They are defined by sites ly.ing on the boundary of the convex hull ofS because just for those sites there existpoints arbitrarily far away but still clos-est.1 No vertices occur if and only if allsites in S lie on a single straight line.Such degenerate configurations also im-ply the existence of regions with only one(unbounded) edge. Otherwise, three ormore edges meet at a common vertex. It

lThe conuex hull of S is the smallest convex polY-gon that contains S.

should be observed that each vertex isthe center of a circle that passes throughat least three sites but encloses no site.

Although n sites give rise to();=

0( n2) separators, only linearly manyseparators contribute an edge to V(S).This can be seen by viewing a Voronoidiagram as a planar graph with n re-gions and minimum vertex degree 3.Each of the e edges has two vertices, andeach of the u vertices belongs to at leastthree 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,the average number of edges of a regiondoes not achieve six; there are less than3 n edges, and each of them belongs toexactly two of the n regions.

The linear behavior of the size of theVoronoi diagram in the plane means that,roughly speaking, this structure is notmuch more complex than the underlyingconfiguration of sites. This is one of themain reasons for the frequent use ofVoronoi diagrams. A second reason isthat V(S) comprises the entire proximityinformation about S in an explicit andcomputationally useful manner. For ex-ample, its applicability to the importantpost-office problem (see below) is basedon the trivial observation that a point xfalls into the region of a site p if andonly if p is closest to x among all sites inS. Moreover, if site p is closest to site q,then reg( p) and reg( q) share a commonedge. This particularly implies that theclosest pair of sites in S gives rise tosome edge of V(S).

Applications in Computer Science

To substantiate the usefulness of theVoronoi diagram in computer science, webriefly describe four situations where thisstructure is used. The practicality y anddiversity of these applications willimpart the appeal of Voronoi diagrams.

Associative File Searching

Consider some file of n two-attributerecords referring, for example, to latitudeand longitude of a city or to age and

ACM Computmg Surveys, Vol. 23, No 3, September 1991

348 ● Franz Aurenhammer

income of a person. Suppose we are givenan additional target record R and wequickly want to retrieve that record ofthe file that matches R the best. If re-trieval occurs frequently on the same file,a supporting data structure is called for.This associative file searching problemwas first posed by Knuth [1973] as atwo-dimensional generalization of theusual (one-attribute) file-searching prob-lem. It possibly is best known in its geo-metric version under the name post-officeproblem: Given a set S of n sites in theplane (post offices), report a site closestto a given query point q (the location of aperson). Note that there exists a trivial0( n)-time solution by computing all ndistances. There are various algorithmsin computational geometry that needthe post office problem as a subroutine[Preparata and Shames 1985].

Shames [1975a] first observed therelevance of Voronoi diagrams to thisproblem. A site p is closest to q if andonly if q falls into the region of p. In apreprocessing step the Voronoi diagramof S is computed. To report a site closestto q, it now suffices to determine theregion that contains q. For this so-calledpoint-location problem, efficient solutionshave been developed by Kirkpatrick[19831, Edahiro et al. [1984], and Edels-brunner et al. [1986]. In particular, pointlocation in a Voronoi diagram with nregions is supported in O(log n) time and0(n) storage overhead. This shows thatthe post-office problem can be solved bymeans of Voronoi diagrams in logarith-mic query time and without increasingthe order of space, which is well knownto be optimal already for usual filesearching.

Cluster Analysis

The problem of automatically clusteringdata arises frequently [Hartigan 19751.Finding clusters means determining apartition of the given set of data intosubsets whose in-class members are simi-lar and whose cross-class members aredissimilar according to a predefined simi-larity measure. In the case of two-

Fig ure 2. Voronoi diagram for two dense clusters.

attribute data, similarity is reflected bythe proximity of sites in the plane. Prox-imity, in turn, is revealed by propertiesof the Voronoi diagram for these sites.For instance, dense subsets of sites giverise to Voronoi regions of small area;regions of sites in a homogeneous clusterwill have similar geometric shape; forclusters having orientation-sensitivedensity, the shapes of the regions willexhibit a corresponding direction sensi-tivity. Figure 2 gives an example. Ahuja[19821 showed how to use these proper-ties for clustering and matching sites.

Voronoi diagrams support variousclustering techniques used in practice.What is required at any stage ofthe clustering process is often littlemore than the retrieval of the nearest-neighbor sites of specified sites. Theycan be reported easily by examining theedges of the regions of the specified sites.This applies to several hierarchicalmethods [Murtagh 1983], partitionalmethods [Asano et al. 1988], and meth-ods involving cluster selection [Aggarwalet al. 1989].

Scheduling Record Accesses

Consider a mass storage system repre-sented by a two-dimensional array of gridpoints, each capable of storing one record.The time for the read/write head to movefrom point (x, y) to the point (u, U) isproportional to I x – u I + I y – u 1, thedistance between these points measured

ACM Computing Surveys, Vol 23, No 3, September 1991


Figure 3. Voronoi diagram and minimum span-ning trees in the LI-metric,

in the L1-metric (i. e., Manhattan dis-tance). One problem that arises in ac-cessing batched requests is the following:How is the head movement to be sched-uled in order to retrieve the requestedrecords in minimum time (distance)?

Finding an exact solution was shownto be NP-complete by Lee and Wong[19801. A satisfactory approximate solu-tion can be obtained quickly by means ofVoronoi diagrams. In an initial step, theVoronoi diagram induced under the Ll-metric is computed in 0( n log n) time bytaking the n requested grid points assites. Now the minimum-Ll-length treespanned by the sites can be constructedquickly: An edge of the tree can onlyconnect sites whose regions are neigh-bored in the Voronoi diagram [Hwang1979; Lee and Wong 1980]. Figure 3 il-lustrates this diagram (solid) and thecorresponding spanning tree (dashed). Toobtain a head movement whose length iswithin a factor of 2 of the optimum, eachtree edge is traversed twice to access allthe sites.

Collision Detection

An important topic in controlling the mo-tion of robot systems is that of collisiondetection. For a robot moving in an ob-stacle environment, one needs to deter-mine collisions between moving robotsubparts and stationary obstacles or be-tween two separately moving subparts ofthe robot. In particular, proximity detec-tion is important in order to be able to

.-. . . . ..jgcJytj?JrJ”

Figure 4. Power diagram for seven circles

stop the system before a collision willhave occurred.

The robot system and the environmentof obstacles are usually modeled bypolygonal objects. For the sake of prox-imity detection, collision-critical pointson the boundary of these objects may becircumscribed by circles whose radii cor-respond to the tolerance threshold of thesystem. This reduces the problem to de-tecting the intersection of circles. Colorsmay be assigned to the circles in order todistinguish between “harmless” inter-sections (among circles stemming fromthe same moving robot part or the sameobstacle) and others. Sharir [1985]pointed out that there will be no inter-section between circles of different colorsif all the connected components formedby the circles are uncolored. Findingconnected components and checking col-ors seem to involve an inspection of eachpair of circles. The problem becomes easy,however, if the power diagram (theVoronoi diagram where the power linesof the circles are taken as separators) isavailable; see Aurenhammer [1988a]where this problem is solved inO(n log n) time for n colored circles. Thecrucial property that can be exploitedis that the points of intersection of twocircles lie on their common power line(Figure 4).


350 “ Franz Aurenhammer

1. HISTORICAL PERSPECTIVE

The history of’ Voronoi diagrams can betraced back to the middle of the nine-teenth century. Although the spectrumof scientific disciplines that include in-terest in Voronoi diagrams is broad, threeaspects have been emphasized:

(1) Their use in modeling natural phe-nomena

(2) The investigation of their mathemat -ical, in particular, geometrical, com-binatorial, and stochastic properties

(3) Their computer construction andrepresentation

Accordingly, Voronoi diagrams areuseful in three respects: As a structureper se that makes explicit natural pro-cesses, as an auxiliary structure for in-vestigating and calculating relatedmathematical objects, and as a datastructure for algorithmic problems thatare inherently geometric. In all threeapplications, efficient and practical algo-rithms for computing Voronoi diagramsare required. Since the first applicationbears the initial seed for their investiga-tion, let us consider the role that Voronoidiagrams play in the natural sciencesfirst.

1.1 Natural Scientist’s Viewpoint

To visualize the appearance of a Voronoidiagram in nature, one could think of thethree-dimensional space being subdi-vided into a manifold of crystals: Fromseveral sites fixed in space, crystals startgrowing at the same rate in all directionsand without pushing apart but stoppinggrowth as they come into contact. Thecrystal emerging from each site in thisprocess is the region of space closer tothat site than to all others. In otherwords, the regions form a Voronoidiagram in three-space.

1. 1.1 Don-rams of Action

Most of the early work on Voronoi dia-grams was motivated by crystallography.

●

Figure 5. Cubic crystal structure.

The objective in this respect was thestudy of regions arising from regularlyplaced sites. Figure 5 illustrates some ofthe regions defined by 15 cubically or-dered sites. In his comprehensive articleon crystal structures, Niggli [1927] callsthem Wirkungsbereiche (domains of ac-tion), a term widely used nowadays andused, with many synonyms, even in the1930s as demonstrated in the clarifyingnote by Nowacki [1933]. Much effort hasbeen devoted to the fundamental crystal-lographical question: Which types of do-mains of action are capable of filling theplane or three space completely if onlycongruent copies (and certain motions)are to be used? Significant work wasdone, among others, by Niggli [1927],Delaunay (Delone) [1932], Nowacki[19761, and by Koch [19751. In fact, thequestion is a deep mathematical one, andwe will come back to it in the discussionof mathematical aspects of Voronoidiagrams.

1. 1.2 Wigner-Seitz Zones

A physiochemical system consists of anumber of distinct sites, typicallymolecules, ions, or atoms. Equilibriumand other properties of the system de-pend on the spatial distribution of the


sites, which can be conveniently repre-sented by dividing the space betweenthem according to the nearest-neighborrule. The resulting Voronoi regions are

often called Wigner-Seitz zones afterWigner and Seitz [1933], who were the

first to use them in metallurgy. Frankand Kasper [1958] use these zones in the

investigation of complex alloy structures;Loeb [1970] bases his survey of cubiccrystal structures on an intimately re -

lated concept, Allotting zones to sites is

of interest to molecular physicists, bio-chemists, material scientists, and physi-cal chemists. Via this approach Brostowand Sycotte [1975] estimate the coordina-tion number of liquid argon, David andDavid [1982] study certain solvationstructures, Brumberger and Goodisman[19831 interpret the small-angle scatter-ing of catalysts, and Augenbaum andPeskin [1985] treat large-scale hydro-dynamic codes—just to name a few.

1.1.3 Johnson-Mehl and Apollonius Model

The nearest-neighbor rule (or equiva-lently, the crystal growth model men-tioned earlier) forces the Voronoi regionsto be convex polyhedra. Allowing thecrystals to start their growth at differenttimes gives rise to hyperbolically shapedregions. The resulting Johnson-Mehlmodel was proposed by Johnson and Mehl[1939] as a more realistic model of struc-

ture for minerals. Figure 6 gives an il-lustration; crystals are augmented with

their date of birth.

crystals growing simultaneously but

at different rates give rise to spherically

shaped regions forming the Apolloniusmodel. This structure can also be ob-

served as cell structures of plants or in

foams made out of soap bubbles [Matzke

and Nestler 1946; Smith 1954; Williams

1968]. It further appears as covering ar-

eas of plants and as areas of best-

received transmitters [Sakamoto and

Takagi 1988]. Weaire and Rivier [1984]

give a comprehensive survey and various

statistical data. Interestingly, the equi-

librium state of a spider web constitutes

a generalized, although still polygonal,


Voronoi diagram. This follows from in-

vestigations by Maxwell [1864]. Figure 7

shows sites for a spider web such that the

distance between neighborhood sites cor-

responds to the tension of the edge they

define.

1. 1.4 Thiessen Polygons

Geographical interest in Voronoi dia-

grams originates with the climatologist

Thiessen [1911], who assigned proximity

polygons to observation sites in order to

improve the estimation of precipitation

averages over large areas. His method

was worked out in detail by Horton [19171who proposed the name Thiessen poly-gons, crediting Thiessen with the idea.As reported in Boots [1979], Thiessen

polygons play four roles in geographical

research: As models for spatial processes,

as nonparametric techniques in point-

pattern analysis, as organizing struc-

tures for displaying spatial data, and for

calculating individual probabilities in

point patterns. In particular, we mention

Tuominen [1949] and Snyder [1962] (ap-

plications to urban planning), Gambini

[19661 and Boots [1979] (market areas;

generalized Thiessen polygons are used

there, in particular, the Johnson-Mehl

and Apollonius models), l~ollison [1977]

(ecological contact models), McLain

[19761 (spatial interpolation), Arnold andMilne [1984] (cartography), and Okabe et

al. [1988] (facility location). For surveysof Thiessen polygons from a geographicaland economic point of view, see Boots

[1986], Eiselt and Pederzoli [1986], and

Sibson [1979].

Suzuki and Iri [1986] report on the

importance of recovering the sites from a

given subdivision of a geographical area.

This inverse process of constructing

Thiessen polygons arises in the optimal

outline of school districts or of voting

precincts. In geographical variation

analysis, connectivity graphs for sites are

a valuable tool [Matula and Sokal 1980].

Among them are the minimum spanning

tree or Prim shortest connection network

(Figure 8), the ~elaunay triangulation,

and the Gabriel graph. These three



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e53

//

/

/’

conceptsThiessenlater.

\/

‘i\\

Figure 6. Johnson-Mehl model.

eo

—-

●

Lo

--\-%

\\

\\

\\

are intimately related to contours obtained in this way with per-

polygons as will be discussed ceptual boundaries studied by Gestaltpsychologists. As pointed out by Blum

1. 1.5 Blum’s (Medial Axis) Transform

Yet another name for the Voronoi dia-

gram is popular in the natural sciences:

Blum’s transform of a set of sites. Con-

cerned with biological shape and visual

science, Blum [1967] used it for modeling

new descriptors of shape. In general, the

main problem in pattern recognition is to

extract characterizing elements from a

given (site) pattern. When no geometri-

cal model of the pattern is available, its

structural description may be based on

the notion of neighborhood of a site and

thus, in particular, on Blum’s transform.

Fairfield [1979] successfully compared

t1973], curvature properties of a given

contour correspond to topological proper-

ties of its transform (compare Figure 9).

The smooth pieces of the contour (solid)

should be viewed as sites of generalized

shape. By definition, each point of the

transform (dashed) is equidistant from at

least two sites. In this context, the trans-

form often is called the medial axis or

skeleton of the contour. The medial axis

of digitized contours was exploited in

image processing by Philbrick [19681,

Montanari [1968], and Lantuejoul and

Maisonneuve [1984].

Although the foregoing list of applica-

tions of Voronoi diagrams in the natural


Voronoi Diagrams “

2a e

26

88 L3

e

55

e

Figure 7.

Figure 8. Minimum spanning tree for the nine

state capitals of Austria.

sciences is not complete, we refrain fromfurther details and turn now to their rolein the mathematical environment.

1.2 Mathematician’s Viewpoint

The very earliest motivation for the studyof Voronoi diagrams stems from the the-ory of quadratic forms. Gauss [18401observed that quadratic forms have aninterpretation in terms of Voronoidiagrams for sites that are parallelohe-drally ordered in space. His idea was

Spider web

353

e

1’

Figure 9. Contour and its medial axis [Philbrick

19681.

exploited by Dirichlet [18501 to establish,among other results, a simple proof ofthe unique reducibility of quadraticforms. A careful generalization to higherdimensions was provided by Voronoi[1908]. To honor the pioneering work ofthese mathematicians, the construct hasbeen referred to as the Dirichlet tessell-ation or the Voronoi diagram, and indeed



these terms are the most popular ones inthe vast body of literature.

The following review provides basicmathematical properties of Voronoi dia-grams and their generalizations (many ofthem will also be crucial in the under-standing of their algorithmic propertiesand applications), as well as points outcertain features of Voronoi diaqams thathave potential applications but may benot familiar to computational geometryresearchers.

1.2.1 Regularly Placed Sites

Mathematical interest in Voronoi dia-grams can be divided according towhether regularly or irregularly placedsites are involved. A main area whereVoronoi diagrams for regularly placedsites have been used for a long time isgeometrical crystallography.

Let R d denote the d-dimensionalCartesian s ace. A tiling of Rd is a cov-

Yering of R by closed sets whose interi-ors are pairwise disjoint. Tilings of R dby convex polyhedra with the propertythat the group of motions mapping thetiling onto itself is a crystallographicgroup [Bieberbach 1912] are of particularinterest. Their polyhedra are necessarilycongruent and were called stereohedraby Federoff [1885] and fundamentaldomains by Schonflies [1891]. One ofthe central questions of geometrical crys-tallography has been to enumerateall stereohedra and to classify themaccording to their crystallographicgroups.

Most progress was made for the sub-class of plesiohedra. They can be inter-preted as Voronoi regions and were calledspecial fundamental domains bySchonflies [1891] and domains of actionby Niggli [1927]. Laves [1930] andDelaunay et al. [1978] enumerated allplesiohedra in R2. Applying a refinedclassification scheme, Delaunay [1932]distinguished among 24 types of plesio-hedra in R3. A complete list ofplanar stereohedra was given byGrunbaum and Shephard [1987].

Delaunay [1932] showed that the num-ber of facets (faces of dimension d – 1) of

Figure 10. Plesiohedron with 18 facets [Griinbaumand Shephard 1980].

a stereohedron in R d is finite; for in-stance, 390 is an upper bound for d = 3.So, by Tarski’s [19511 decidability theo-rem, all spatial stereohedra can be “ef-fectively” determined. In spite of thisfact, their enumeration is still incom-plete. Also, an enumeration method forplesiohedra mentioned in Delaunay[19631 does not yield a practical algo-rithm. Among others, Nowacki [1976],Koch [1973], and Engel [1981] discoveredspatial plesiohedra with 18, 23, and 38facets, respectively. Figure 10 depicts aplesiohedron with many facets. It seemssurprising that its congruent copies arecapable of filling three-space completely.

The methods used to obtain plesio-hedra with large numbers of facets areprobably as interesting as the numbersthemselves. After carefully choosing aset of regularly placed sites (dependent ofseveral variable parameters), certainparts of its Voronoi diagram are con-structed via computer [Engel 1981]. Forexcellent sources on tilings, refer toGrunbaum and Shephard [1980, 1987],

It is worthwhile to mention thatVoronoi diagrams for regularly placedsites also find other applications inmathematics. They apply to numericalintegration [Babenko 1977], to packing



and covering problems for congruentspheres [Rogers 1964], and to statisticalinvestigations of lattice systems [Besag1974; Mollison 1977; Conway and Sloane1982].

1.2.2 Irregularly Placed Sites

Let us now turn our attention to Voronoidiagrams arising from sets of irregularlyplaced sites. A tiling of R d by polyhedrais called a cell complex in R d if the tilingis facet to facet, that is, each facet of apolyhedron is also a facet of some otherpolyhedron in that tiling.2 Every tilingof Rd by plesiohedra is a cell complex.The Voronoi diagram for irregularlyplaced sites is still a cell complexalthough its regions are no longercongruent polyhedra.

This fact gives rise to various ques-tions. Motivated by the problem of find-ing densest sphere packings, Rogers[19641 posed the following extremal prob-lem: How big is the smallest possibleVoronoi region with t facets and definedby sites with minimum distance two? Es-timates of this and related quantitieswere derived by Muder [1988a, 1988b] forplanar regions.

Much work on general Voronoi dia-grams is concerned with their combinato-rial properties and, in particular, withtheir size, that is, their numbers of facesof various dimensions as a function of thenumber n of sites considered. The size ofa Voronoi diagram is an important quan-tity since it relates the amount of spaceneeded to store this structure to the in-put size. Answering Crum’s problem,Dewdney and Vranch [1977] exhibit anarbitrarily large set of Voronoi polyhedrain R 3 each pair of which shares a facet.The example they give shows that thesize of a Voronoi diagram in R3 is 0( n2);

this was also observed by Preparata[1977]. As already mentioned, a Voronoidiagram in the plane has size 0(n) sinceits edges and vertices form a planar

2 In fact, a tiling if facet to facet if and only if it isface to face for faces of any dimensions less than d

[Gruber and Ryskov 19871.

/. ,? ‘I,

(–)lfl’,’1”111

l,\

4,,11,,1

,,1

,,1 /’I.

\ : ‘ .’”

Figure 11. Voronoi diagrams are related to convexhulls

graph. The size of higher dimensionaldiagrams has only recently been ana-lyzed completely. Klee [1980] and Brown[19801 showed that Voronoi diagrams inR d are equivalent to certain convexpolyhedral surfaces in R ‘f 1 in a strongsense. Hence, known results on the sizeof polyhedra carry over, in particular,the so-called upper and lower bound the-orems [Brondsted 19831. Exploiting thisrelationship, exact bounds on the num-bers of individual faces of d-dimensionalVoronoi diagrams were derived by Seidel[1982] and by Paschinger [1982]. Fig-ure 11 illustrates Brown’s original trans-form that relates the Voronoi diagramin R2 to a convex hull in R3 viastereographical projection.

1.2.3 Generalized Voronoi Diagrams

Interestingly, not every convex polyhe-dral surface in R‘+ 1 is related to aVoronoi diagam in R ~. For a more gen-eral class, a one-to-one correspondencecan be established. Paschinger [1982] andAurenhammer [1987a] showed thatpower o!iagrams are equivalent to theboundary projection of convex polyhedralsurfaces. 3 This generalization assigns aparticular weight w to each of the givensites p and replaces the euclidean dis-tance 6(x, p) between a point x and p by

3 Such projections are commonly called Schlegeldiagrams in discrete geometry [Griinbaum 19671,

ACM Computing Surveys, Vol. 23, No 3, September 1991


Figure 12. Apollomus model [Aurenhammer and Edelsbrunner 1984].

62( x, p) – w. Intuitively speaking, w ex-presses the capability of p to influence

its neighborhood. In particular, the pair

(p, w) may be interpreted as a spherewith center p and radius ~ if w > 0.As an important property, the separatordefined by two weighted sites in R2 is

still a straight line: the power line of twocircles. Figure 4 shows a powerdiagram in the plane. Power diagramsalready appeared in Dirichlet [18501—therefore, often being called generalized

Dirich let cell complexes— and continuedto be objects of interest, especially insphere packing [Rogers 1964], illumin-ating spheres [Linhart 19811, andthe geometry of numbers [Gruber andLekkerkerker 1988].

The concept of weighting the sites givesrise to several other useful types of dia-grams. Weighting the euclidean distanceby a multiplicative constant yields theApollonius model, investigated byAurenhammer and Edelsbrunner [1984].The separator of two sites in the planedescribes their Apollonius circle. FromFigure 12 it can be seen that regionsmay be disconnected and may partitionthe plane into G( n2) connected compo-nents. Additiue weights give rise to theJohnson-Mehl model (compare Figure 6).For many additional types, see Ash andBolker [1986].

At this point, let us modify the conceptof Voronoi diagram in another way. Theorder-h Voronoi diagram of n sites is apartition of R d into regions such thatany point within a fixed region has thesame k closest sites. Its regions are con-vex polyhedra that form a cell complexin R‘. For k = 1, the classical Voronoidiagram is obtained. In the case of k =

Figure 13. Planar order-2 Voronoi diagram

[Shames19781.

n – 1, the structure is often referred toas the furthest site Voronoi diagram sincenow the region of a site contains all pointsin space furthest from it. Figure 13 showsan order-2 Voronoi diagram for 8 sites inthe plane. Note that only 15 of the 28pairs of sites define a region and that aseparator may yield more than one edge.(An example of a furthest site Voronoidiagram is given in Figure 24.)

Early treatments of order-k diagramsare found in Miles [1970] and in Shamesand Hoey [1975]. Lee [1982al succeededin proving exact upper bounds on themaximum numbers of regions, edges, andvertices of a planar order-k Voronoidiagram. Asymptotically these num-bers are @(k( n – k)). The fascinatingrelationship between Voronoi dia-grams, weighted diagrams, order-k dia-grams, and other geometric objects wasworked out by Edelsbrunner et al.[19861, Aurenhammer [1987 al, and



Aurenhammer and Imai [1988]. Withinthis context, the concepts of power dia-gram and hyperplane arrangement playa central role. Section 3.1 is devoted to adetailed description of this material. Inparticular, various results on the size andon the computational complexity of con-structing such diagrams are obtained.However, still little is known on the sizeof the order-k Voronoi diagram in higherdimensions; consult [Edelsbrunner andSeidel 1986]. Only for the furthest-sitediagram exact upper bounds are avail-able [Seidel 1987]. The number of re-gions of an order-k Voronoi (or power)diagram is closely related to the num-ber of k-sets of a finite point set. Deter-mining these numbers belongs to themain open problems in combinatorialgeometry.

Once the usefulness of known types ofVoronoi diagrams had been realized, fur-ther generalizations were attempted.Motivation mostly came from applica-tions in computational geometry. In or-der to meet practical needs, generaldistance functions like the LP-metrics[Lee 1980a] or convex metrics [Chewand Drysdale 1985] were taken to definea Voronoi diagram. Also, the shape ofsites was varied while using a canonicalextension of the euclidean distance func-tion. Line segments [Kirkpatrick 1979],curve segments [Yap 1987], and disks orgeneral convex sites [Leven and Sharir1986, 1987] have been considered. If thesegments form a closed curve, their dia-gram is just the medial axis of a contour(compare Figure 9). Disks can be viewedas centers weighted additively by theirradii, thus giving rise to the Johnson-Mehl model. This list is not meant to beexhaustive; we shall be concerned withmathematical and computational proper-ties of varous generalized Voronoidiagrams in later sections.

Let us mention one more way of gener-alization: changing the underlying space.Ehrlich and Im Hof [1979] investigatedthe behavior of Voronoi regions inRiemann manifolds. Brown [1980],Paschinger [1982], and Yap [1987] ob-served that Voronoi diagrams on the

sphere and on the torus, respectively, areclosely related to their equivalents in theeuclidean space of the same dimension.Diagrams on three-dimensional poly-hedral surfaces and on the three-dimensional cone have been treated byMount [1985] and by Dehne and Klein[19871, respectively.

1.2.4 Delaunay Triangulations

Hand in hand with the investigation ofVoronoi diagrams goes the investigationof related constructs. Among them, theDelaunay triangulation is most promi-nent. It contains a (straight-line) edgeconnecting two sites in the plane if andonly if their Voronoi regions share acommon edge. The structure was intro-duced by Voronoi [1908] for sites thatform a lattice and was extended byDelaunay [1934] to irregularly placedsites by means of the empty-circle me-thod: Consider all triangles formed bythe sites such that the circumcircle ofeach triangle is empty of other sites.The set of edges of these trianglesgives the Delaunay triangulation ofthe sites.

The planar Voronoi diagram and theDelaunay triangulation are duals in agraphtheoretical sense, Voronoi verticescorrespond to Delaunay triangles,Voronoi regions correspond to sites, andedges of both types correspond by defini-tion. From Figure 14 it can be seen thatDelaunay edges (solid) are orthogonal totheir corresponding Voronoi edges(dashed)–but do not necessarily inter-sect them— and that the boundary ofthe convex hull of the sites consists ofDelaunay edges. The duality immedi-ately implies upper bounds of 3 n – 6 andof 2 n – 4 on the number of Delaunayedges and triangles, respectively (com-pare Introduction). The Delaunay tri-angulation and its duality to Voronoidiagrams generalize to higher dimen-sions in an obvious way. By resultsby Dewdney and Vranch [1977], theDelaunay triangulation in R3 mayalready be the complete graph on

()

nn sites, thus having

2edges. For a



\ /‘i\ //

‘. /’I /

e1’ //

\ /

‘Y’\/’ \_- A

--! , -.

---- \ ~y/ ‘.\

\’~ x.‘.

Y’ \ . .

/’ \/ \/ \/ \

Figure 14. Voronoi diagram and Delaunay trian-gulation are duals,

catalog of properties of higher dimen-sional Delaunay triangulations arisingfrom site lattices see Gruber andLekkerkerker [19881.

Several interesting properties areknown for the Delaunay triangulationin the plane. It was first observed bySibson [1977] that this triangulationis locally equiangular. This propertyholds when, for any two triangles whoseunion is a convex quadrilateral, the re-placement of their common edge by thealternative diagonal does not increase theminimum of the six interior angles con-cerned. Actually, the Delaunay triangu-lation is the only one with this propertythat particularly shows its uniqueness.For sites being in general position (theDelaunay triangulation may containmore-sided faces if four sites are cocircu-lar), Edelsbrunner [19871 showed that lo-cal equiangularity is equivalent to globalequiangularity. The triangles define thelexicographically largest list of sortedangles. A similar result holds if only thesmallest angle of each triangle is consid-ered [Lawson 1977]. Note that the Delau-nay triangulation thus maximizes theminimum angle over all triangulationsof a given set of sites. On the other hand,a simple example shows that the maxi-mum angle is not minimized. It is inter-esting to note that, by the empty circleproperty, any triangulation withoutobtuse angles must be Delaunay. Tri-

angulations without “extreme” an-gles are desirable in finite elementand interpolation methods.

Lawson [1972] gave a counterexampleto Shames and Hoey’s [1975] conjecturethat the Delaunay triangulation hasminimum total edge length. It does noteven approximate the shortest triangula-tion [Manacher and Zobrist 1979], and infact it may be as long as any triangula-tion [Kirkpatrick 1980]. It is, however,close to optimal on the average [Lingas1986a]. A different criterion of optimal-ity is mentioned in McLain [1976]. Foreach triangle, all its points should be atleast as close to one of its defining sitesas to any other site. This property is notshared by the Delaunay triangulation, asis claimed there.

As an important fact, the Delaunaytriangulation is a supergraph of severalwell-known and widely used graphsspanned by a set of sites in the plane.Among them are the minimum spanning

tree (or Prim shortest connection net-work ) introduced by Kruskal [1956] andPrim [1957], the Gabriel graph intro-duced by Gabriel and Sokal [1969], andthe relative neigh borhood graph intro-duced by Toussaint [19801. Section 2.4gives definitions of these graphs anddescribes their interrelations.

The Delaunay triangulation can be ex-ploited to find certain linear combina-

tions among its defining sites. Each sitenot on the convex hull can be repre-sented as a weighted mass center of thesites adjacent in the triangulation; seeSibson [1980] who mentions applicationsto surface smoothing. Aurenhammer[1988b] generalized this result to powerdiagrams and their order-k modifica-tions. In particular, Gale transforms of

the sites may be derived in this way[Aurenhammer 1990bl. Gale transformsare a versatile tool in the investigation ofhigh-dimensional convex polyhedra[Grunbaum 19671.

The generalized Delaunay triangula-tion obtained from power diagrams gainsin importance from the following recog-nition problem: Given some cell complex,can it be interpreted as a Voronoi dia-gram? A cell complex in R d may be


viewed as a power diagram if and only ifthe cell complex admits a certain dualconstruct, called the reciprocal figure byMaxwell [1864], Crapo [1979], Whiteley[1979], and Ash and Bolker [1986]. Thereciprocal figure is completely character-ized by the properties of the Delaunaytriangulation for power diagrams. Usingreciprocal figures as a criterion,Aurenhammer [1987b] showed that sev-eral well-known types of cell com-plexes are power diagrams. Moreover,Voronoi diagrams can be recognizedand their defining sites can be re-stored in an efficient way using recip-rocal figures [Aurenhammer 1987c]. Fora nice survey on reciprocal figures,see Ash et al. [1988]. Essentially distinctcriteria for cell complexes to be classicalor to be multiplicatively or additivelyweighted, Voronoi diagrams were pro-posed by Ash and Bolker [1985, 1986].Blum [1967] first suggested that gener-ally shaped sites could be reconstructedfrom the shape of the regions they de-fine. Calabi and Hartnett [1968] elabo-rated on this question in some detail.Recognition problems of this kind findapplications, aside from geography andeconomics, in statics and in computerscience.

This list of geometric and combina-torial properties of Delaunay triangu-lations is not complete. We shallsee various others while discussingapplications in computational geometry.

1.2.5 Stochastic Properties

So far, we have surveyed a good deal ofresearch on geometric and combinatorial.aspects of Voronoi diagrams. Anothersignificant stream of investigations con-cerns the determination of statistical dataabout the diagrams obtained from ran-dom distributions of sites. One of theearliest and strongest motivations forstudying stochastic properties of Voronoidiagrams stems from their practical rele-vance to physical and chemical processes,especially in metallurgy and crystallog-raphy [Johnson and Mehl 19391. Accord-ingly, most efforts concentrated on sitesdistributed in R2 and in R3. In his valu-


able paper, Meijering [1953] derivedmeans for the volume, the total boundaryarea, and the total edge length of theregions of a Voronoi diagram in R 3 aris-ing from a Poisson field of sites, as wellas the average number of vertices, edges,and facets. Several of these quantitiesare also given for the Johnson-Mehlmodel. Further progress was made byGilbert [19611 who determined the vari-ances for the volumes of such Voronoiregions. He also found some variancesassociated with plane or line sectionthrough regions. Based on experimentalresults, Kiang [19661 gave a formula forthe random size distribution of Voronoiregions in R 1, along with a conjecture forits generalization to R2 and R3. Theconjecture is, however, incorrect accord-ing to results in Gilbert [1961]. Figure 15shows his obtained density functions inR1 (solid), in R2 (dashed), and in R3(dotted). Observed volume over meanvolume is scaled.

Miles [1970] made a thorough study ofVoronoi diagrams induced by a planarhomogeneous site process. Expectations ofthe edge number, edge length, perimeter,and area of the regions are given.Further results concern the Delaunaytriangulation and the order-k family ofdiagrams. Relevant experimental datacan be found in Crain [19781. His ob-served frequencies of edges per polygonare of particular interest since no theo-retical results are presently available.An important result by Dwyer [1989]shows that the expected number of ver-tices of the classical (closest or furthestsite) Voronoi diagram in d dimensions isonly 0(n) if the n sites are uniformlydrawn in a hyperball. At this place, wemention only marginally that several re -suits have been obtained that concernthe behavior of Voronoi diagrams for sitesbeing introduced in random order ratherthan being drawn by some probabilitydistribution. We shall see such results inSection 1.3.

Space constraints preclude mentioningall related work. To name a few authors,we refer to Newman et al. [19831 (num-ber of nearest neighbors in d di-mensions), Weaire and Rivier [19841


360 . Franz Aurenhammer

Figure 15. Empirical density function of the volume of regions [Kiang 19661.

(generalized diagrams), Besag [19741,Mollison [19771, Cruz Orive [19791,Conway and Sloane [1982] (sitelattices; applications to spread ofepidemics and to coding theory aregiven), and Aurenhammer et al.[19911 (probabilistic distance functions).The reader interested in geometricalprobability in general and stochasticproperties of cell complexes in particularmay consult the expository papers byMoran [1966, 19691, Little [19741, andBaddeley [1977].

1.3 Computer Scientist’s Viewpoint

We have documented the remarkable rolethat Voronoi dia~ams play in the math-ematical and applied natural sciences.Yet for a long time their practical useful-ness suffered from the absence of reason-ably simple and efficient methods fortheir computation. This section reviewsmethods for the computer constructionand representation of Voronoi diagrams.Algorithmic applications of Voronoidia~ams and of related structures arediscussed in Section 2.

1.3.1Early Algorithms

Since Voronoi diagrams have been usedfor decades by natural scientists, many

intuitive construction rules were pro-posed. The earliest diagrams were drawnwith pencil and ruler; see Horton [1917]or Kopec [19631 who mentions problemsof ambiguity if many sites lie on a com-mon circle. Probably the most obviousapproach is to delineate the diagram inthe plane region by region, by singlingout those separators that contribute toedges of the current region. A provi-sional and admittedly inefficient versionof such an algorithm was described byRhynsburger [19731. Other early algo-rithms build up the diagram vertex byvertex [Brassel and Reif 1979] or by in-cremental insertion of sites (i. e., of theirregions) [Green and Sibson 19771.

Figure 16 illustrates the insertion of asite p that involves two tasks. First, weneed to find the current region in whichp falls. Let q be the site defining thisregion; the separator of p and q thenwill contribute an edge, e, to p’s region.Second, the boundary of p’s region (bold)is created edge by edge, starting with e.During this process, the parts of the olddiagram closest to p (dashed) aretraversed and deleted. These parts arespecified by lying on p’s side ofthe separators of p and its new neigh-bors. If appropriately implemented, thesecond task requires time proportional to



\\\\

,“\

e

e

Figure 16. Inserting a Voronoi region.

the number of edges deleted. This num-ber is 0(i) in the worst case (i denotesthe number of sites inserted so far) sincewe have to delete a planar graph with atmost i regions. The first task requires0(i) time in the worst case, but a simpleheuristic will achieve 0(V) expectedtime. Experimental results showed thatthis method is quite efficient and thus isapplicable to rather large sets of sites. Asfor the other algorithms mentionedabove, however, the worst-case behavioris 0( nz) for n given sites.

Several intuitive methods are alsoavailable for the Voronoi diagram in R 3.They compute the induced cell complexfacet by facet [Brostow et al. 1978], ver-tex by vertex [Finney 1979], or via itsdual, the Delaunay triangulation[Tanemura et al. 19831. Potentially, theyrequire 0( n4) time in the worst case,although they may perform quite well forvarious distributions of sites. For d >4dimensions, insertion strategies workingdirectly [Bowyer 19811 or based on theDelaunay triangulation [Watson 1981]have been used to construct a suitablecombinatorial representation of theVoronoi diagram. Their time complexityis O(nl+ll~) and 0(n2– I[d), respectively,provided the sites are “well distributed. ”This should be contrasted, however,against the maximal size of a Voronoi

diagram in R d that grows exponentiallywith d. A three-dimensional implemen-tation of Watson’s algorithm is discussedin Field [1986]. Avis and Bhattacharya[1983] propose an O(n rdlzl ‘1) time algo-rithm for determining all vertices of thed-dimensional diagram. They also out -line a linear programming method forcalculating the d-dimensional Delaunaytriangulation but give no concretecomplexity analysis.

Only recently have optimal or near-optimal algorithms for constructingVoronoi diagrams been devised. The rea-son is that only in the last few yearshave powerful algorithmic techniquesbeen fully developed and exploited forcomputational geometry purposes.

1.3.2 Speeding Up Insertion

The process of building a Voronoi dia-gram in the plane by incremental inser-tion of sites stands out by its simplicity.Originally having an 0( n ~) expectedperformance, the Green-Sibson algorithmmay be polished up to run faster for sev-eral distributions of the sites. By intro-ducing suitable orderings of the sites, theexpected time for finding the region thenext site falls in and for integrating thenew region can be lowered to O(1) [Ohyaet al. 1984a, 1984b].4 This gives an ex-pected runtime of 0(n). Clearly this isthe best we can hope for because the sizeof the diagram is a trivial lower boundfor the time needed to compute it.

An alternative approach to speedingup insertion is randomization. Based ona general result by Clarkson and Shor[19881, Mehlhorn et al. [19901 showed thatinserting the sites in random order yieldsan 0( n log n)-time performance withhigh probability. This complexity is inde-pendent of the distribution of the sites;expectation is taken over all possiblepermutations of the sites. The algorithmextends to the class called abstract

4 Recall in this context that the average number ofedges of a region is less than six.


362 * Franz Aurenhammer

Voronoi diagrams by Klein [1989] with-out increase of runtime. This generalconcept includes power diagrams and di-agrams defined by line segments or byLP-metrics. Guibas et al. [1990] proposean even more practical version of ran-domized incremental construction of theclassical Voronoi diagram.

As with other geometrical algorithms,the problem of numerical errors arises inthe construction of Voronoi diagrams. Forexample, sites nearly lying on a commoncircle define vertices that tend to ap-proach arbitrarily close. A proposal formaking the strategy of insertion robustagainst numerical errors is outlined inSugihara and Iri [1988].

The process of inserting Voronoi re-gions extends nicely to R3. Regions areconvex polyhedra that can be constructedfacet by facet by intersecting existingregions with separators that are planesin this case. A region cannot have morethan n – 1 facets (one for each differentsite) and thus by Euler’s relation, has0(n) edges and vertices. Therefore, oneneeds 0( n + t) time per region if t facets,

edges, or vertices are deleted during

its insertion; see Aurenhammer and

Edelsbrunner [1984] who treat insertions

in a more general cell complex in R3.Since each component deleted has to beconstructed first, an 0( n2 )-time algo-rithm results. This is worst-case optimalsince a Voronoi diagram in R3 may havea size of 61(n2).

1.3.3 Divide and Conquer

A widely used method to design fast al-gorithms is divide and conquer. Shamesand Hoey [19751 observed that thismethod applies well to problems in com-putational geometry and, in particular,to the construction of a Voronoi diagramin the plane: The given set of n sites isdivided into two subsets by a verticalline. The diagrams for these subsets arecomputed recursively and are then“merged” in the conquer step to form thetotal diagram. Figure 17 shows the chainof edges (bold) to be constructed duringthe process of merging the diagram for

e“{!- ● “./ :.- .,,,. .,----- ----- I,

.

-.. .:.’---,‘, ● >. :0

‘. // .,.‘y .’ .,..-. .,

“ )--: ‘1/ . -{, * .: ! “ ‘“ .,/’ .’

//

,{ ,,,

Figure 17. Merging two Voronoi diagrams

[Shames 19781.

left sites (dashed) with the diagram forright sites (dotted).

Section 3.2 is devoted to a detaileddescription of the method and its gener-alizations. In particular, two diagramscan be merged in 0(n) time, which im-plies an 0( n log n) time algorithm if thedivide step is carried out in a balancedway. This is worst-case optimal since anyalgorithm that constructs an explicit de-scription of a Voronoi diagram may beused to sort: Interpret an input sequenceof n real numbers as a set of sites on thex-axis and construct their Voronoi dia-gram. Regions of sites for consecutivenumbers will share an edge, so the sortedsequence can be obtained in 0(n) time

by scanning through the regions.5

Although theoretically fast and ele-

gant, the divide-and-conquer approach

has certain disadvantages. Implementa-

tion details are somewhat complicated,

and numerical errors are likely by

5 Voronoi diagrams do not help for sorting general

sets of sites. Seidel [19851 proved that sorting nsites in the plane with respect to their x-coordi -nates takes Q( n log n) worst-case time even whentheir Voronoi diagram is part of the input. It is

unclear whether sorting helps for Voronoi dia-

grams Fortune [1988], however, pointed out thatpresorting the sites in two different directions low-ers the additional time for constructing the

diagram in the L1-metric to 0( n log log n),


construction: Dividing the plane intonarrow slabs forces the vertices definedby sites within a slab to approach infin-ity. Moreover, the expected behavior isstill 0( n log n). An 0( n)-expected-timealgorithm that applies divide and con-quer to a certain subset of sites wasoutlined by Bentley et al. [1980].

Divide and conquer is the basis of alarge class of algorithms for computinggeneralized Voronoi diagrams in theplane. Let us briefly list some of them inorder to give credit to the authors whoconsidered them first. Shames and Hoey[19751 reported that their algorithm alsoapplies to the furthest site Voronoi dia-gram in the plane. Hwang [1979], Leeand Wong [1980], and Lee [1980] consid-ered point sites under the LP-metrics,whereas even more general distancefunctions were treated by Widmeier etal. [1987], Chew and Drysdale [1985], andKlein and Wood [19881. Imai et al. [1985]treated sites under the Laguerre dis-tance; that is, their power diagram.Concerning the divide-and-conquer con-struction of Voronoi diagrams for sitesmore general than points, we refer toKirkpatrick [1979] (line segments), Leeand Drysdale [1981] (line segments ornonintersecting discs), Sharir [1985](arbitrary disks), Leven and Sharir [19871(planar convex bodies), Yap [1987] (curvesegments), and Lee [1982b] (medial axisof a simple polygon). It should be notedthat most of the algorithms just men-tioned are worst-case optimal.

1.3.4 Higher Dimensional Embedding

Transforming geometrical problems intomore easily understood and solved onesplays an important role in computationalgeometry (e.g., see [Edelsbrunner 19871).Brown [1979] first perceived the possibil-ity of transforming Voronoi diagrams inR2 into convex hulls in R3: The sites aremapped, via stereographical projection,into points lying on a sphere (compareFigure 11). This takes 0(n) time.The convex hull of the resulting three-dimensional point set is computed usingthe algorithm by Preparata and Hong


[19771 that requires O(n log n) time. Theconvex hull is dual to the Voronoi dia-gram of the given sites in a geometricalsense. Hence 0(n) additional time suf-fices for deriving the latter. This elegantapproach matches the optimal 0( n log n)time bound, although it suffers from theproblem of processing three-dimensionalobjects.G

An important feature of the embeddingmethod is its easy generalization tohigher dimensions. Methods for deter-mining higher dimensional convex hullsare well established [Seidel 1981, 19861,hence efficient worst-case algorithms forcomputing the d-dimensional Voronoi di-agram become available. Their runtime,however, increases exponentially with daccording to the maximal size of a dia-gram. This should be contrasted with arecent result by Dwyer [1989], showingthat 0(n) expected time suffices for com-puting the Voronoi diagram in constantdimensions d for uniformly distributedsites.

Brown’s idea was further developed byEdelsbrunner et al. [1986] for construct-ing the order-k Voronoi diagram familyand by Aurenhammer [1987al for con-structing the power diagram and itsorder-k family in general dimensions.See also Edelsbrurmer [19861, Aggarwalet al. [1989a], and Aurenhammer [1990alwhere order-k diagrams are obtained indifferent ways from their higher dimen-sional embeddings. We refrain from anydetails here and refer to Section 3.1 for acomprehensive discussion of this mate-rial. For different approaches to the com-putation of order-k diagrams in the planeconsult Lee [1982al and Chazelle andEdelsbrunner [19871. Clarkson [19871speeds up a combined version of theseapproaches using random sampling,achieving O(knl”) expected time and

6 In fact, since Preparata and Hong construct con-

vex hulls by divide and conquer, this is just athree-dimensional translation of a divide-and-conquer construction. By Brown’s result, simpleconvex hull algorithms [Clarkson and Shor’s 1988]lead to simple Voronoi diagram algorithms,however.


364 ● Franz Aurenham mer

space (for any c > O) that is nearlyoptimal.

Embedding in higher dimensions yieldsalgorithms for various different typesof diagrams. See Aurenhammer andEdelsbrunner [1984] for the multipli-catively weighted Voronoi diagram,Aggarwal et al. [1989a] for the medialaxis of a convex polygon, Edelsbrunneret al. [19891 for cluster Voronoi dia-grams, and Aurenhammer and Stockl[1988] for the peeper’sdiagram.

1.3.5 Plane-Sweep Construction

Another powerful technique in

Voronoi

computa-tional ge~metry is the pjane-sweep ‘tech-nique [Preparata and Shames 19851. Incontrast to the embedding method, thistechnique decreases the dimension of theproblem. Intuitively speaking, the staticproblem of computing a Voronoi diagramin the plane is reduced to the dynamicproblem of maintaining the cross sectionof the diagram with a straight line. Thealgorithm simulates sweeping a lineacross the plane from below. At any pointin time, the portion of the diagram belowthe sweep line is complete (Figure 18).Fortune [1985, 1987] first observed thatupdates on the sweep line can be imple-mented to cost O(log n) time if a certaincontinuous deformation of the diagram istreated. From this deformation, the orig-inal diagram can be constructed in 0(n)time. The method, its application, and itsmodification for generalized Voronoidiagrams are described in detail inSection 3.3.

The plane-sweep approach to Voronoi

dia~ams combines simplicity and effi-

ciency. It achieves the optimal 0( n log n)

time and 0(n) space bounds for the clas-sical type, the additively weighted type,and the dia~am for line segments[Fortune 1987]. The method has been

successfully applied by Seidel [1988]

to Voronoi diagrams with line segment

obstacles joining sites (and their con-

strained Delaunay triangulations). This

type had been previously attacked by

means of rather involved divide-and-

Figure 18. Maintaining the intersection with aline.

conquer algorithms [Lee and Lin 1986;Lingas 1986b; Chew 1989a; and Wangand Schubert 19871. As was reported byRosenberger [19881, plane sweep alsoperforms well for planar order-k dia-grams even when the sites are weightedadditively.

1.3.6 Delaunay Triangulation Algorithms

Since the Voronoi diam-am and theDelaunay triangulation ‘are duals, thecombinatorial structure of either struc-ture is com~letelv determined from itsdual. Conse~uentfy, the Delaunay trian-gulation of n sites in the plane can beobtained in O(n) time after an0( n log n)-time precomputation of theVoronoi diagram. Several practical ap-plications, however, solely exploit combi-natorial properties of the Delaunaytriangulation. This has led to theinvestigation of methods for constructingthis structure directly, thus avoidingthe need to calculate and store thecoordinates of Voronoi vertices.

Lawson [1972] proposed an algorithmfor constructing locally equiangular tri-angulations by local improvement. Start-ing with an arbitrary triangulation ofthe sites, edges are “flipped” accordingto the equiangularity criterion (stated inSection 1.2 .4) until no more such ex-changes are required. Sibson [19771 suc-ceeded in proving that this procedure


Voronoi Diagrams w 365

Figure 19. Equiangularity and empty circle prop-erty.

actually constructs the Delaunay tri-angulation. Lawson’s original methodwas somewhat speeded up by Lee andSchachter [1980] who proceed incremen-tally and apply the edge flipping proce-dure after each insertion of a new site.McLain [19761 uses the empty circleproperty to construct the Delaunay trian-gulation by successively adding triangleswhose circumcircles are empty of sites.In fact, Figure 19 illustrates the equiva-lence of the empty circle property andthe equiangularity criterion. Althoughbeing simple to implement, all these al-gorithms have an 0( nz)-time behavior inthe worst case. On the other hand, Maus[19841 reports that a modification ofMcLain’s algorithm based on preparti-tioning exhibits an 0(n) expected run-ning time for uniformly distributed sites,

Lee and Schachter [1980] gave an0( n log rz)-time Delaunay triangulationalgorithm; it uses divide and conquer andis similar to Shames and Hoey’s primalVoronoi algorithm. A modified imple-mentation using prepartitioning wasshown by Dwyer [19871 to run inO(n log log n) expected time while re-taining the optimal 0( n log n) worst-casetime; it generalizes to the LP-metrics forp > 2. Drysdale [19901 modifies thedivide-and-conquer approach to compute,in 0( n log n) time, Delaunay triangula-tions generated by general convex dis-tance functions. A recent algorithm byGuibas et al. [19901 inserts (point) sitesin random order. Using a structure simi-lar to the Delaunay tree [Boissonnat andTeillaud 19861, earlier versions of the

triangulation are maintained to facili-tate the determination of the trianglecovering the next site to be inserted. Thispractical algorithm exhibits a random-ized running time of 0( n log n) and issimilar in spirit to Clarkson and Shor’s[1988] convex hull algorithm.

The question arises as to which of theplanar Delaunay algorithms can be ex-tended to higher dimensions. We havealready seen some early (although widelyused) algorithms at the beginning of thissection. It should be further observed thatthe method of higher dimensional em-bedding actually first constructs theDelaunay triangulation (via a convexhull), then derives the Voronoi dia-gram via dualization. Recently, Rajan[19911 proved a property of d-dimensional Delaunay triangulations re-lated to equiangularity. It implies anincremental algorithm that makes thetriangulation locally Delaunay after eachinsertion of a new site by applying atriangle-flipping procedure similar toLawson’s method in the plane. The algo-rithm is worst-case optimal for odd d.Whether every d-dimensional triangula-tion can be made Delaunay by localimprovement remains open.

1.3.7 Storage Representation and Dynamization

The Voronoi diagram maybe viewed as adata structure that organizes its definingsites in a prescribed manner. A host ofapplications of this data structure incomputational geometry will be given inSection 2. Out of a number of possibili-ties to store the planar Voronoi diagramor, equivalently, its dual, the Delaunaytriangulation, the quad-edge structureproposed by Guibas and Stolfi [19851 isparticularly practical, Essentially, eachedge is stored as a pair of directed edges.Each directed edge, in turn, stores thevertex it originates from and pointers tothe previous and to the next (directed)edge of the region to its left, The quad-edge data structure provides a cleanseparation between topological and geo-metrical aspects and supports the imple-mentation of standard techniques for

ACM Computing Surveys, Vol 23, No. 3, September 1991

366 0 Franz Aurenhammer

computing Voronoi diagrams, such as di-vide and conquer or incremental inser-tion. The structure has been generalizedto higher dimensions by Dobkin andLaszlo [19891 (facet-edge structure inR3) and by Brisson [1989] (cell-tuple structure in R ~). These data struc-tures represent the incidence andordering information in a cell com-plex in a simple uniform way. See alsoAurenhammer and Edelsbrunner [1984]and Edelsbrunner et al. [1986] fordifferent data structures representingcell complexes.

Like any data structure, the Voronoidiagram can be dynamized, that is,maintained for a set of sites that variesover time by insertion or deletion. Thecase of site insertion is covered by thealgorithms mentioned before that buildup the diagram on line by introducingnew sites. Integration of a new regioninto the Voronoi diagram for n sites inthe plane clearly costs 0(n) time in theworst case since the region may have n

edges. Gowda et al. [1983] handle inser-tions and deletions of sites in 0(n) timeby means of the so-called Voronoi-treethat occupies 0( n log log n) space. Thistree records the history of a divide-and-conquer construction of the diagram(Figure 20). Its leaves hold the sites inlexicographical order. Each inner node isassociated with the diagram that comesfrom combining its sons’ subdiagrams.The idea originates with Overmars [1981]who, however, did not give details of themethod or its storage requirement. By aresult by Aggarwal et al. [1989a], dele-tion of a given site can be performed intime proportional to the number of edgesof its region.

Using the Delaunay triangulation, aplanar set of sites can be organized into ahierarchical data structure, called theDelaunay tree by Boissonnat andTeillaud [1986]. This structure is semidy -namic; it allows the insertion of sites andsupports the retrieval of the triangle inwhich a given point falls in efficientexpected and worst-case time. TheDelaunay tree reflects the historyof an incremental construction of the

Delaunay triangulation (Figure 21).Any triangle T destroyed by insert-ing a new site gets as sons (solidpointers) the new triangles sharingan edge with T. Remaining triangles ad-jacent to T in such an edge get the re-spective new triangle as a stepson (dottedpointers). Guibas et al. [19901 furtherdeveloped this structure. Among otherresults, they showed that the expectednumber of structural changes of theDelaunay tree is 0(n) if n sites areinserted in random order into an (ini-tially empty) Delaunay triangulation.

A different way of dynamiting Voronoidiagrams or Delaunay triangulations isto allow movement of sites (continuousupdates) rather than their insertion ordeletion (discrete updates). The Voronoidiagram of continuously moving sites willchange its shape continuously, althoughonly at “critical” points in time will thecombinatorial structure of the diagramchange. Aside from degenerate cases,such combinatorial changes always arelocal (Figure 22). An edge between tworegions (here 2 and 4) collapses to a ver-tex and reappears between two others(1 and 3) just in the moment when thefour sites involved get cocircular. In thedual environment, this process corre-sponds to flipping the diagonals of thequadrilateral spanned by these four sites,which is the union of two Delaunay tri-angles. This suggests the definition ofthe Delaunay history of a set of movingsites as the chronologically ordered list ofedge flips in their Delaunay triangula-tion. Tokuyama [1988] considered thecase of two rigidly moving sets of n andm sites, respectively. Their Delaunayhistory has length 0( nm) and can becomputed and preprocessed in0( nm log nm) time in order to retrievethe Delaunay triangulation (and theVoronoi diagram) in 0( n + m) time at

any given moment. For n sites movingin fixed but individual directions andwith constant individual velocity, theDelaunay history has length 0( n3).This result by Imai et al. [1989] shouldbe contrasted with the intuitive

argument that():

cocircularities may


Voronoi Diagrams “ 367

k~L●

le*3 .5

97 e’2°

6°

A B

(-)c

.. ___ ‘-----

\\

\

c

\

‘)--’)\\\

~F(\\

6 \

\

‘---

“.. El ElFigure 20. Voronoi tree [Gowda et al. 19831.

occur among n sites. Guibas et al. [19911showed that the bound remains nearlycubic if the sites are moving continu-ously along rather general trajectories.See also Aonuma et al. [1990] for relatedresults.

1.3.8 Computing Voronoi Diagrams in Parallel

Parallelizing algorithms in computa-tional geometry usually is a complicated

task since many of the techniques used(incremental insertion or plane sweep,for instance) seem inherently sequential,Since it is one of the fundamental struc-tures in this area, the Voronoi diagramhas been among the first geometricalstructures whose construction has beenparallelized. The underlying model ofcomputation mostly has been the CREWPRAM (concurrent-read, exclusive-writeparallel random access machine), where



. ..<.. .\‘.

>

Figure 21. Delaunay tree.

I I

---:!$---“;+.../ 3

// I

Figure 22. Combinatorial change.

processors share a common memorythat allows concurrent reads but nosimultaneous writes to the same cell.

As has been mentioned, !2( n log n) is alower time bound for sequentially con-structing the Voronoi diagram for n sites

in the plane. Therefore, O(log n) paralleltime is the best that can be achievedwhen only 0(n) processors are to be used.

In her pioneering work, Chow [1980]showed that convex hulls in three spacecan be computed in 0(log3 n) time using0(n) processors. By their duality toplanar Voronoi diagrams, the complexitycarries over. Aggarwal et al. [1988] im-proved the Voronoi diagram constructionto 0(log2 n) parallel time by applyingdivide and conquer. Their approach criti-cally depends on restricting the domainthat can contain the “merge chain”(compare Figure 17) and has been im-proved recently by Cole et al. [1990]using a tailor-made data structure. Anoptimal parallel construction of Voronoidiagrams is still outstanding, however.

Goodrich et al. [1989] construct theVoronoi diagram for line segment sitesin 0(log2) time using 0(n) processors.Their approach to constructing this im-portant generalization is similar in spiritto Yap’s [1987] curve segment algorithm.AS a related question, Schwarzkopf


[1989] addresses the problem of comput-ing the digitized image of a planarVoronoi diagram using certain processornetworks.

2. ALGORITHMIC APPLICATIONS

The precomputation of a Voronoi dia-gram is the initial step of various algo-rithms in computational geometry. Wehave already mentioned some of them inthe Introduction, In this part of thesurvey we demonstrate the broadscope of algorithmic applications ofVoronoi diagrams and of closely relatedstructures.

2.1 Closest-Site Problems

According to their definition, Voronoi di-agrams apply naturally to various prox-imity problems. Determining closest sitesplays a dominant role in this contextsince it appears as a subroutine of manygeometric algorithms. Applications in-clude clustering and contouring and var-ious other problems whose relation toproximity may not be so obvious.

2.1.1 Nearest-Neighbor Queries

Probably the most popular problem inthis area is the post-office problem. Howshould a fixed set of n sites in the planebe preprocessed in order to determinequickly the site closest to an arbitrarilychosen point (the query point)? The post-office problem is motivated in theIntroduction as a two-dimensionalfile-searching problem. Another impor-tant application stems from a commondata classification rule that requires as-signment of new data points to the sameclass in which their nearest neighbor lies.An early solution yielding O(log n) querytime but requiring 0( nz) storage wasprovided by Dobkin and Lipton [19761.Shames [19751 pointed out that theVoronoi diagram of the sites divides theplane into regions of equal answer withrespect to the post-office problem: By def-inition, the region of a site contains allpoints closer to this site than to all other


sites. This reduces the problem to findingthe region that contains the query point,an approach that became powerful onceefficient data structures were developedfor the latter problem: point location inplanar straight-line graphs [Kirkpatrick1983; Edahiro et al. 1984; andEdelsbrunner et al. 1986]. Actually,logarithmic query time and linearstorage are achieved. This is asymp-totically optimal since it matchesthe information-theoretical lower bound.

Shames’ approach in conjunction withKirkpatrick’s was worked out in a geo-metrically dual setting by Edelsbrunnerand Maurer [1985]. Usage of a structurerelated to Boissonnat and Teillaud’s[19861 Delaunay tree even obviates theneed of postprocessing the Voronoi dia-gram for point location; Guibas et al,[1990] show that post-office queries aresupported in 0(log2 n) expected time bythis structure. An off-line solution to thepost-office problem (all query points aregiven in advance rather than being spec-ified on line) was presented by Lee andYang [19791.

In principle, higher dimensional vari-ants of the post-office problem may besolved via the Voronoi diagram, too.Their practical relevance is evident fromthe equivalent file-searching problem formultiattribute data mentioned in the In-troduction. Although Chazelle [1985]showed that point-location in a Voronoidiagram in R3 can be performed effi -ciently, the approach suffers from a con-siderable storage overhead of @(n2 ) thatcomes from the worst-case size of the dia-gram. In d >3 dimensions, the pro-hibitively large size of the diagram maybe reduced to some extent for thatpurpose [Dewdney 1977].

Generalizations of the post-office prob-lem for nonpoint sites or for modifieddistance functions may be solved bymeans of an appropriate Voronoi dia-gram as well. We particularly mentionMount [1986] (geodesic distance on apolyhedral surface) and Aurenhammeret al. [19911 (probabilistic distances inthe plane) as two generalizations leadingto interesting types of diagrams.



2. 1.2 k-Nearest-Neighbor Queries

An obvious and important generalizationof the post-office problem is to ask for thek closest sites for a query point. It wasobserved by Shamos and Hoey [1975] thatthe order-k Voronoi diagram divides theplane into regions of equal answer withrespect to this problem. Postprocessingthe diagram for point-location yields adata structure that handles queries inO(log n + k) time and requires O(k2 n)storage. [The diagram may contain@(k( n – k)) regions for each of whichthe k closest sites are stored.] If k isspecified in the input, an 0( n3)-spacedata structure yielding the same querytime was obtained by Edelsbrunner et al.[19861 via embedding the whole family oforder-k Voronoi diagrams for the sitesinto an arrangement of planes. A tech-nique for compacting order-k Voronoi di-agrams recently developed by Aggarwalet al. [19901 drastically reduces the stor-age requirement of these approaches [to0(n) for fixed k and to 0( n log n) if k isin the input] while still achieving theoptimal O(log n + k) query time.

A related problem is the circular re-trieval problem that requires a report ofall sites covered by a given query disk.In facility location, for example, onemight be interested in all sites beinginfluenced by a new facility of given ra-dius of attraction. First steps toward asatisfactory solution were undertaken byBentley and Maurer [19791. Their methodis based on a certain sequence of order-kVoronoi diagrams of the given sites andhas been refined by Chazelle et al.[1986b] and recently by Aggarwal et al.[19901. The latter refinement yields adata structure using 0( n log n) space andachieving O(log n + t) query time, wheret is the number of sites in the query disk.

2. 1.3 Closest Pairs

Finding the closest pair among n sitescertainly belongs to the most fundamen-tal proximity problems. One obvious ap-plication arises in collision detection; thetwo closest sites are in the greatest dan-

@

/1, /I 1, /

/

1’/

/\

\ v’

<“ ‘1 i>----

--- +“ =..l’, ‘.

-.I \

I 1\

; \1

/’ I

I

Figure 23. All closest pairs and the largest emptycircle.

ger of collision, In other applications, forexample, in clustering, one needs to findthe closest site for each of the sites. Thestraightforward approach clearly resultsin calculating 6( nz) distances. In theplanar case, both questions were settledin optimal time 0( n log n) by Shamosand Hoey [19751 by precomputing theVoronoi diagram of the sites. See Figure23 depicting all closest pairs by bold seg-ments. Since the region of a site and theregion of its closest site always share anedge, one only needs to visit each edgeand calculate the distance between thetwo sites whose separator yields thatedge. This clearly takes 0(n) time oncethe diagram is available. Recently,Hinrichs et al. [1988a, 1988bl showedthat it suffices to maintain certain partsof the diagram during a plane sweeprather than to compute the diagram ex-plicitly in order to solve both problems inoptimal time. Curiously, the furthest pairof sites cannot be derived directly fromthe furthest site Voronoi diagram butrequires some additional point-location[Toussaint and Bhattacharya 19811.Finding the furthest pair means de-termining the diameter of the under-lying set of sites and thus has severalapplications.



2.2 Placement and Motion Planning

Voronoi diagrams have the property thateach point on a Voronoi edge maximizesthe distance to the closest sites. Al-though being straightforward, this prop-erty is essential for the usefulness ofVoronoi diagrams in placement andmotion planning problems.

2.2.1 Largest Empty Figures

The largest empty figure problem is thefollowing: Given n point sites in theplane, where should a figure of pre-scribed shape be placed so that its area ismaximized but no site is covered? If thefigure is a circle, the problem can bereformulated as finding a new site beingas far as possible from n existing ones.Shames and Hoey [1975] mentioned therelevance of this facility location problemto operations research and industrial en-gineering. They also showed how to finda largest empty circle in 0( n log n) time:The center of such a circle is either avertex of the Voronoi diagram of the sitesor the intersection of a Voronoi edge andthe boundary of the convex hull of thesites; see Figure 23 illustrating the oc-currence of the latter case. Thus atten-tion can be restricted to 0(n) possibleplacements of the circle,

The smallest enclosing circle of thesites is obtainable in 0( n log n) time in asimilar manner. Shames [1978] claimedthat this circle always will be centered ata vertex of the furthest site Voronoi dia-gram. This was proved true by Toussaintand Bhattacharya [19811. From Figure24 it can be seen that the edges of thisdiagram form a tree structure. This re-stricts the maximum number of vertices,and thus of placements of the circle, ton – 2.7 On both problems, there exists

7 It is easily seen that the regions of the furthestsite diagram are either unbounded or empty, Thecorresponding furthest site Delaunay triangulationthus is an outer planar graph (with at most n sitesas vertices), which is well known to contain at most2 n – 3 edges and n – 2 triangles.

\/

/“/ \

\\

Figure 24. Furthest site Voronoi diagram and

smallest enclosing circle.

a considerable literature [Preparataand Shames 1985]. In particular, theproblems are solvable in 0(n) time bymeans of linear programming as wasdemonstrated by Megiddo [1983].

The Voronoi diagram approach gener-alizes—while still achieving the0( n log n)-time bound—to other figuresprovided these are “circles” with respectto the underlying distance function. Forexample, the largest empty axis-parallelsquare can be found using the Voronoidiagram in the L~-metric. Chew andDrysdale [1985] find largest emptyhomothetic copies of generally shapedconvex figures via diagrams that areinduced by a rather general class ofdistance functions. The determination ofthe largest empty axis-parallel rectangleturns out to be somewhat more compli-cated; the aspect ratio of the rectangle,and thus the distance function to be used,is not known in advance. This problemhas an obvious application. If we aregiven a piece of fabric or sheet metal andthe sites are flaws, what is the largestarea rectangular piece that can be sal-vaged? Chazelle et al. [1986a] gave an0( n log3 n)-time solution. They use aninteresting type of Voronoi diagram thatcan be intepreted as the power diagramof a certain set of circles [Edelsbrunnerand Seidel 1986; Aurenhammer and Imai1988]. The running time above was

ACM Computing Surveys, Vol. 23, No, 3, September 1991


improved to 0( n log n) by Aggarwal andSuri [1987] using different methods. Theproperties of the Voronoi diagram for linesegment sites have been exploited byChew and Kedem[1989] to find the largestsimilar copy of a convex polygon suchthat no segment is overlapped. As a re-lated question, Aonuma et al. [1990] placea given convex polygon inside and asfar as possible from the boundary of anarbitrary polygon. Their algorithm relieson Voronoi diagrams dynamized withrespect to certain site movements.

2.2.2 Translational Motions

Voronoi diagrams are an aid in the plan-ning of collision-free motions of a figurein the presence of obstacle sites. Prob-lems of this kind arise in robotics. TheVoronoi diagram approach plays an im-portant role in this area and is called theretraction method. Intuitively speaking,when moving on the edges of the Voronoidiagram the robot always keeps as faras possible from the neighboring ob-stacles. This was first perceived byRowat [19791; several authors pursuedthis idea further.

For many applications it is feasible toapproximate the scenario of obstacles bypolygonal sets. O’Dunlaing and Yap[19851 plan the motion of a disk betweentwo given points in this environment. Ina preprocessing step, the Voronoi dia-gram of the obstacles is computed by tak-ing their n boundary segments as itsdefining sites. The graph formed by the0(n) straight or parabolic edges of thediagram can be computed in 0( n log n)time [Kirkpatrick 1979; Fortune 19871.Figure 25 depicts the graph (dashed) forfour line segment obstacles (bold). Ob-serve that each point on an edge isequidistant from its two closest obsta-cles. Therefore, for each edge, it is easyto compute the minimum clearance whenmoving on it. To produce a collision-freemotion (if it exists), one first finds theedges e and e’ closest to the center of thedisk in initial and final position, respec-tively, and then searches the graph start -ing with e until either e’ is reached or

,- .-,~—. _---–/ > -——- -——/’ /.’

/,’///

/“;- L..

.— ------ ---/’ -_

/’ ---/

/’. /

/’.

Figure 25. Voronol diagram for line segments.

the clearance falls short of the diskradius. All of that can be done inadditional 0(n) time.

Compared to others, this simple ap-proach has the additional advantage thatthe disk radius need not be known dur-ing the preprocessing phase. Extendingthe approach, Leven and Sharir [19871plan translational motions of a convexplanar robot in 0( n log n) time. Notethat generally shaped robots may be cir-cumscribed by a convex figure, and pathsfeasible for the figure will do for therobot as well.

2.2.3 Rotational Motion

Allowing the robot to rotate adds a thirddegree of freedom to the problem. Muchattention has been paid to moving—amidst polygonal obstacles— a linesegment that is allowed to rotate. Theposition of a segment (of given length) inthe plane is determined by the triple p =(x, y, p) indicating the coordinates of oneendpoint and the orientation. The set ofall triples p such that the correspondingsegment L(p) does not collide with anyobstacle is a subset of the underlyingthree-dimensional configuration space.Within this subset, we may define aVoronoi diagram as the set of all p suchthat -L(p) is equidistant from its twoclosest obstacles; closeness is with re-spect to the minimum distance betweensegment and obstacle.

O’Dunlaing et al. [1986, 1987] pro-vided a thorough analysis of this dia-gram (which seems to belong to one ofthe most complex types). In particular,they showed that the diagram can be



constructed, and a motion be planned in0( n2 log n log* n) time, thus being a sig-nificant improvement over earlier tech-niques. Sifrony and Sharir 11986] speededup the method—by introducing a relatedgraph-to run in O(t log n) time, wheret = 0( n2) is a parameter dependent onthe segment length. Canny and Donald[19881 simplified the method by relaxingthe definition of the diagram.

There are several other versions of themotion planning problem to whichVoronoi diagrams apply [Rohnert 19881.An axiomatic characterization of theboundary properties of a generally shapedthree-dimensional scenaro in order to de-fine a Voronoi diagram suited to motionplanning was proposed by Stifter [19891.For a more detailed introduction and re-view of the use of Voronoi diagrams inmotion planning, the interested readermay consult Schwartz and Yap [1986] orAlt and Yap [19901.

2.2.4 Path Planning

A special case of planning collision-freemotions, where the robot is taken to be asingle moving point, is path planning.Finding euclidean shortest paths in thepresence of polygonal (or polyhedral) ob-stacles has received particular interest.For example, consider the problem ofdetermining that location, for each of mfixed locations within a polygonal factoryfloor, reachable on the shortest path. Thisproblem can be solved in 0( n +m)log2 ( n + m) time by first constructingthe Voronoi diagram of n sites under thegeodesic metric interior to a simple poly-gon with m edges [Aronov 1989]. Thesite yielding the shortest path to a par-ticular site defines a neighboring regionin the diagram. This improves an earlierresult by Asano and Asano [19861. Re-lated questions can be solved by means ofthe furthest site geodesic Voronoi dia-gram inside a polygon; Aronov et al.[1988] showed how to construct this

structure efficiently.

The following important question is

still unsolved. Is it possible to compute

the geodesic Voronoi diagram for n sites

------ T-.

9--/.‘\ ‘. .‘\

\ ,/- ----

/’ -. -.%-..

. .

Figure 26. Geodesic separator.

and a collection of disjoint and con-vex polygons with a total of m edges in0( n + m)log( n + m) time? The mainproblem in this respect is that com-puting geodesic distances is not aconstant-time operation, Figure 26 showsthe separator (dashed) of two sites underthe geodesic distance among two convexpolygons (bold). It is composed of straightline and hyperbola segments meeting atintersections with visibility lines (solid).

Path-planning problems in three-dimensional polyhedral spaces have beenstudied, by among other authors, Mount[19861, Mitchell et al. [1987], andBaltsan and Sharir [1988]. For example,finding shortest paths lying entirely on a

(generally nonconvex) polyhedral surface

is of interest in areas like autonomous

vehicle navigation, where hilly terrains

are being modeled. Efficient solutions are

obtained by using Voronoi diagrams de-

fined in such spaces. Moreover, since

finding shortest paths is closely related

to determining visibility between obsta-

cles, Voronoi diagrams involving visibil-

ity constraints have been exploited in

this context. An example is the peeper’s

Voronoi diagram used by Baltsan and

Sharir [1988] and by Aronov [1989]. Each

point in the plane is assigned to the clos-

est visible site, and visibility is con-

strained to a segment on a line avoiding

the convex hull of the sites (see Figure

36). This structure can attain a size of

@ n2) and is constructible in 0( TL2) time

[Aurenhammer and St6ckl 1988].

2.3 Triangulating Sites

Intuitively speaking, triangulating nsites in the plane means partitioning the



plane into triangles whose vertices areall and only the sites. Triangulations areimportant because they (as being planargraphs) on the one hand have only linearsize and, on the other hand, establishconnectivity information among the sitesthat suffices for many applications.

2.3.1 Equiangular Triangulations

An early and major application is theinterpolation of functions of two vari-ables, where function values initially areknown only at irregularly placed sites.Practical evidence stems from finiteelement methods or from processinggraphical data, for example, in terrainmodeling. Given a triangulation ofthe sites, the function value at an arbi-trary point can be computed by interpo-lation within the triangle containing thatpoint. Lawson [1972] and McLain [1976]have reported that a triangulation is wellsuited to interpolation if its triangles arenearly equiangular. The Delaunay trian-gulation is the unique triangulation thatis optimum in this sense [Sibson 1977].Sibson [19801 further suggests the use oflocal coordinates obtained from sites be-ing adjacent in this triangulation in or-der to compute smooth surfaces. Theremight, however, occur more-sided poly-gons (rather than triangles) if varioussites are cocircular. Mount and Saalfeld[19881 showed how to triangulate suchpolygons in order to retain equiangular -ity. This process does not affect the 0( n

log n) time needed for constructing theDelaunay triangulation.

Given a set of sites and a particularfunction value (height) at each site, anytriangulation of the sites defines a trian-gular surface in space. The “roughness”of such a surface T may be measured bythe Sobolev seminorm

where I A I denotes the area of the spa-tial triangle A, and a~ and 6A denote theslopes of the plane containing A. By ex-ploiting equiangularity, Rippa [1990]showed that the Sobolev seminorm is

Figure 27. Constrained Delaunay triangulation.

minimized if the underlying triangula-tion of the sites is Delaunay. This re-sult is somewhat surprising since theDelaunay triangulation–although itselfclearly being independent of the heightat each site —optimizes a quantity thatdepends on these heights.

2.3.2 Constrained Triangulations

In a constrained Delaunay triangulation,prescribed edges are forced in as part ofthe triangulation. See Section 3.3 for aformal definition. Figure 27 depicts sucha triangulation with two prescribed edges(bold), This structure can be constructedin 0( n log n) time [Seidel 1988] and hastwo major applications. First, it providesa more realistic approach to modelingterrain surfaces since salient elementslike mountain ridges or valleys may beprescribed. Lee and Lin [1986] showedthat the equiangularity criterion is stillfulfilled by constrained Delaunay trian-gulations, Since triangles with small an-gles produce a poor computer graphicsdisplay, the best possible visualization isachieved. Triangulations inside a simplepolygon are an important special case.The boundary edges of the polygon willnot be edges of the (unconstrained)Delaunay triangulation of the polygonvertices, in general, and thus have



to be prescribed.8 Chew [1989b]used constrained Delaunay triangula-tions to generate triangular meshesinside polygons where all angles arebetween 30° and 120°.

Second, constrained Delaunay triangu-lations are an aid in path planning. Inorder to plan a path from A to B thatavoids a set of n line segment obstacles(and thus arbitrarily shaped polygonalregions, for instance) one might con-struct this triangulation-by taking Aand B and the segment endpoints as sites

and the segments as prescribededges— and then search the graph formedby triangulation edges. This was ob-

served by Chew [19861, who also provedthe following interesting result: Betweenany two sites there is a path in thistriangulation whose length is at most

~ times the geodesic distance betweenthem, provided the L1-metric (instead ofthe euclidean) is taken to define the tri-angulation. This yields an 0( n log n)-time algorithm for approximating theoptimal path, which is a significantimprovement over exact methods.

Dobkin et al. [1990] proved a factor ofabout 5 for (nonconstrained) euclideanDelaunay triangulations and the eu-clidean distance. Their result has beenimproved to a factor of about ~ by Keiland Gutwin [19891. Results of this kindare rather surprising since they show theexistence of a sparse graph being almostas “good” as the complete graph on nsites, independently of n.

Let us mention another application.Given n sites and some prescribed (non-crossing) edges between them, a diagonalis a new edge that does not cross any

prescribed one. A popular heuristic fortriangulations with minimum total edgelength is the greedy method. This method

6 It is an interesting question how many new siteshave to be placed on the polygon boundary in order

to force it in as a subgraph of the Delaunay trian-gulation. Schwarzkopf (personal communication,1990) observed that !l(nz) sites may be necessaryfor an n-vertex polygon. A known upper bound is2 o(m log ~), The O(~) bound ~laimed in Boissonnat

[19881 involves a constant depending on the anglesof the polygon.

iteratively adds shortest diagonals byconsidering the current (and initially

empty) set of edges as prescribed. Lingas

[19891 reports that a shortest diagonalcan be found in 0(n) time given the

constrained Delaunay triangulation. This

gives an 0( n2 log n)-time and 0( n)-space

algorithm for constructing such greedytriangulations. Previous methods re-

quired 0( n2) space. Whether a minimum

length triangulation for n sites in the

plane can be computed in polynomial

time is an important open question.

2.3.3 3D Triangulations

The problem of “triangulating” pointsites in three-space arises in the decom-

position of three-dimensional objects, for

example, of computer-generated solid

models or of objects specified by planar

cross sections. Field [19861, Boissonnat

[1988], and others report on advantages

of tetrahedral decompositions and pro-

pose the use of the Delaunay triangula-

tion. Although there is some evidence for

the quality of thus produced tetrahedral,

some important properties of the two-

dimensional Delaunay triangulation do

not carry over.

First, the number of tetrahedral maybe

E)( n2), hence leading to a storage re-

quirement that is prohibitive for many

applications. This fact suggests the use

of postprocessing. ~hazelle et al. [1990]

showed the following somewhat counter-

intuitive result: The number of tetrahe -

dra may always be lowered to about n G

by adding about & new sites. This pro-cess will also eliminate elongated (andthus undesirable) tetrahedral, In fact,adding 0(n) (well-chosen) sites will al-ways reduce the size of the triangulationto 0(n), with the additional effect thatonly constantly many edges emanate

from each site [Bern et al. 19901.

Second, the question of finding a

three-dimensional analog to the equian -

gularity property is still unsettled. Rajan

[19911, however, recently succeeded in

proving a related property. The three-

dimensional Delaunay triangulation

minimizes the largest containment



radius of the tetrahedral; that is, the ra -dius of the smallest sphere containing atetrahedron. This sphere coincides withthe circumsphere if and only if its centerlies within the tetrahedron. Note thatthe containment radius measures the te-trahedron size more appropriately thanthe circumradius since small but elon-gated tetrahedral may have arbitrarilylarge circumradii. The Delaunay trian-gulation is the most compact triangu-lation in this sense. This property isquite remarkable as—unlike the two-dimensional situation—the number of te-trahedral is not independent of the way oftriangulating the given sites but mayvary from linear to quadratic.

There is another property of Delaunaytriangulations that turns out to be usefulin three-dimensions. For a set of trian-gles in three-space, an in-front/behindrelation may be defined with respect to afixed viewpoint. Generalizing a result byDe Floriani et al. [19881, Edelsbrunner[19891 proved that this relation is acyclicfor the triangles in a three-dimensionalDelaunay triangulation, no matter wherethe viewpoint is chosen. This is relevantto a popular algorithm in computergraphics that eliminates hidden surfacesby first ordering the three-dimensionalobjects with respect to the in-front/behind relation and then drawing themfrom back to front, thus over-paintinginvisible parts. For Delaunay triangles,such an ordering always will exist, whichis quite important in view of theirfrequent use in practice. In particular,so-called a-shapes are made of suchtriangles. a-shapes have been used byEdelsbrunner et al. [19831 to model theshape of point sets.

2.4 Connectivity Graphs for Sites

The Delaunay triangulation contains, assubgraphs, various structures with far-ranging applications. We will brieflydiscuss some of them in the sequel.

2.4.1 Spanning Trees

Given n point sites in the plane, a(euclidean) minimum spanning tree is a

straight-line connection of the sites withminimum total edge length. See Figure 8for an illustration. Not surprisingly, thisstructure plays an important role intransportation problems, pattern recog-nition, and cluster analysis. Construc-tion methods working on generalweighted graphs have been known for along time; see Kruskal [19561, Prim[19571, and, more recently, yao [19751whose algorithm works in 0( m log log n)time for a graph with m edges. All thesealgorithms use a basic fact: For any par-tition of the sites into two subsets, theshortest edge (edge of minimum weight)between the subsets will be present inthe tree. Applying Yao’s algorithm di-rectly to the complete euclidean distancegraph for the sites results in an Q( nz)running time, however.

Shames and Hoey [19751 recognizedthat the edges of a minimum spanningtree must be Delaunay edges. This inter-esting and important property holdssince, by the fact mentioned above, eachtree edge is the diameter of a disk emptyof sites and, by definition, each triangu-lation edge is a chord of some disk emptyof sites. From the computational view-point, Yao’s algorithm just needs to beapplied to the Delaunay triangulation— agraph with 0(n) edges. In fact, 0(n) timesuffices to derive the minimum spanningtree from the Delaunay triangulation[Preparata and Shames 19851. This gives

an 0( n log n)-time algorithm, which isoptimal by reduction to sorting n realnumbers.9

The relationship above extends to moregeneral metrics. For example, Hwang[1979] extends it to the L1-metric (com-pare Figure 3). One may be tempted toconjecture that the (euclidean) maximumspanning tree can be obtained from thefurthest site Delaunay triangulation ofthe sites (the dual of the furthest site

9 The minimum spanning tree should not be con-fused with the minimum Steiner tree of a set of

sites, a concept where the addition of new points isallowed in order to minimize the total edge length.The problem of constructing a minimum Steinertree is NP-complete [Garey et al. 1976].



Voronoi diagram). Shames [1978] showedthat this conjecture is erroneous.Eppstein [1990], however, pointed outthat two sites are connected in one ofthe k smallest spanning trees only ifboth belong to the same subset of sitesdefining a region of the order-(k + 1)Voronoi diagram. This leads to animproved algorithm for finding the ksmallest spanning trees.

The efficient construction of the mini-mum spanning tree in three space hasturned out to be a diffkult problem. Pre -computation of the Delaunay triangula-tion does not help in general since thecomplete graph may be the output in theworst case. Efficient approximation andexpected-time algorithms were given,among other authors, by Vaidya [1988a]and Clarkson [1989]. The first algorithmrunning in subquadratic worst-case timeis due to Yao [1982]; it can be speeded upto run in 0( n log n)15 time by usingVoronoi diagrams of carefully chosengrOUpS of sites. By exploiting—in addi-tion—an interesting relation to a certainclosest point problem, Agarwal et al.[19901 were able to bring down the com-plexity to 0(n413 log4/3 n). It still re-mains open whether an algorithmmatching the best currently known lowerbound of !2( n log n) can be developed.

2.4.2 Spanning Cycles

A graph connecting n sites is calledHamiltonian if it contains some cyclepassing through all the sites. The ques-tion of whether the Delaunay triang-ula-tion in the plane is Hamiltonian arose inpattern recognition problems wherea reasonable simple curve throughthe given sites is desired. Althoughcounter-examples were given byKantabutra [19831 and Dillencourt[1987a, 1987b], Delaunay triangulationshave been used with success for thisproblem. This gives evidence that thisgraph is Hamiltonian with high proba-bility. Dillencourt [1987cI supports thisthesis by proving a result concerning theconnectivity of Delaunay triangulations.

A (euclidean) traveling salesman touris a minimum-length cycle spanned by n

Figure 28. Delaunay triangulation and salesmantour [Dillencourt 1987].

sites. Constructing traveling salesmantours is a prominent problem of combina-torial optimization. Papadimitriou [1977]showed its NP-completeness. The prob-lem becomes tractable for tours beingclose to optimal. It follows from the abovediscussion that the Delaunay triangula-tion will not contain a traveling sales-man tour, in general. 10 See also Fig-ure 28 where the triangulation and thetour are drawn solid and dashed, re-spectively. Rosenkrantz et al. [19771,however, observed that traversing theminimum spanning tree twice will pro-duce a tour that is within a factor of2 of the optimum: Since removing anedge from the traveling salesman tourleaves a spanning tree of the sites, thistour must be longer than the minimumspanning tree. Note that this approxima-tion algorithm takes only 0( n log n)

time. By partitioning the minimumspanning tree into paths, then construct-ing a minimum-length matching of theirendpoints, Christofides [1976] improvedthis to a factor of 1.5, with the expense ofan 0( n2 A log4 n) construction time due

10 This is even true for special traveling salesman

tours called necklace tours that are cycles realiz-able as the intersection graph of disks around thesites. Necklace tours can be found in polynomialtime in case of their existence [Edelsbrunner et al.1987].



Figure 29. Relative neighborhood graph andGabriel graph.

to the best known matching algorithm byVaidya [1988bl.11

2.4.3 Relative Neighborhood and Gabriel Graph

It is worth mentioning two graphs thatare “in between” the minimum spanningtree and the Delaunay triangulation inthe sense of making explicit more prox-imity information than does the formerand less than does the latter. Amongthem is the relative neighborhood grap~that connects two sites provided no othersite is closer to both of them than theirinterpoint distance. In Figure 29, theedges of the relative neighborhood graphare shown solid. Toussaint [19801 re -ported the usefulness of this structure inpattern recognition, and Supowit [19831succeeded in constructing it in 0( n log n)time, given the Delaunay triangulation.Yao (personal communication, 1988) an-nounced that 0(n) time suffices for de-riving the relative neighborhood graphfrom the latter. Interestingly, this graphretains its linear size in higher dimen-sions (provided a general placement ofthe sites). This stresses the importance ofits rapid construction, especially in threedimensions, since it would imply fastminimum spanning tree algorithms. Un-

11A minimum-length matching of 2 n sites is agraph with n edges joining pairs of sites, such thatthe pairs are distinct and the total edge length isminimized. Akl [19831 showed that a minimum-length matching is not necessarily a subgraph ofthe Delaunay triangulations of the sites.

fortunately, no algorithm with signifi-cantly subquadratic running time isknown.

A similarly defined construct is theGabriel graph. It contains an edge be-tween two sites if the disk having thatedge as its diameter is empty of sites.This concept proved useful in processinggeographical data [Gabriel and Sokal1969; Matula and Sokal 19801. Howe[1978] observed that the Gabriel graphconsists of just those Delaunay edges thatintersect their dual Voronoi edges. Basedon the same observation, Urquhart [19801gave an 0( n log n) time algorithm for itsconstruction. It is easily seen that theGabriel graph is a supergraph of the rel-ative neighborhood graph. The edges ofthe former are shown solid or dashed inFigure 29.

2.5 Clustering Point Sites

Clustering a set of n point sites in theplane means determining partitions ofthe set that optimize some predefineclustering measure. This measure usu-ally is a function of the interpointdistances of the set to be clustered.

2.5.1 Hierarchical Methods

A single linkage clustering [Hartigan19751 hierarchically clusters the sites asfollows. Initially, the sites are consideredto be clusters themselves. As long asthere is more than one cluster, the twoclosest clusters are merged. The distanceof two clusters is defined as the mini-mum distance between any two sites, onefrom each cluster. This process can becarried out in a total of 0( n log n) timeby maintaining, at each stage, the fol-lowing Voronoi diagram: Each point ofthe plane belongs to the region of theclosest cluster, and closeness is with re-spect to the closest site within a cluster.The regions of the two closest clusterswill always have a common edge in thediagram. That edge corresponds to anedge of the minimum spanning tree ofthe entire set of sites [Shames and Hoey1975].



////

\ \.,\

‘\

/’/\ \ \T\\\\\I ‘.

Figure 30. Complete linkage Voronoi diagram.

The model of complete linkage cluster-ing is similar, except that the distance oftwo clusters is defined as the maximumdistance between any pair of sites, onefrom each cluster. The correspondingVoronoi diagram, where closeness of apoint in the plane is with respect to thefurthest site in a cluster, deserves specialattention. Its size is fI( rzz) for generalclusters and 0(n) provided the convexhulls of the clusters are pairwise disjoint[Edelsbrunner et al. 1989]. Figure 30shows the diagram induced by four two-site clusters, indicated by bold segments.For each cluster, the portion of the bisec-tor of its two sites that lies within itsregion is drawn dashed, The diagram canbe constructed in 0( n log n) random-ized time for convex-hull disjoint andconstant-sized clusters and— in thissetting— applies to the retrieval ofexpected-nearest sites [Aurenhammeret al. 1991]. Efficient maintenance of thediagram during the clustering processwould lead to an improved clustering al-gorithm but remains open.

2.5.2 Partitional Strategies

As another example, consider the prob-lem of partitioning a set of n sites into afixed number of t clusters such that theminimum of the (single linkage) dis-tances between clusters is maximized.The following elegant 0( n log rz)-time so-

lution was proposed by Asano et al.[19881. Construct the minimum spanningtree of the sites and remove its t – 1

longest edges. The resulting t subtreesalready give the desired clustering. Us-ing similar methods, Asano et al. [19881obtain an equally efficient algorithm forpartitioning the sites into two clusters sothat their maximum diameter is mini-mized. This problem becomes consider-ably harder if partitions into more thantwo clusters are sought. 12 Partitions intoseparable clusterings have been investi-gated by Dehne and Noltemeier [19851and Heusinger and Noltemeier [19891,who exploited the fact that clusters of ksites being separable from the remainingsites by a straight line define unboundedregions in the associated order-k Voronoidiagram.

2.5.3 Optimum Cluster Selection

The k-variance problem asks for select-ing, from a given set S of n sites, acluster of k sites with minimum vari-ance, the sum of squares of all intersitedistances in the cluster. Aggarwal et al.[1989b], who cite applications in patternrecognition, observed the following inter-esting fact. If C is k-cluster of minimalvariance, then the region of C in theorder-k Voronoi diagram of S isnonempty, Thus it suffices to examinethe variances of sets of k sites associatedwith Voronoi regions. This can be donein time proportional to the number,O(k(n – k)), of regions by exploiting thefact that sets of neighbored regions differin exactly two sites. Thus, the mosttime-consuming step is the precomputa-tion of a higher order Voronoi diagram.A similar approach leads to improve-ments if the cluster measure to be mini-mized is not variance but diameter. If Cis a k-cluster with minimum diameter,then C is contained in some set of 3k – 3sites whose region in the order-(3k – 3)Voronoi diagram of S is nonempty;

12Asano et al. [1988] claimed its NP-completeness.Rote (personal communication, 1990), announced apolynomial algorithm in the case of three clusters.



see Aggarwal et al. [1989bl wherean 0(k2 n ~k log n)-time selectionalgorithm is obtained in this way.

Several other clustering algorithmsthat rely on Voronoi diagrams have beendescribed in the literature; some of themare mentioned in the Introduction.

3. SELECTED TOPICS

A thorough understanding of the geomet -ric, combinatorial, and topological prop-erties of Voronoi diagrams is crucial forthe design of efficient construction algo-rithms. This third part of the surveypresents these properties in a unifiedmanner and discusses their algorithmicimplications. We start with a descriptionof the various relationships of Voronoidiagrams to objects in one more dimen-sion. We continue with some topologicalproperties that are particularly useful forthe divide-and-conquer construction of alarge class of Voronoi diagrams in theplane. Finally, we investigate a continu-ous deformation of the planar Voronoidiagram being well suited to construc-tion by the plane-sweep techniqueand extend it to generalized Voronoidiagrams.

3.1 Geometry of Voronoi Diagrams: Their

Relationship to Higher Dimensional

Objects

Voronoi diagrams are intimately relatedto several central structures in discretegeometry. This section exhibits these re-lationships in a comprehensive mannerand discusses some of their geometricand algorithmic consequences. Althoughour main interest is in the classical typeof Voronoi diagram, it is advantageous tobase the discussion on the more generalconcept of power diagram since the geo-metric correspondences to be describedextend to that type in a natural way. Weshall refer to d dimensions in orderto point out the general validity of theresults.

3. 1.1 Convex Polyhedral Surfaces

Consider a set S of n point sites in R dsuch that each site p G S is assigned an

Figure 31. Power line of two circles.

individual real number w(p) called itsweight. ‘I’he power function of a pointx e Rd with respect to p is specified by

pow(x, p) = (.x- p) T(x -p) - w(p).

For W(p) >0, one may think of theweighted site p as a sphere in R d with

radius - around p; for a point xoutside this sphere, pow( x, p) > 0, and

v“- expresses the distance of xto the touching point of a line tangent tothe sphere and through x. In the plane,the locus of equal power with respect totwo weighted sites p and q is commonlycalled the power line of the correspond-ing circles. If the circles are nondisjoint,their power line passes through theirpoints of intersection; see Figure 31. Ingeneral dimensions, the locus of equalpower is a hyperplane in R‘, called thechordale of p and q or chor( p, q), forshort. chor( p, q) is orthogonal to thestraight line connecting p and q butdoes not necessarily separate p from q,For W(p) = w(q), chor(p, q) degeneratesto the symmetry hyperplane of p and q.Let h( p, q) denote the closed halfspacebounded by chor( p, q) and containing the



/

Figure 32. Spatial interpretation of chordale [Aurenhammer 1987a].

points of less power with respectThe power cell of p is given by

cell(p) = ,E~~fP}h(p, q).

to p. thehyperplane

7r(p): Xd+l = 2xTp –pTp + w(p)

in R‘+ 1 [Aurenhammer 1987a]. This factis best explained by reference to Fimre

In analogy to classical Voronoi regions,the power diagram of S, PD(S) is theconvex polyhedral complex defined in R dby these cells; see Figure 4 for the cased = 2. PD(S) coincides with the classicalVoronoi diagram of S in the case ofequally weighted sites. Note that powercells may be empty if general weightsare used.

For technical reasons let us identifyR d with the linear subspace of Rd+ 1orthogonal to the (d + l)s~ coordinateaxis xd+ ~. The key observation for theexistence of objects in R‘+ 1 that are re.lated to power diagrams is that the powerfunction pow( x, p) can be expressed by

32. Let p and q -be two sites in- R2whose weights are indicated by circles.Projecting the circles onto the paraboloid

2 defines twoof revolution xs = x; + X2planes T(p) and T(g) that intersect theparaboloid in these projections. It is aneasy analytical exercise to prove that theline T(p) fl x(q) projects orthogonally tothe chordale chor( p, q) in the xl xz-plane.This correspondence holds for arbitrarydimensions. As an immediate conse-quence, PD(S) is the orthogonal projec-tion of the boundary of the polyhedronthat comes from intersecting the half-spaces of Rdw~ above m(p), for all p c S.In fact, m is a bijective mapping between

ACM Computing Surveye, Vol. 23, No. 3, September 1991


weighted sites in R d and hyperplanes inR‘+ 1. Thus, for any intersection of upperhalf-spaces there exists a correspondingpower diagram in one dimension lowerand vice versa.

This general result has far-reachingimplications. Most important, it showsthat power diagrams are in a geometricand combinatorial sense equivalent tounbounded convex polyhedra or, moreprecisely, to their boundaries, that is, toconvex polyhedral surfaces. Convex poly-hedra are well-understood objectsin discrete geometry, hence so are powerdiagrams. Exact upper and lower boundson the numbers of their faces of variousdimensions are known. In particular,PD(S) in Rd realizes at most n cells, L

j-dimensional faces (1 = j < d – 1),andf. – 1 vertices, for

n–d+i–2~=~o(~)( i )

‘io(d-~+l)(n-d: i-2)

This follows from the so-called upperbound theorem [Brondsted 19831. Thenumbers ~ are O(n Fdlzl ) for O < j sd–1.

As for the facets of a convex polyhe-dron, the power cells are convex but pos-sibly unbounded polyhedra. Not everyhalf-space within a given set needs tocontribute to a facet, so power cells maybe empty or degenerate. d + 1 or morehyperplanes in R d+ 1 intersectin a com-

mon point (unless parallelism occurs), soat least d + 1 power cells meet at eachvertex of PD( S). In particular, there areno vertices if the cardinality n of S doesnot exceed d. These and many otherproperties of PD(S) can be read off con-veniently from its (d + 1)-dimensionalembedding,

3. 1.2 Hyperplane Arrangements

In order to exploit the situation fully,we turn our attention to diagrams ofhigher order [Edelsbrunner 1987];

Aurenhammer [1987a]. To this end, theconcept of a power cell is generalized tomore than one site, Let T be a subset ofk sites in S. The power cell of T isdefined as

cell(T) = o h(p, q).PGT, q=S– T

The complex of all (nonempty) power cells

arising from the()

~ subsets of S with

fixed cardinality k. is known as theorder-k power diagram of S, k-PD(S).Clearly 1-PD(S) = PD(S) holds. Notethat k-PD( S) is just the order-k Voronoidiagram of S provided all sites in S areequally weighted. ( n – 1)-PD(S) is alsocalled the furthest site power diagramof s.

Now let Z denote the intersection ofthe half-spaces below the hyperplanesT(p) for all p e T and above the hyper -planes x-(q) for all q e S – T. Due tothe foregoing reasoning, Z projects or-thogonally onto cell(T) in R‘. Thisobservation draws the connectionbetween order-k power diagrams and so-called hyperplane arrangements. The lat -ter are cell complexes that arise fromdissecting the space by a finite number ofhyperplanes. Each cell in an arrange-ment is the intersection of some half-spaces below or above its defining n hy -perplanes. If we let a k-level consist of allcells lying below k and above n – k hy -perplanes, we can conclude that the k-level induced by {T(p) I p = S} projects tothe cells of k – PD(S) for each k be-tween 1 and n – 1.This shows that eacharrangement in R‘~ 1 corresponds to acomplete family of higher order powerdiagrams in R ~ and vice versa. The cor-respondence between order-1 power dia-grams and convex polyhedral surfacesdiscussed earlier is a special case of thissituation.

Figure 33 depicts a cell (left) being theintersection of five upper and three lowerhalf-spaces, that is, a three-level cell. Itsorthogonal projection onto the xl xz-planeis an order-3 power cell among eight sites(right) whose boundary is indicated bybold lines. Note that the power cell is


Voronoi Diagrams * 383

e .“●.. /

“.

/-O“’’’’&’’’’:+p~o”I

ii

i

i

; i

Figure 33. Projecting an arrangement cell [Aurenhammer 1987a].

●

carved into subcells by the furthest sitepower diagram of the three emphasizedsites and by the power diagram of theremaining five sites.

The correspondence to arrangementsstraightforwardly yields an upper boundon the overall size of all higher orderpower diagrams of a fixed set of n sites:the maximal number of faces of an ar-rangement of n hyperplanes in Rd+ 1.This number is precisely

,=2:+1( “ )(:)d–~+1

for faces of dimension j, which is inO(n~+l) for 0 = j < d + 1 [Alexandersonand Wetzel 1978]. Edelsbrunner et al.[1986a] developed an algorithm for con-structing such arrangements in 0( nd+ 1)time. Consequently, the complete familyof higher order power (or Voronoi) dia-grams of a given set of sites can becomputed in optimal time and space.Edelsbrunner [1986] showed how to con-struct the k-level of an arrangementin R 3, and thus a single order-kpower (or Voronoi) diagram in the plane,at a cost of 0(6 log n) per edge.Recently, Mulmuley [19911 proposed arandomized algorithm for constructinglevels of order 1 to k in d >4 dimen-sions in worst-case optimal time.

We state another interesting conse-quence. Let w’(p) denote the image ofreflection of the hyperplane T(p) throughRd. The cells in the k-level of { w’( p) I p eS} correspond bijectively to the cells inthe (n – k)-level of {r(p) I p c S} by re-versing “upper” and “lower” for theirsupporting half-spaces. Clearly, any twocorresponding cells yield the same or-thogonal projection onto Rd. That is tosay, they yield the same power cell thatis simultaneously of order-k and order-(n – k) for fixed k. This shows that anyorder-k power diagram for n sites can beinterpreted as some order-( n – k) powerdiagram. In particular, the furthest sitepower diagram of S is the order-1 powerdiagram of some set of sites (distinct fromS, in general). We point out that this isnot true for the subclass of classicalVoronoi diagrams, as can be seen fromthe maximal size of their closest site andfurthest site counterparts that differ evenin R2; see the Introduction and Section2.2.1.

Finally, let us mention a result onpower cells in R2. Let T c S and I T I =k. By definition, cell(T) is the intersec-tion of k( n – k) half planes and thusmight have that many edges. On theother hand, cell(T) is the projection of apolyhedron in R3 that has at most nfacets and thus only 0(n) edges by



Euler’s relation. It follows that cell(Z’)has O(n) edges and can be constructedin 0( n log n) time by intersecting nhalf-spaces of R3.

3. 1.3 Convex Hulls

The convex hull of a finite point set isthe intersection of all half-spaces con-taining this set. Next, we study the dual-ity of (order-1) power diagrams andconvex hulls [Aurenhammer 1987al.Duality is usually defined for con-vex polyhedra. Let Z and Z’ betwo convex polyhedra in R‘+ 1. Z and Z’are said to be dual if there is a bijectivemapping ~ between the j-dimensionalfaces of Z and the (d – j)-dimensionalfaces of Z’ such that f G g, for any twofaces f and g of Z if and only if ~(g) ~4(f). Since power diagrams in Rd areprojections of convex polyhedral surfacesin Rd+ 1, the notion of duality carriesover in a natural manner.

We now consider, for each site p E S

with weight W(p), a point in Rd+ 1:

( PA(p) =

)PTP – W(P)

From the formula expressing the hyper-plane T(p) itcan be seen that A(p) andT(p) are related via polarity with re-spect to the paraboloid of revolution. A(p)is the pole of m(p) and x(p) is the polarhyperplane of A(p). If T(p) happens tointersect the paraboloid, then polarityhas the geometric interpretation dis-played in Figure 34. The hyperplanestouching the paraboloid at its points ofintersection with 7(p) all concur at A(p).

Recall that the intersection, Z, of thehalf-spaces above ~( p), for all p e S, pro-jects to PD(S). Take an arbitraryj-dimensional face of Z that lies in theintersection of, say, T( pi), . . , 7r(pn),m 2 d —~“ + 1. That is, there exists somepoint on these but below all other hyper -planes defined by S. Since polarity pre-serves the relative position betweenpoints and hyperplanes, there is a hyper -plane with A(pl), . . . . A(p~) on it andA(q) above it, for all remaining sites

\ /

Figure 34. Polarity between points and lines.

q~s. This, however, means that

‘(PI), . . . , ~(P~) span a (d – j)-dimensional face of the 10wer part of theconvex hull of the point-set {A(p) I p e S}.This lower part consists of all boundaryfaces of the hull that are visible from thepoint on the xd+ ~-axis at – co. In otherwords, we have shown the duality of Z,and thus of PD(S), to that lower convexhull part.

By the same reasoning, the duality ofthe furthest site power diagram of S andthe upper part of the convex hull of{A(p) I p e S} can be established. Thisreveals another interesting link betweenclosest site and furthest site power dia-grams and, particularly, classical Voronoidiagrams. Most important, these dia-grams become constructible in generaldimensions d via convex-hull algo-rithms. Running times of 0( n log n) for

d = 2 and of 0(nrdi21 ) for odd d areachieved, both of which are worst-caseoptimal; see Preparata and Hong [19771and Seidel [1981], respectively. For evend >4, these diagrams can be computedat logarithmic time per face [Seidel 1986].

By projecting down the edges of thelower and upper part of the convex hullthat arises from a Voronoi diagram, oneobtains the closest site and furthest siteDelaunay triangulation, respectively. Insuch a triangulation, each edge is orthog-onal to its dual facet. Conversely, theexistence of some set of sites allowing a


Voronoi Diagrams 9 385

triangulation whose edges are orthogo-nal to the facets of a given cell complexin R ~ is a necessary and sufficient condi-tion for that cell complex to be a powerdiagram or, equivalently, to be the pro-jection of a convex polyhedral surface[Aurenhammer 1987b, 1987cI. If thegiven cell complex is simple (i.e., exactlyd + 1 cells meet at each vertex), a projec-tion surface can be reconstructed, if itexists, in time proportional to the num-ber of facets of that cell complex. It iswell known that such a surface alwaysexists except for d = 2. This implies thatany simple complex of n cells in d > 3dimensions, which might have a largenumber of lower dimensional faces, canbe specified by storing only n half-spacesof Rd~~.

Surprisingly, even higher order powerdiagrams are dual to some convex hull inRd+ 1 [Aurenhammer 1990a]. For anysubset T of S with k sites let A(T)denote the point in Rd+ 1 whose ortho-gonal projection onto Rd and whose(d + l)S’ coordinate are

~p and ~’[pup- W(P)],PET

respectively. This definition conforms tothe case k = 1 since A(T) = A(p) whenT = { p]. Using arguments similar as fork = 1, one may verify that the lower partof the convex hull of the point-set

Sk={ A(T)l TcS,l Tl=k}

is dual to k-PD(S). As one corollary,that very lower part has to be dual to theintersection of some corresponding upperhalf-spaces and hence to an order-1 powerdiagram. To be more precise, k-PD(S)coincides with PD( S&), for S& being the(accordingly weighted) orthogonal projec-tion of S’h onto R‘. This implies that theclass of power diagrams is closed underorder-k modification— another result thatis not true for the more restrictive classof classical Voronoi diagrams. From thealgorithmic viewpoint, k-PD(S) becomesconstructible via convex hull algo -

rithms. For example, the planar Voronoidiagram of order-k can be obtained fromits order-(k – 1) predecessor inO(kn log n) time by simply computing aconvex hull in R 3. Insertions and dele-tions of sites in an order-k Voronoi dia-gram also amount to computing certainconvex hulls; these operations even canbe handled on-line in an efficient manner[Aurenhammer and Schwarzkopf 19911.

3.1.4 k-Sets

Let Q be a set of n points in Rd+l. Ak-set of Q is a subset of k points of Qthat can be separated from the remain-ing n – k points by some hyperplane ofRd+ 1. The discussion in the foregoingsection reveals a correspondence betweenk-sets and order-k power diagrams.

By polarity, points in Q can be trans-formed into hyperplanes; each point q e Qbijectively corresponds to a weighted sitep in R d such that A(p) = ~ and thus tothe hyperplane T(p) in R +1. Now, letM be a k-set of Q, and consider a hyper -plane h that separates M from Q – M.Without loss of generality, M lies belowh. Equivalently, the pole of h lies belowall polar hyperplanes of the points in Mand above all polar hyperplanes of thepoints in Q – M. Hence the intersectionof the corresponding lower and upperhalf-spaces of R ‘+ ~ bounded by thesepolar hyperplanes is nonempty. This in-tersection, being a cell in the k-level of ahyperplane arrangement, projects to acell of the order-k power diagram of a setof weighted sites, { p e Rd I A(p) c Q}.

We have obtained a one-to-one corre-spondence between k-sets in R d+ 1 andpower cells in R‘. This allows us to ap-ply known results on the number of k-setsto the analysis of the size of order-k powerdiagrams. Let fd( k, n) denote the maxi-mum number of k-sets of any set of npoints in Rd. Known asymptotic valuesof this function are fz(k, n) = Q( n log k)and fz(k, n) = 0( n ~) [Edelsbrurmerand Welzl 1985], f~(k, n) = Q(nk log k)

and fa(k, n) = 0(ns13 log5/3 n)[Edelsbrunner et al. 1986; Aronov et al.1990, respectively]. Not surprisingly, a



better understood quantity is

Chazelle and Preparata [1986] showedg~(k, n) = O(nk5) and Clarkson [19871showed g~(k, n) = O(nl+’k2), for an~e > 0. Note that g~(n – 1,n) = O(n )

according to the maximum number ofcells of a hyperplane arrangement inRd. Recently, Clarkson and Schor[1989] succeeded in proving g~(k, n) =@(n L~/zJ k FW21 ) as n/k ~ m for fixed

d. Determining exact or at least asymp-totically tight bounds on ~d(k, n) stillremains an important open problem.

3. 1.5 Related Diagrams

The central role of power diagrams withinthe context of Voronoi diagrams becomeseven more apparent by the observationsdescribed below [Aurenhammer and Imai1988]. Let p, q, and r denote threeweighted sites. For the three chordalesthey define, chor( p, q) fl chor( q, r) C

chor( p, r) necessarily holds. It is not dif-ficult to see that this condition is alsosufficient for three hyperplanes to be thechordales defined by three sites if wekeep in mind the relationship betweenchordales in R d and intersections of hy -

d+ I On the other hand,perplanes in R .the condition is trivially fulfilled by theseparators for any Voronoi diagram withpolyhedral regions since the regionswould not form a cell complex otherwise.We thus conclude that any Voronoi dia-gram whose separators are hyperplanesis a power diagram. This result appliesto diagrams defined by the generalquadratic-form distance

Q(x, p) = (~-p) %(x-p) - ZO(p),

with M a nonsingular and symmetric(d x d)-matrix and thus to many partic-ular distance functions considered in theliterature. Note that Q equals the powerfunction if M is the identity matrix.

One is tempted to believe that powerdiagrams are related solely to objects in

higher dimensions. Below we will showthat a certain rather general type ofVoronoi diagram in Rd - 1 can be embed-ded into a power diaend we identify RFy%h~~~~~~~~sub;pace Xd = O of R‘. Let p be somepoint site in R‘ - 1 and consider an arbi-trary distance function f( x, p) for p. Forsome strictly increasing function F onR, we define the cone of p with respectto F as

cone~(p)

—— {(:@Rd-’,xd=F(f (x, p))}.

A Voronoi diagram for point sites inR‘ - 1 under the distance function f istermed transformable if there exists anF that forces the affine hull, U( p, q), ofcone~( p) n cone~( q) to be a hyperplanein R d for any distinct sites p and q.Transformable diagrams represent pro-jected sections of power diagrams in R‘.A point x c Rd-~ satisfies f( x, p) <

f( X, q) if and only if its projection, x’,onto cone~( p) lies in a fixed (open) half-space, h(p, q), bounded by Q(P, q). Asa consequence, x belongs to p’s regionexactly if

x’~z= (1 ~(P,~),qes – { p}

for S being the underlying set of sites inRd- 1. But the d-dimensional polyhedronZ is a power cell since a(p, q) n cr(q, r)c a( p, r) holds so that these affine hullscan be interpreted as chordales. In con-clusion, the boundary of p’s region is theorthogonal projection onto Rd-1 of

cone~( p) intersected with a power cell inRd.

Transformable Voronoi diagrams occurfor the additively weighted distance,

a(x, p) =~(x, p) – w(p)

with F(a) = a,

and the multiplicatively weighted dis-tance,

6(X, p)m(x, p) = with F(m) = 2 m2.

w(p)


Voronoi Diagrams 0 387

_.-— — -— —-- . . A

“i i----____—.y-——B J-f-

\N,

~,——— —-——— —

‘+\ /

/“q \,\ v/

v

\

Figure 35. Cones for two distance functions.

The former gives rise to hyperbolicallybounded and star-shaped regions (theJohnson-Mehl model, Figure 6), whereasthe latter induces spherically boundedregions (the Apollonius model, Fig-ure 12). Its regions are disconnected, ingeneral, which already reflects the majorcomplexity of the latter type.

Figure 35 displays cones for the addi-tively weighted distance (above) and themultiplicatively weighted distance (be-low). They are cones of revolution in theformer case—intersecting in a hyper-bola that projects down to a hyperbolic

separator— and paraboloids of revolutionin the latter case— intersecting in an el-lipsis that projects down to a circularseparator. In both cases, the affine hullof cone intersections is a plane.

Upper bounds on the size of both typesof diagram in d dimensions can be de-rived from the maximum size of a powerdiagram in R ~+ 1. Concerning the algo-rithmic aspects of this relationship, theonly known algorithms for constructingthese diagrams in d > 3 dimensions areobtained. The Apollonius model in theplane can be computed in 0( n2) time and


388 e Franz Aurenhammer

space, which is worst-case optimal. Wemention that the notion of embeddabilitygeneralizes nicely to order k.

3.1.6 Upper Envelopes

Let fl, ..., f, be piecewise linear d-variate functions on Rd. In the follow-ing, we will not distinguish between afunction on Rd and its graph in Rd+ 1.Consider the pointwise maximum, U, ofthese functions:

u(x) = mm, f,(x).

U is called the upper envelope Of

f,,... > L. We have already observed thatcertain upper envelopes are related topower diagrams. For any set S ofweighted sites in R‘, the upper envelopeof the hyperplanes T(p), p ~ S, projectsto PD(S). Below we briefly consider twotypes of Voronoi diagrams where this re-lationship generalizes particularly well.

Let S be a set of n point sites in theplane, and consider a partition of S intosubsets Cl, . . . . C~ called clusters. TheHausdorff distance of a point x to thecluster C, is defined as

h is also called the complete linkage dis-tance; compare Section 2.5.1. Each clus-ter C, may be associated with a convexpolyhedral surface cap(C,) in R3, beingthe boundary of the polyhedron thatcomes from intersecting the half-spacesbelow T(p) for all p e C,. Easy argu-ments show the following. For any pointx in the plane, h(x, C,) < h( x, C~) if andonly if the vertical line through x inter-sects cap(Ci) in a point lying abovecap(C7). This implies that the upper en-velope of cap(C,), . . ., cap(CJ projects tothe Voronoi diagram induced bycl, . . ., Ct and h. For an illustration ofthe Voronoi diagram for two-site clusterssee Figure 30.13

To obtain another example, assumethat visibility of the sites is constrained

13 Observe that the upper envelope of cap(T), for allsubsets 2’ of k sites of S, consists of faces of thek-level of {T(p) I p c S} and thus projects to theorder-k Voronoi diagram of S.

to a window, W, a segment on a lineavoiding the convex hull of S. A sitep G S and a point x are called visible ifthe line segment joining x and p inter-sects W. The peeper’s Voronoi diagramof S and W assigns each point in theplane to the region of the closest sitevisible from it. See Figure 36 where thewindow is shown between two bold linesegments. The diagram is composed ofperpendicular bisectors of sites (dashed)and of rays through sites and windowendpoints (solid).

Each site p may be associated with anunbounded convex polygon plate(p) inR3, being the projection onto T(p) of theset of all points visible from p. Again itis easy to see that a point x falls into theregion of a site p just if the vertical linethrough x intersects plate(p) in a pointthat lies above plate(q) for all q e S –{ p}. We conclude that the upper enve-lope of plate( p), p e S, projects to thepeeper’s Voronoi diagram of S.

Upper envelopes of piecewise linearfunctions on R2 are well-studied geomet -ric objects. Path and Sharir [1989] showedthat they can attain a size of @(n2a(n)),where n is the total number of trianglesneeded in partitioning the linear piecesof these functions and where a denotesthe inverse of Ackermann’s function.Construction algorithms exist that runin time proportional to their worst-casesize; see Edelsbrunner et al. [19891 whoalso give a lower bound of fl( n2 ) on thesize of a cluster Voronoi diagram, Re-cent results on upper envelopes byHuttenlocher et al. [1991] imply that thediagram for t clusters with a total of nsites has a size of O(tna( tn)). The upperenvelope arising from the peeper’sVoronoi diagram has a worst case size of0( n2 ) [Aurenhammer and Stockl 1988].Note that the superlinear size of adiagram implies the existence ofdisconnected regions.

3.2 Topology of Planar Diagrams:

Divide-and-Conquer Construction and Its

Variants

By far not all types of Voronoi diagramsconsidered in the literature can be



--

.-

\\

\\\\

9----

Pa

—.-

0

●

-’\a\--—- ——.———-

P

b--., . .

--._~

Figure 36. Peeper’s Voronoi diagram.

brought into connection with geometricobjects having nice algorithmic proper-ties, like convex hulls or hyperplane ar-rangements. So there is need for findingfeatures common to such “ungeometric”diagrams. We do this by extracting topo-logical properties, being guided by theirrelevance to the divide-and-conquer con-struction of Voronoi diagrams. Attentionis restricted to the plane as the instanceof most applications and easiest analysis.

The concept of diagram we are inter-ested in here may be characterized asfollows. The n sites are points in R2.The distance function varies by differentshapes of its “circles” that may dependon local properties of R2. It thus mayvary for the individual sites.

3.2.1 Convex Distance Functions

An important type fitting into the con-cept above is generated by the convex

distance function or the Minkowski dis-tance [Chew and Drysdale 1985]. Let Cdenote the boundary of some compact andconvex subset of R2 with the origin o inits interior. The distance with respect toC of o to some point x e R2 is given by

6(X, o)dc(x, O) =

fs(x’, o)

for x’ being the point of intersection of Cand the ray from o to x. Clearly dc canbe defined with respect to any site q of agiven set S by translating C so that qoccupies the former position of o, Ob -serve that dc( x, q) depends on the eu -clidean distance ti( x, q) as well as on thedirection of the ray from q to x. Convex

distance functions include, among others,the general LP-metric

~p(~,~)=vlql -x,lp+lqz-xglp,



with

()91 ()xl

‘= 92and x = X2 ‘

whose most important instances are forp = 1 (the Manhattan metric; C is asquare rotated by 450), p = 2 (the eu -clidean distance; C is a circle), and p = co(the Maximum metric; C is an axis-parallel square). Note that dc is a met-ric only if C is point-symmetricwith respect to q since dc( q, r) mightdiffer from dc(r, q) otherwise. However,the triangle inequality

dc(q, r) + d~(r, s) ~ dc(q, s)

can easily be shown to hold even if C isnot point-symmetric.

Let us now investigate the Voronoi di-agram defined by dc and a set S of nsites in R2. From the classical type weadopt the notions of separator sep( q, r) ofq and r, dominance dom( q, r) of q overr, and region reg( q) of q; compare theIntroduction. Clearly sep( q, r) need notbe a straight line as is the case foreuclidean distance. An even more un-pleasant phenomenon is that sep( q, r)fails to be one dimensional in general. Toremedy this shortcoming, the notionsabove may be redefined slightly by refer-ring to the lexicographical order (sym-bolized by < ) of the sites in S [Kleinand Wood 19881. The dominance of qover r is redefked as

Dom(q, r)

{

dom(q, r) –sep(q, r), ifq<r——dom(q, r), otherwise.

The new separator, Sep( q, r), is the in-tersection of the boundaries of Dom( q, r)and Dom( r, q), which is obviously one-dimensional. Figure 37 shows a two-dimensional separator in the L1-metric(left) and its redefined version (right).For a Voronoi diagram in the L1-metric,see Figure 3. The region of a site q nowcan be rewritten as

Reg(q) = ,e~~{,} Dom(q, r).

J*q*r

Figure 37. Redefining a separator.

That is, each point in R2 belongs to theregion of the lexicographically leastsite among its closest sites in S. Thisensures that the interiors of the regionscover the plane up to a set of Lebesguemeasure zero.

Defined in this way, the Voronoi dia-gram, VC(S), for a convex distance func-tion dc has the nice property that itsregions are star shaped; Reg( q) containsthe straight-line segment between q andx for each x e Reg( q), To prove this bycontradiction, assume the existence of apoint y on this segment with dc( y, q) >dc(q, r), for r ~ S – { q}. This impliesdc(x, q) = dc(x, y) + dc(y, q) > dc(x,

y) + dc( y, r). By the triangle inequality,the latter sum is greater than or equal todc(x, r) so that dc(x, q) > dc(x, r), thatis, x # Reg( q), follows.

The star shapedness of the regions im-plies their simple connectedness. Clearly,regions cannot vanish since q ~ Reg( q)holds for all q c S. So VC(5’) can be

viewed as a planar graph with exactly nregions. Since vertex degrees are at leastthree, there are 0(n) edges and vertices.We conclude a linear bound on the size ofVc(s).

3.2.2 Divide-and-Conquer Construction

Our next aim is to show that Voronoidiagrams for convex distance functionscan be constructed efficiently by adiuide-and-conquer algorithm. Divideand conquer splits the problem at handinto two smaller subproblems, computestheir solutions recursively (unless they



are very small and can be solved bytrivial methods), and finally combines thepartial solutions to the global one. TheVc type of diagram is well suited to at-tack by this strategy. The underlying setS of sites is partitioned into two subsetsS’l and S2 of nearly equal cardinal-ity. VC(SI) and VC(S2) are computedrecursively and are then merged toVc(s).

The merge step deserves the main at-tention since it actually constructs thediagram. We will discuss this process insome detail and under the simplifyingbut computationally unrestrictive as-sumption that S1 and S2 are separableby a vertical line. Since VC(SI) as well asVC(S.J defines a partition of R2, anypoint x ~ R2 falls into the region Regl(q)of VC(SI) for some site q e S1 and intothe region Reg2(r) of VC(S2) for somesite rs S2. Now observe that x e Reg(q)of VC(S) if dc(x, q) < CZc(x, r), and x cReg(r) of VC(S) otherwise. So we have tocut off some part from Regl( q) and fromReg2(r) by means of Sep(q, r) in order toobtain Reg( q) and Reg( r). Carrying outthis task for all relevant pairs of sitesconstitutes the merge process. The unionof the newly integrated pieces of separa-tors will be called the merge chain,

M(Sl, S2).The efficiency of an implementation of

the merge process critically depends onthe topological properties of the mergechain. We first show that M(SI, S2 ) be-haves well if the euclidean distance 8 istaken for dc [Shames and Hoey 1975].As a matter of fact, each horizontal lineintersects M(SI, S2 ) in exactly one pointin this case, that is to say, M(SI, S2) is(vertically) monotone. M(SI, S2) is a

polygonal line composed of edges of theclassical diagram V(S) that are portionsof perpendicular bisectors of sites in S1and S2, respectively. Figure 17 gives anillustration. If there were a horizontalline intersecting the merge chain in twoor more points, some site q ● SI would beforced to have a larger xl-coordinate thana site r~S2. This contradicts the as-sumption that S’l and S2 are separableby a vertical line; compare Figure 38.

\\\

+

e

\\ ----

--“q

ee

rIII

Figure 38. Impossible merge chain

The monotonicity of M(SI, S2) impliesits connectedness and unfoundedness.Hence the merge chain may be con-structed edge by edge, starting with anunbounded one. As is mentioned in theIntroduction, unbounded edges separatesites on the convex hull of S. So it suf-fices to determine a site in S1 and a sitein S2 that are neighbors on the boundaryof the convex hull of S. This can beaccomplished in 0(n) time by finding aline tangent to the convex hulls of re-spective S1 and S2. Each further edge ofM(SI, S2) is found by tracing the bound-aries of the current regions of V(SI) andV(S2) in an appropriate direction untilthey intersect the actual separator. Easycounting arguments show that only 0(n)edges are processed in total: Each edgetraced is either discarded or shortened toa new edge of V(S). This implies anoverall runtime of 0(n) for the mergeprocess. By the recurrence relationl“(n) = 2T(n /2) + O(n) that resultsfrom using divide and conquer, thetime complexity T(n) for computing aclassical Voronoi diagram is 0( n log n).

Let us now come back to the distancedc for general convex shapes C [Chewand Drysdale 1985]. As in the euclideancase, M(SI, S2 ) is composed of those partsof the boundaries of regions of VC(S)that separate sites in S1 from sites in S2.Thus M(SI, S2) is one dimensional. Po-tentially, this merge chain may consist ofcyclic and acyclic (but then necessarilyunbounded) topological curves in R2.



Examples show that M(SI, SQ) is in factdisconnected in general. Its connectedcomponents all, however, are acyclic andthus unbounded. Let us prove that thereare no cycles by assuming the existenceof some cycle M and deducing a contra-diction. Without loss of generality, Mencircles a subset T of sites in SI.Clearly, then M does not encircle anysite in Sz. Furthermore, there exists aleftmost point x e M with x e Sep(r, s)for some r, s e Sz. Since the regions ofVC(S) are star shaped, the three openline segments from x to r, s, and someq e T, respectively, do not intersect M.Thus, as r and s lie to the right of q, xhas to lie to the left of q. This, however,contradicts the convexity of the underly-ing shape. C encircles x and passesthrough q, r, and s since the last threepoints are equidistant from x with re-spect to dc. Figure 39 illustrates thissituation; M and C are shown bold anddashed, respectively.

In order to construct the acyclic mergechain M(S1, S2),one needs to detect someunbounded edge of each of its connectedcomponents. The previously describedprocedure of tracing boundaries of re-gions may then be applied. Unfortu-nately, the convex-hull method thatworks successfully in the euclidean casefails to be correct for dc. To get an alter-native method, we observe that each un-bounded edge has to be contained in theintersection of two unbounded regions ofVC(SI) and VC(S2), respectively. By acircular scan through these regions, eachrelevant pair Regl( q) and Regz(r) can befound and intersected with Sep( q, r) in atotal of 0(n) time. If Regl(q) fl Regz(r)n Sep( q, r) extends to infinity, then thisedge is taken as the starting edge for aconnected component of M(SI, S2).

The linear-time behavior of the mergeprocess thus can be maintained for gen-eral convex distance functions dc. Con-sequently, VC(S) requires O(n log n)time and 0(n) space for construction. Ofcourse, the complexity analysis is basedon the assumption that dc is computa-tionally simple. That is, dc( q, x) can becalculated in O(1) time for all q e S and

//

/

/ 0,

/

I\x

\

\\

\\

,

Figure 39. Cycles do not occur.

x~R2, and, in addition, its separatorscan be computed and intersected in 0(1)time. This may not be the case forcomplex shapes C, however.

3.2.3 Nice Metrics

We have seen that divide and conquer isa powerful approach to the constructionof generalized Voronoi diagrams. It alsoapplies well to sites more general thanpoints; see Section 1.3.3. We do not pur-sue these modifications here, however,since they do not fit into the presentconcept of a diagram. Instead, we at-tempt to characterize metrics in Rz thatpreserve the two main properties of aVoronoi diagram exploited in the pre-ceding discussion [Klein 1989; Klein andWood 1988]: the connectedness of regions



that implies an 0(n) size and the ab-sence of cycles in the merge curve thatallows an 0( n log n) time construction.

Let d be any metric in R2 that in-duces the euclidean topology. A curve ~in R2 is termed d-straight if, for anythree consecutive points x, y, z on -y, theequality

d(x, y) + d(~, Z) = d(X, Z)

holds. Clearly &straight curves arestraight-line segments, for ~ being theeuclidean metric. By arguments similarto that used for convex distance functions(namely using the triangle inequality), itcan be verified that any Voronoi regionReg( p) for d is d-star-shaped: y c Reg( p)holds for the d-straight curve -y from pto any x e Reg( p). The d-star-shapednessof Reg( p) does not necessarily imply itsconnectedness since d-straight curves donot always exist. If, however, for any twopoints x, z e R2 there is some y distinctfrom x, z with d(x, y) + d(y, z) =d( x, z), then the existence of -y is guar-anteed. Hence the Voronoi regions underthose metrics are connected.

Posing an additional restriction on themetric d makes the Voronoi diagrameven more well behaved: Assume that,for any point m e R2 and for any positivenumber r, the generalized disk

D={xc R21d(x, m) <r}

with center m and radius r under themetric d is a simply connected set. Forfixed r, the shape of D may vary withthe position of m provided D remainssimply connected. Symmetric convex dis-tance functions dc (i.e., with point-symmetric shape C) constitute a specialcase of translation-invariant disks. And,indeed, the proof that merge chains areacyclic can be extended from convex dis-tance functions to d.

To underline the usefulness of this con-cept, let us give some examples for d. LetK be any point in R2. The Moscow met-ric with respect to K induces d-straightcurves composed of pieces straight to-ward K or radially around K of minimaltotal euclidean length. Figure 40 depicts

a d-straight curve (dashed) between twopoints x and y and a Voronoi diagram(solid) for eight sites in this metric. Forthe geodesic distance among polygonalobstacles, d-straight curves are the mini-mum euclidean length connections avoid-ing any edge of the polygons (compareFigure 26). Also of interest are compositemetrics where, for instance, distances aremeasured in the L1-metric in some por-tion of R2 and in the L2- (i.e., euclidean)metric in the complement. Here, d-straight curves are composed of those inLI and L2 such that curve length is min-imized. In all three examples, the respec-tive disks are simply connected. As aconsequence, the resulting Voronoi dia-grams-except for the geodesic distancefunction that fails to be computationallysimple— can be computed in optimal0( n log n) time and 0(n) space by divideand conquer. Thereby, the underlying setof point sites has to be divided into sub-sets by curves whose intersections withany disk for d is either connected orempty. Consult Klein [1989] for furtherdetails. We pose it as an open question tofind equivalent characterizations formetrics whose Voronoi diagrams areconstructible by divide and conquer inoptimal time.

3.3 Deformation of the Voronoi Diagram:

Plane-Sweep Technique

The variety of techniques that can beused for constructing Voronoi diagramsis possibly as fascinating as the diagramsthemselves. The construction of theclassical planar Voronoi diagram bymeans of divide and conquer or viathree-dimensional convex hulls arewell-investigated algorithmic problems.Although the achieved complexity boundsare worst-case optimal, researchers con-tinued considering alternative methodsof construction. Aside from the incre-mental insertion strategy described inSections 1.3.1 and 1.3.2, the plane-sweeptechnique to be discussed now stands outby its conceptual and computational sim-plicity. It extends nicely to variousgeneralized types of Voronoi diagrams.



e

‘~e @‘)/

/

Figure 40. Voronoi diagram in the Moscow metric [Klein 1989].

3.3.1 Deformation

Let S denote a set of n point sites in R2and let V(S) be their Voronoi diagram.As we will see, there are serious reasonsfor refraining from a direct constructionof V(S) via the plane-sweep technique.Therefore, first we introduce a continu-ous deformation of V(S) [Fortune 19871.Then, we describe the plane-sweep tech-nique and its use for computing thisdeformation.

To explain the mechanism of deforma-tion, the euclidean distance function withrespect to the sites is interpreted in thefollowing way. With each site p G S, acone

cone(p)

={(X, Z) GR31X~R2, Z=8(X, p)}

is associated. Cone(p) is an upwardly di-rected cone in R 3 with vertical axis ofrevolution, with apex p, and with aninterior angle of m/2.14 Cones may beviewed as bivariate functions on R2. Wedefine the lower envelope of {cone(p) I

14Note that the definition of cone(p) agrees withthat of coneF( p) used for transformable Voronoidiagrams in Section 3.1.5; F is the identityfunction.

p c S} as the pointwise minimum of thesefunctions or, equivalently, as the surfacecomposed of that portion of each conethat lies below all other cones. From thedefinition of cone(p) it is evident thatthis lower envelope projects verticallyto V(S) onto R2.

For our purposes it is preferable toproject the envelope onto R2 in a differ-ent way, namely under an angle of T/4in positive xz-direction. This alreadygives the desired deformation V*(S) ofV(S). To be more precise, the deforma-tion * maps each point x e reg( p), say

()xl (xl

x=X2 ‘

into x* =)x2+a(x, p) “

Let us study what happens in this pro-cess. Clearly, sites are invariant under *.Each separator sep( p, q) is deformed intoa hyperbola with bottommost point p ifp is below q and bottommost point q,otherwise. (This hyperbola degeneratesto a vertical half line if p and q have thesame xz-coordinate.) Consequently, thedeformed region reg*( p) of p is the in-tersection of hyperbolically bounded halfplanes of R2. Reference to the upper en-velope model shows that * preserves thetopological properties of a Voronoi dia-gram. For instance, the interiors of the


\ I

+’\\\,f.,.*,. ) ,.’

. . . . ,/”, / ---. . -.*. ‘. *..

‘.\. . . ~__-—-

Figure 41. Deforming a Voronoi diagram [Fortune1987].

deformed regions cover R2 up to a set ofLebesgue measure zero. The algorithmicadvantage of the deformation over theoriginal is that the bottommost point ofeach region is its defining site. Excep-tions are regions for the bottommost sitesin S that necessarily are unbounded.Figure 41 shows the Voronoi diagram ofsix sites (left) and its deformation (right).

3.3.2 Plane-Sweep Technique

The properties of V*(S) are now ex-ploited for its construction using theplane-sweep technique. Generally, thistechnique proceeds as follows. A horizon-tal line L is swept across the object to beconstructed from below by keeping theinvariant that the portion of the objectbelow L is complete at any point in time.During the plane sweep, the cross sectionof L with this object has to be updated atcertain critical points. We thus have tohandle a one-dimensional dynamic prob-lem instead of a two-dimensional staticproblem.

The applicability of the plane-sweeptechnique crucially depends on whetherthe “critical points” can be predicted.This is, for example, not the case if theVoronoi diagram of S is to be constructedin its original: A site p e S might lieabove L while its region reg( p) alreadyextends to below L. And it is importantto know when regions begin to intersectL in order to integrate them properlyinto the cross section. We do not run intosuch problems when constructing V*(S),however, since the bottommost point of adeformed region is just its defining site.


Initially, when L is below S, the por-tion of V*(S) below L contains no de-formed edge. Hence the cross section isempty. As L moves upward the crosssection has to be updated, either by start-ing a new region when L hits a site or bystarting a new edge when L hits a vertexthat comes from intersecting two de-formed separators. The properties ofV*(S) described above will ensure thecorrectness of this method. Efficiency isgained by organizing points of intersec-tion of L with edges of V*(S) (the crosssection) in a dictionary ordered by xl-coordinates and by organizing sites andalready detected vertices (the criticalpoints) in a priority queue ordered byxz-coordinates. There are at most n +(2 n - 4) critical points since V(S), andthus V*(S), has at most 2 n – 4 vertices;compare the Introduction. Further, thereare at most n – 1 points in the crosssection since L cannot intersect a regionof V*(S) twice. An update causes only aconstant number of operations on bothdata structures each of which costO(log n) time. In conclusion, V*(S) iscomputed in 0( n log n) time and 0(n)space. It is clear that the originaldiagram V(S) can be derived from itsdeformation in 0(n) time.

Fortunately, this simple constructionmethod extends to several generalizedVoronoi diagrams [Fortune 1987]. It nat -urally applies to the additively weightedtype since the correspondence to lowerenvelopes extends provided cone(p) istranslated downward by the weight W(p)for each site p. The intersection ofcone(p) with R2 is then a circle withcenter p and radius w ( p), and indeedthis type may be interpreted as theVoronoi diagram for circles under theeuclidean distance function (Johnson-Mehl model, Figures 6 and 35). The timebound of 0( n log n) and the space boundof 0(n) clearly remains unaffected.

An important type suited for a plane-sweep attack is the Voronoi diagram fornoncrossing straight-line segments in R2.The distance of a point x from a segments is measured by min{~(x, y) I y e S}. Theregion of a segment can be deformed so



as to have the lower endpoint of thatsegment as its bottommost point. Al-though details are more complicated, areasonably simple 0( n log n) time and0(n) space algorithm is obtained. In par-ticular, the medial axis of a simple poly-gon is constructible via plane-sweep: Theedges of the polygon are taken as linesegment sites, and the parts of the dia-gram outside of the polygon are ignored.Note that the separator of two segmentsconsists of portions of straight lines andparabolas; compare Figure 25.

3.3.3 Constrained Voronoi Diagrams

A further type of Voronoi diagram, andone to which the plane-sweep techniqueapplies particularly nicely, is the classi-cal diagram constrained by a set of linesegments [Seidel 1988]. Let S denote aset of n point sites in R2 and let T be aset of m noncrossing line segments withtheir endpoints in S. Note that m = O(n)since the segments form a planar graph.We view the segments in T as obstaclesand define the bounded distance betweentwo points x, y ~ R2 as

b(x, y)

{

a(x, y), if~fl T.Ql,——co, otherwise.

The b-regions

reg~(~)

={xe R21b(x, p)= b(x, q), qeS}

for all sites p ● S define the boundedVoronoi diagram, V~(S, T), of S and T.b-regions of endpoint sites are generallynonconvex (near the corresponding ob-stacle), whereas b-regions of nonendpointsites have to be convex. Both types ofb-regions, and necessarily the formertype, might have portions of obstaclesrather than of perpendicular bisectors asedges. Clearly, V~(S, T) is a polygonalcell complex in R2.

The bounded Voronoi diagram is nowmodified to the constrained Voronoidiagram, VC(S, T), of S and T by modi-fying, for each obstacle t ~ T, the b-

regions it supports. Let t support‘egb(pl), . . . . reg~(p~) frOm the left andreg~(ql), ..., reg ~(ql) from the right. Theformer b-regions are extended to the leftof t as if only sites pl, . . ., ph were pres-ent, and the latter b-regions are ex -tended to the right of t as if only sitesql,..> ql were present, Note that p, = qjhas to occur twice, namely, for the end-points r and s of t. So reg~(r) and reg~(s)are extended to both sides of t.The c-re-

gion, reg.(p), of a site p e S is defined asthe union of reg ~(p) and its extensionswith respect to all obstacles. c-regionsclearly may overlap, so they fail to definea cell complex and the graph defined bytheir edges is nonplanar. See Figure 42illustrating the bounded Voronoi dia-gram (solid) and the constrained Voronoidiagram (solid or dashed or dotted) for 11sites and 2 obstacles (bold).

The significance of VC(S, T) is dueto its dual structure, the constrainedDelaunay triangulation of S and T.Intuitively, this triangulation is as simi-lar as possible to the classical Delaunaytriangulation but integrates the obsta-cles in T, a structure with several practi -cal applications. Formally, it consists ofthe obstacle segments in T and, in addi-tion, of all edges between sites p, q e Sthat have b( p, q) < m and that lie on acircle enclosing only sites r ~ S with atleast one of b(r, p), b(r, q) = m. The in-terested reader may verify that two sitesare connected in this triangulation if andonly if their perpendicular bisector con-tributes an edge to the constrainedVoronoi diagram. Note that the bisectorof the endpoint sites of an obstacle maycontribute two edges (on either side ofthe obstacle) rather than only one. SinceS contains n sites, T contains m obsta-cles, and no triangulation of n points canhave more than 3 n – 6 edges, weconclude that VC(S, T) realizes at most3 n + m – 6 edges and thus has a size of0(n). Figure 27 shows the constrainedDelaunay triangulation that correspondsto the constrained Voronoi diagram inFigure 42.

Let us now come to the plane-sweepconstruction of VC(S, T). To aid the


I

..”

.

/

Figure 42. Bounded

gram [Seidel 1988].and constrained Voronoi dia-

intuition, we think of VC(S, T) as beingembedded into m + 1 parallel planes inthe following way. One plane, call it Ho,contains the bounded Voronoi diagramV~(S, T). Each of the m planes, 11~, forthe obstacle t e T contains the diagram,Vt, induced by the extensions with re-spect to t for all b-regions supported byt.Clearly, this embedding comprises thesame information as is inherent inVC(S, T). The idea now is to sweep HOand all Ht simultaneously by (essen-tially one and the same) horizontal lineL in order to construct Vb(S, T) and allVt, for tE T, separately. Again these m+ 1 diagrams are not constructed di-rectly but via their deformationsV:(S, T) and V~* that come from map-ping upward, within the respective plane,each point ~ inside an (extension of a)b-region reg~( p) by 8(x, p). Since an ex-tended b-region is just a classical Voronoiregion, its deformation has the definingsite as the bottommost point (unless thesite itself is bottommost for the respec-tive plane). In analogy to the classicalVoronoi diagram, regions are created assites are hit by L, and edges are createdas L hits intersections of deformed bisec-tors. One must, however, take care thatedges are assigned correctly to their em-bedding planes. When a deformed edgein Ho meets a deformed obstacle t*,thenits construction is continued within Htand vice versa. The number of critical


points arising during the construction ofV$(S, T) and all V,* is proportional tothe size of VC(S, T) and thus is 0(n).

Using appropriate data structures sup-porting the m + 1 plane-sweeps allowsus to implement the operations at eachcritical point in O(log n) time so an0( n log n) time and 0(n) space algo-rithm for computing VC(S, T) is obtained.In particular, the constrained Delaunaytriangulation of S and T is computablewithin these bounds since it can be ob -tained in O(n) time from VC(S, T) byexploiting duality.

ACKNOWLEDGMENTS

This survey was completed while the author was at

the Institut fur Informatik, Freie Universitat

Berlin, Germany. Work was partially supported by

the ESPRIT II Basic Research Program of the EC

under contract no. 3075 (project ALCOM).

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Received February 1989, final rewslon accepted January 1991.

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