Abstract Voronoi Diagrams Revisited

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1 Abstact Voonoi Diagams Revisited Rolf Klein Elma Langetee Zaha Nilfooushan Abstact Abstact Voonoi diagams [21] wee designed as a unifying concet that should include as many concete tyes of diagams as ossible. To ensue that abstact Voonoi diagams, built fom given sets of bisecting cuves, ae finite gahs, it was euied that any two bisecting cuves intesect only finitely often; this axiom was a conestone of the theoy. In [12], Cobalan et al. gave an examle of a smooth convex distance function whose bisectos have infinitely many intesections, so that it was not coveed by the existing AVD theoy. In this ae we give a new axiomatic foundation of abstact Voonoi diagams that woks without the finite intesection oety. Keywods: Abstact Voonoi diagams, comutational geomety, distance oblems, Voonoi diagams. 1 Intoduction Voonoi diagams belong to the most inteesting and useful stuctues in geomety. Dating back to Descates [13], and known to mathematicians eve since (see, e. g., Gube [16]), Voonoi Diagams wee the toic of a seminal ae by Shamos and Hoey [32] that heled ceating a new field, comutational geomety. The geneal idea is uite natual. Thee is a sace on whom some objects, called sites, exet a cetain influence. Each oint of the sace belongs to the egion of that site whose influence is stongest. Most often influence is eciocal to distance. Meanwhile, CiteSee lists moe than 4800 elated aticles on Voonoi diagams. Suveys focussing on thei stuctual and algoithmic asects wee esented by Auenhamme [6], Auenhamme and Klein [7], Fotune [15], and, fo genealized Voonoi diagams, by Boissonnat et al. [9]. Beyond thei value to comute science, Voonoi diagams have imotant alications in many othe sciences; ominent examles can be found in Held [17] and in Okabe et al. [30]. Fo many yeas, comutational geometes have studied Voonoi diagams in the lane that diffeed by the tyes of sites and distance measues used. Tyically, algoithms wee handtailoed to fit a aticula setting. This situation called fo a unifying view. An elegant stuctual aoach was by Edelsbunne and Seidel [14] who suggested to define geneal Voonoi diagams as lowe enveloes of suitable cones. Indeendently, Abstact Voonoi Diagams (AVDs) wee intoduced by the fist autho in [20], as a unifying concet fo both, stuctue theoy and algoithmic comutation. The basic obsevation behind AVDs was that Voonoi diagams ae built fom systems of bisecting cuves that have cetain combinatoial oeties in common, wheeas the natue of the sites and of the distance function ae of seconday imotance. A challenge was in finding a small set of simle axioms fo bisecting cuve systems. They should ensue that a Voonoi diagam fomed fom such a cuve system has desiable stuctual oeties (like being a finite lane gah of linea comlexity), and that it can be efficiently comuted. At the same time, this aoach should be as geneal as ossible. Univesity of Bonn, Institute of Comute Science I, D Bonn, Gemany. Amikabi Univesity of Technology, Faculty of Mathematics and Comute Science, Tehan, Ian. 1

2 In ode to achieve the goals just mentioned, AVDs wee defined in the following way. Fo any two elements, of a set S of indices, also efeed to as sites, a cuve J(, ) was given, that slits the lane into two unbounded oen domains. One of these domains was labeled by, the othe by ; these labels wee at of the definition of J(, ). The cuve itself was added to one of the two domains accoding to some global ode on S. The Voonoi egion of was defined as the intesection of all sets associated with ; detailed definitions will be be given in Section 5. Now thee oeties wee euied of the given cuves and of ode. Voonoi egions should be ath-connected, and thei union should cove the whole lane. Moeove, any two cuves J(, ), J(, s) should intesect only finitely often. These euiements ae met, fo examle, by the Euclidean Voonoi diagams of oints o line segments, additive weights, owe diagams, and all convex distance functions whose cicles ae semi-algebaic. It tuned out that these axioms wee stong enough to ensue that abstact Voonoi diagams have many of the oeties found in diagams based on concete sites and distance functions, and that they can be constucted efficiently. The finite intesection assumtion was instumental in analyzing the stuctue of abstact Voonoi diagams. It was alied twice. Fist, in oving the toological fact that in a neighbohood of any oint v, the bisecting cuves assing though v fom a sta; see the iece of ie Lemma [21]. This fact allowed a local view on which combinatoial definitions could be based. Second, the finite intesection oety was exlicitly used to guaantee that abstact Voonoi diagams ae finite lana gahs; see Lemma [21]. Thee asymtotically otimal AVD algoithms have been develoed, each fo a cetain subclass of AVDs. A deteministic O(n log n) divide & conue algoithm [21], based on wok by Shamos and Hoey [32] and by Chew and Dysdale [10], fo situations whee ecusive atitions with cycle-fee bisectos ae guaanteed; a deteministic linea time algoithm [23] fo situations esembling geneal convex osition, based on the techniue by Aggawal et al. [2], and an O(n log n) andomized incemental constuction algoithm [24] fo AVDs whose egions have ath-connected inteios, based on wok by Clakson and Sho [11]. McAlliste et al. [5], Ahn et al. [3], Kaavelas and Yvinec [19], Abellanas et al. [1], Aichholze et al. [4], and Bae and Chwa [8] esented new tyes of Voonoi diagams that wee unde the umbella of the AVD concet. The notion of abstact Voonoi diagams has been genealized to futhest site diagams by Mehlhon et al. [28], to dimension 3 by Lê [25], and to a dynamic setting by Malinauskas [26]. A slightly simlified vesion of abstact Voonoi diagams has been imlemented in LEDA by Seel [33]. But Cobalan et al. [12] gave an examle of a convex distance function whose bisectos have an infinite numbe of intesections; its unit cicle is smooth, but not semi-algebaic. The existing AVD concet, with its definitions and oofs elying on the finite intesection oety, did not cove this examle. The uose of this ae is in oving that abstact Voonoi diagams can be defined and constucted without the finite intesection assumtion. In fact, the othe two axioms that Voonoi egions be ath-connected and cove the lane ae just stong enough to imly what is needed. Although the oof of this fact euies new techniues uite diffeent fom those used in [21, 24], we think this effot is well-invested. Fist, the class of concete Voonoi diagams coveed by the AVD concet gows; in aticula, all convex distance functions ae included now. Second, with one axiom less to check, alying AVDs becomes easie. Thid, thee is scientific value (and aesthetic leasue) in minimizing axiomatic systems. Some cae is necessay in dealing with geneal cuves that can intesect each othe infinitely often. To kee the analysis simle, we euie, in this ae, that not only the Voonoi egions, 2

3 but also thei inteios, ae ath-connected. This was also ostulated fo the andomized incemental algoithm in [24]. It heled to avoid the comlications in [21] that wee caused by the fact that Voonoi edges and vetices could fom connections between seveal ats of one Voonoi egion. Ou euiement also allows us to abandon the ode, which was used to distibute the bisecting cuves among the sites. The est of this ae is oganized as follows. In Section 3 we state the new set of axioms, and deive some eliminay facts. The main at is Section 4, whee we show that AVDs based on the new axioms ae finite lane gahs, without esoting to the finite intesection assumtion. This is accomlished in the following way. Fist, we ove that a bisecting cuve J(, ) cannot moe than twice altenate between the domains seaated by some cuve J(, ), without disconnecting a Voonoi egion; see Lemma 6. 1 This allows us to analyze how J(, ) and J(, ) can behave in the neighbohood of an intesection oint, without having a iece of ie lemma available. Fo sets S of size 3 we show, in Lemma 8, that each oint w on the bounday of a Voonoi egion is accessible fom this egion. That is, thee exists an ac α with endoint w such that α without w is fully contained in the Voonoi egion. 2 Using an elegant agument by Thomassen [34], accessibility imlies that an abstact Voonoi diagam of thee sites contains at most two oints that belong to the closue of all Voonoi egions. Fom this one can diectly conclude that AVDs of many sites ae finite lane gahs; see Theoem 10. Now a iece of ie lemma can be shown at least fo the Voonoi edges meeting at a Voonoi vetex, which is sufficient fo ou uoses. In Subsection 4.3 we show that a cuve system fo index set S fulfills ou axioms iff this holds fo each subset S of size thee. This fact was obseved in [22] fo the old AVD model; the oof given in Subsection 4.3 is new and moe geneal. In Section 5 we addess the constuction of abstact Voonoi diagam based on the new axioms. With the finite intesection assumtion and ode emoved, the class of cuve systems to which andomized incemental constuction can be alied, is now stictly lage than in [24]. Divide & conue can be alied if acyclic atitions ae ossible, as in [21]; but cuve systems causing the inteio of Voonoi egions to be disconnected ae no longe admissible. By this estiction, the divide & conue algoithm becomes consideably simle. 2 Acknowledgement The authos would like to thank the anonymous efeees fo thei caeful eading of the fist vesion of this ae, and fo thei valuable comments and suggestions. 3 The new AVD axioms We ae given a finite set S of indices (eesenting sites), and, fo any two indices of S, a cuve J(, ) =J(, ) that slits the lane into two unbounded domains, labeled D(, ) and D(, ). 3 These labels ae assigned to the two domains as at of the definition of J(, ); see Figue 1. We define, fo each S, the set VR(, S) := D(, ), (1) S\{} 1 One should obseve that both cuves ae associated with the same site,. In the old AVD model [21], this obsevation was an easy conseuence of the fact that AVDs ae finite lane gahs. 2 By the Jodan cuve theoem and its invese, Jodan cuves ae chaacteized by accessibility; see Theoem 4. 3 Infomally, D(, ) and D(, ) will sometimes be called the half-lanes defined by J(, ). 3

4 D(, ) J(, ) D(, ) Figue 1: A bisecting cuve. and let V (S) := R 2 \ S VR(, S). (2) Now we state which axioms the given cuves must fulfill. Definition 1 The cuve system J := {J(, ); S} is called admissible if the following axioms ae fulfilled. A 1 ) Each cuve J(, ), whee, S, is maed to a closed Jodan cuve though the noth ole by steeogahic ojection to the shee. 4 A 2 ) Fo each subset S S and fo each S, the set VR(,S ) is ath-connected 5 A 3 ) Fo each subset S S, we have R 2 = S VR(,S ). Hee, A denotes the toological closue of a set A in the Euclidean toology. Definition 2 Fo an admissible cuve system J we call the set VR(, S) the Voonoi egion of with esect to S, wheeas V (S) is called the Voonoi diagam of S. Examle. Figue 2 (i) shows an admissible cuve system fo S = {,, }, and the esulting Voonoi diagam (ii). An index laced closely to a bisecting cuve J(, ) indicates on which side of J(, ) domain D(, ) is located. In this examle we obseve some henomena that cannot occu fo Euclidean bisectos of oints. Thee ae oints like a in the intesection of two bisecting cuves J(, ) and J(, ) that do not lie on the thid cuve, J(, ). A oint like w that is included in all thee bisecting cuves need not be a Voonoi vetex. The intesection of two bisecting cuves, like J(, ) and J(, ), consists of an infinite numbe of connected comonents, cuve segments o single oints. These comonents may have accumulation oints. In fact, in Figue 2 thee is an infinite seuence of intesection oints a i J(, ) J(, ) that convege towads a J(, ) J(, ), such that each segment of J(, ) between a i and a i+1 is disjoint fom J(, ). Theefoe, one must be caeful not to seak of the fist oint of J(, ) on J(, ) to the left of oint a, etc.. 4 Moe ecisely, the ojected image is continuously comleted, by the noth ole, to a closed Jodan cuve. 5 We need not distinguish between ath-connectedness and ac-connectedness because the Euclidean lane is Hausdoff. Thus, two oints of a ath-connected set can be connected not only by a ath, which is a continuous image of [0, 1], but even by an ac which is image of [0, 1] unde a homeomohism, that is, of a bijective, bi-continuous maing. 4

5 a i a i+1 a v w (i) VR(,S) v VR(,S) VR(,S) (ii) Figue 2: An admissible cuve system (i) and the esulting Voonoi diagam (ii). 3.1 Peliminaies In this section, and in the following one, we assume that J is an admissible cuve system fulfilling axioms A 1,A 2,A 3 of Definition 1, unless stated othewise. Fist, we obseve that the Voonoi diagam V (S) can also be chaacteized in the following way; comae Lemma in [21]. Lemma 3 Let J be a system of admissible cuves fo index set S. Then, V (S) = VR(, S) J(, ) S = S VR(, S) VR(, S) Poof. : If z V (S) then z VR(, S) \ VR(, S) fo some S, by axiom A 3 and the definition of V (S). Hence, thee exists a site such that z / D(, ). Assume z D(, ). As this set is oen, it would contain a whole neighbohood of z. But this contadicts z VR(, S) D(, ). Theefoe, z J(, ), thus oving the ue inclusion. The lowe one is shown by contadiction. If no set VR(, S), whee, contained z then, by finiteness of S, fo a neighbohood U(z), U(z) VR(, S) c = VR(, S) c VR(, S) would hold, contadicting z/ VR(, S). : Because of VR(, S) VR(, S) D(, ) D(, ) J(, ) 5

6 we need to conside only the ue inclusion. Let us assume that z VR(, S) J(, ). If z wee contained in some Voonoi egion VR(, S) then it would be an inteio oint of this egion. Because of z VR(, S) this would imly =. But if z lies in the inteio of the Voonoi egion VR(, S), it cannot be situated on J(, ), contadicting ou assumtion. Hence, z V (S). In ode to ove the finiteness of the Voonoi diagam as a gah, we shall emloy some oeties of lane cuves that ae sometimes stated as at of the Jodan cuve theoem. Theoem 4 Let C be a lane cuve, homeomohic to a cicle. Then R 2 \ C consists of two domains D 1,D 2 with common bounday C. Fo each oint z C thee exists a neighbohood U whose bounday is homeomohic to a cicle, such that U \ C consists of exactly two connected comonents. Point z is accessible fom each domain U D i, that is, fo each U D i thee exists an ac α fom to z such that α minus its endoint z belongs to U D i. Theoem 4 is a diect conseuence of the stonge Jodan-Schönflies theoem, which states that a homeomohism between a cicle and a closed cuve, C, in the lane can be extended to the whole lane, such that the inteio of the cicle is maed onto the inteio domain of C, and the cicle s exteio to the exteio of C; cf. Rinow [31], fo examle. Theoem 4 also holds fo closed cuves on the shee. Hence, it holds fo the bisecting cuves J(, ) we ae dealing with, because they ae maed to Jodan cuves though the noth ole unde steeogahic ojection. Fo simlicity, a homeomohic image of the cicle, o a homeomohic image of the line that bisects the lane, will be called a Jodan cuve in the seuel. As a tivial conseuence of Theoem 4, evey neighbohood of a oint z on a Jodan cuve C contains oints of both domains D i. It is inteesting to obseve that the convese of Theoem 4 is also tue. If C is a comact set whose comlement in the lane consists of two connected comonents, such that each oint of C is accessible fom both, then C is a closed Jodan cuve; see Thomassen [34] fo a simle oof. The following tansitivity lemma will be a handy tool. Its claim would be tivial if we could ead z D(, ) as z is close to than to. Fo the old AVD model [21], a simila statement with a slightly diffeent oof was made in Lemma Lemma 5 Let,, S. Then D(, ) D(, ) D(, ) holds. Poof. Let z D(, ) D(, ). Point z must be contained in one of D(, ),J(, ),D(, ). If z wee contained in D(, ), it could not lie in any of the closed Voonoi egions VR(, S ) D(, ) =D(, ) J(, ) VR(, S ) D(, ) =D(, ) J(, ) VR(, S ) D(, ) =D(, ) J(, ) fo S := {,, }. This is imossible since these thee sets cove R 2 by axiom A 3. Suose z J(, ). By Theoem 4, thee exists an ac α with endoint z such that α\{z} D(, ). With z, even a neighbohood U of z is contained in the oen set D(, ) D(, ). Inside U, ath α contains a oint z D(, ) D(, ) D(, ), which leads to the same contadiction as befoe. Conseuently, the thid case alies, that is, z D(, ). We note that Lemma 5 neithe holds fo the closues of the sets D(, ), no fo the bisecting cuves themselves. 6

7 4 The gah stuctue of V (S) The main goal of this section is in oving that V (S), whee S = n, is a finite, lane gah with n faces, even though ou bisecting cuves do not fulfill the finite intesection oety. To this end, we conside fist an abstact Voonoi diagam of thee sites, and show, in Lemma 8, that each oint w on a egion bounday is accessible fom this Voonoi egion. As we do not have a iece of ie lemma available, which would gant us a clea view to a neighbohood of w, this euies some local analysis on the bisecting cuves assing though w. This analysis will be based on the following Lemma 6, suoted by Lemma 7. Then the oof oceeds as follows. By Lemma 3, each oint of V (S) lies on the boundaies of at least two egions. We show that only finitely many oints can be situated on thee o moe egion boundaies; see Lemma 9. Fom this fact, the finiteness of V (S) will be deived in Theoem Thee sites Let us conside the Voonoi egion of a site in the diagam V (S) whee S = {,, } consists of only thee sites. Fo convenience we may assume that J(, ) is a hoizontal line, and that D(, ) euals the lowe half lane. The following lemma states that J(, ) can change at most twice between D(, ) and D(, ). Lemma 6 Thee cannot be fou oints consecutively visited by J(, ) that belong altenately to D(, ) and D(, ). Poof. Suose that ou claim is wong, and that J(, ) does visit fou oints a 1,a 2,a 3,a 4, in this ode, such that a 1,a 3 D(, ) and a 2,a 4 D(, ). The following facts will be helful in deiving a contadiction. By π we denote the elative inteio of an ac π, that is, the ac without its endoints. Facts. We can connect 1. oints a 2 and a 4 by an ac π such that π is contained in VR(, S), 2. oints a 2 and a 4 by an ac ρ such that ρ VR(, S), and 3. oints a 1 and a 3 by an ac σ such that σ VR(, S), see Figue 4. Poof. 1.) By Theoem 4, oint a 2 is accessible by an ac α 2 fom D(, ). Since a 2 is an inteio oint of D(, ), α 2 can be shotened to lie in D(, ). Then, α 2, without its endoint a 2, is contained in D(, ) D(, ) = VR(, S). Similaly, thee is an ac α 4 accessing oint a 4 fom VR(, S). W. l. o. g., α 2 and α 4 ae disjoint. Thei esective endoints a 2 and a 4 in VR(, S) can be connected by an ac α entiely unning in VR(, S), by axiom A 2. Should α 2 intesect α in a oint diffeent fom a 2, let a 2 denote the fist oint of α met when tavesing α 2 fom a 2 towads a 2.6 We cut both α 2 and α at oint a 2, and efom simila sugey on α 4, if necessay. The concatenation of the thee esulting acs yields an ac with the oeties desied; see Figue This oint is well-defined. Indeed, if f(t) is a aametization of α 2 satisfying f(0) = a 2, then t := su{t 0; f([0,t]) α c } exists, and f(t )=a 2 holds. 7 One should obseve that in Figues 3 and 4 domain D(, ) is deicted to be above cuve J(, ). Ou oof does not make use of such an assumtion. 7

8 a 2 a 4 J(, ) a 2 a 2 α 4 a 4 α 2 α Figue 3: Constucting an ac that uns though VR(, S) and connects a 2 to a 4. 2.) Point a 2 is also accessible fom D(, ) by an ac β 2. Clealy, β2 lies in D(, ) D(, ) D(, ), by Lemma 5, hence in VR(, S). The est of the oof is analogous to the oof of (1). 3.) Point a 1 is accessible fom D(, ) by an ac γ 1 contained in D(, ). Thus, γ1 is contained in D(, ) D(, ) D(, ), hence in VR(, S). The same holds fo a 3, and we continue as befoe. This concludes the oof of the thee facts. σ ρ a 1 a 3 D J(, ) a 2 π a 4 J(, ) Figue 4: Domain D contains oint a 3, but not a 1. Hence, ath σ must intesect eithe ρ o π, in ode to connect a 1 and a 3. Both altenatives ae imossible because all acs ae contained in diffeent Voonoi egions. To comlete the oof of Lemma 6, we ague as follows. Togethe, acs π VR(, S) and ρ VR(, S) fom a closed Jodan cuve; let D denote its inteio domain, as shown in Figue 4. We obseve that J(, ) cannot intesect the elative inteios of eithe ath π o ρ, because these ae contained in D(, ) and D(, ), esectively. Being a simle cuve, J(, ) can ass though oints a 2 and a 4 only once. On the othe hand, J(, ) must ass though D to seaate π fom ρ. Theefoe, the segment of J(, ) between a 2 and a 4 is fully contained in domain D, while the two unbounded comlementay segments of J(, ) stay outside. Conseuently, we have a 3 D and a 1 D c. But the ath σ connecting a 1 to a 3 is contained in VR(, S) and, theefoe, unable to intesect the bounday of D, which belongs to the closues of the egions of and. This contadiction comletes the oof of Lemma 6. The next ste is in oving that, fo Voonoi diagams of thee sites, each oint w on the bounday of a Voonoi egion is accessible fom this egion; see Lemma 8 below. To this end, we need to discuss the diffeent ways in which two bisecting cuves J(, ) and J(, ) can intesect at some oint w. Let g(t) be a aametization of J(, ) such that D(, ) lies on the ight hand side of J(, ) as t tends to +. Suose that g(0) = w holds. As t aoaches 0 fom below, the oints g(t) cannot altenate between D(, ) and D(, ) infinitely often, thanks to Lemma 6. Thus, thee exist δ, δ > 0 such that G := g(( δ, 0)) is included in D(, ) o in D(, ). The same holds fo G + := g((0,δ )). Analogously, thee ae two segments F,F + of J(, ) befoe and afte w, each of which is contained in one of the sets D(, ) o D(, ); see 8

9 Figue 5 fo an examle. Let us fist assume that none of the segments G,G + is fully contained G F w F + G + Figue 5: Hee G stays in D(, ), while G + uns in D(, ). Moeove, F D(, ) and F + D(, ). in J(, ), and none of F,F + in J(, ). 8 To facilitate ou analysis, we fist conside cuve J(, ) and distinguish between the following cases. A : F, F + D(, ) B : F D(, ) and F + D(, ) C : F D(, ) and F + D(, ) D : F, F + D(, ) Analogous cases A, B, C, D ae ossible fo J(, ); they esult fom elacing with, and F with G, in this definition. In incile, 16 combinations AA, AB, AC,..., DC, DD of these cases should be consideed. Howeve, we obseve that XY coesonds to YX unde the symmety Σ:, F G. Of the emaining 10 combinations, only 4 ae geometically ossible, as we shall conclude fom the following lemma. Intuitively, it states that the facts J touches J at w and J cosses J at w ae symmetic in J and J. Lemma 7 (i) If G, G + ae contained in the closue of the same half-lane defined by J(, ), the same holds fo F,F + with esect to J(, ). (ii) If F D(, ) and F + D(, ) then G D(, ) and G + D(, ). The same holds with + and evesed. Poof. (i) Let U be a neighbohood of w, chosen by Theoem 4, such that J(, ) U is one connected segment contained in the union of G,G +, and {w}. We can make U small enough to guaantee that the at of J(, ) assing though U is also contained in F,F +, and {w}, but ossibly disconnected; see Figue 6. By way of contadiction, assume that G, G + D(, ), but that thee ae oints z D(, ) F and z D(, ) F + close to w on J(, ). The segment H of J(, ) U that contains z, w, z, divides domain U into two domains, U 1 to the left of H, and U 2 to the ight of H. Both have Jodan cuves as boundaies. Since z, z ae accessible fom U 1, thee exists an ac α 1 U 1 connecting them. We can assume that α 1 stays in the oen half-lane D(, ). Namely, each of the (at most countably many) excusions of α 1 to D(, ) can be elaced with cicula acs in D(, ), as deicted in Figue 6. 9 Since ac α 1 connects oints z, z fom both sides of J(, ), it must meet J(, ) at some oint y U 1 D(, ). But each oint y w of J(, ) in U belongs to G o to G +, which ae contained in the closue of D(, ), by assumtion. Contadiction! 8 Unde this assumtion, G D(, ) imlies the following. Moving along G towads w, one neve entes D(, ). One always meets anothe oint of D(, ), and ehas oints of J(, ) in between. The latte may accumulate. 9 Only finitely many cicula acs ae needed to this end. This can be seen as follows. Let I denote the inteval of J(, ) connecting an exit and e-ente oint of α 1. Fo each oint of I a cicula neighbohood is contained in U 1. Since I is comact, finitely many of these cicles cove I. 9

10 U J(, ) y α 1 H z w z U 1 U 2 Figue 6: Illustating the oof of Lemma 7. (ii) Now assume F D(, ) and F + D(, ), and let z D(, ) F and z D(, ) F + as in the oof of (i). In addition to α 1, thee exists an ac α 2 U 2 D(, ) connecting z to z. Cuve J(, ) must intesect both acs, α 1 and α 2 ; hence, it must visit both D(, ) and D(, ). Since both G and G + ae fully contained in the closue of one of these half-lanes, thee ae but two ossibilities. Eithe G D(, ) and G + D(, ), which is what we claim, o G D(, ) and G + D(, ), which leads to a contadiction, because z D(, ) cannot be situated to the left of J(, ). Clealy, Lemma 7 emains tue unde symmety Σ intoduced befoe Lemma 7. A uick insection shows that this leaves us with only 4 combinations, namely AD, DD, AA, and BC. They ae illustated in Figue 7. It emains to account fo those cases whee the cuves J(, ), J(, ) shae one o two segments close to thei intesection oint w. We shall now demonstate how to view all situations ossible as secial cases of the configuations AD, DD, and BC dislayed in Figue 7. The euality and uneuality signs shown in this figue indicate which cuve segments may coincide and which ae suosed to be diffeent. The case analysis given in Lemma 8 will be in accodance with these oeties. The situations whee both G and G + ae contained in D(, ) ae included in AD o DD, esectively, deending on the oientation of G. Othewise, one of the segments G,G + lies in D(, ) but not in F, while the othe segment is at of F. We conside both cases in tun. If G D(, ) and G + F, two subcases ae ossible: G + = F, which educes to the subcase (F + = G and F D(, ) ) of BC unde symmety Σ, and G + = F +, which educes, unde Σ, to the subcase (F + = G + and F D(, ) ) of AD. If G + D(, ) and G F, we have to conside the subcase G = F, which educes, unde Σ, to the subcase (F = G and F + D(, ) ) of AD, and the situation whee G = F +, which is itself a subcase of BC. Now we ae eady to ove the main esult of this subsection. Lemma 8 Let w be a oint on the bounday of VR(, S), whee S =3. Then thee exists an ac α with endoint w such that α \{w} VR(, S) holds. Poof. Fist, we assume that w is contained in only one bisecting cuve, J(, ). Fo w to belong to the bounday of the egion of, it cannot be in D(, ). Thus, w D(, ). By Theoem 4, alied to J(, ) on the shee, thee is an ac α accessing w fom D(, ) D(, ) = VR(, S). Now we assume that w is contained in both, J(, ) and J(, ). By the evious discussion, we need only insect the fou cases sketched in Figue 7. AD) Hee an ac accessing w fom D(, ) is also contained in D(, ), afte shotening, hence in VR(, S). DD) In this case D(, ) D(, ) is emty, in a neighbohood of w, so that oint w cannot be on the bounday of VR(, S), in contadiction to ou assumtion. 10

11 G G + G + = = F w + F = = w F G F + AD DD F G + a AA w b G F + F G BC w = = F + G + Figue 7: Fou ways fo J(, ),J(, ) to ass though a oint w. AA) Hee the Voonoi egion of would be disconnected. Fomally, we could find oints a, b D(, ) on G + and G, esectively, and connect them with aths π VR(, S) and ρ VR(, S) as in the oof of Lemma 6. The aguments esented thee show that the domain D bounded by these aths must contain oint w, so that ρ has to ass above w while π stays below J(, ). This is in conflict with the suosed oientation of J(, ) at w. BC) This is the most inteesting case. Let us walk along G + backwads, towads w. Suose that we encounte two oints b, d of F that aea in the ode (d, b) on F, but in the ode (b, d) on G + ; see Figue 8 (2). Then the Voonoi egion of would be disconnected, by the same fomal oof as fo AA. Thus, all oints of F encounteed on the backwad walk along G + towads w, must be situated in the same ode on both oiented segments, F and G +, as shown in Figue 8 (1). This means, each new oint of F we meet must be to the left of its edecesso on F, and, theefoe, fathe away fom w. Conseuently, a subsegment of G + with endoint w must be wholly disjoint fom F ; let us denote it by G + again. Now let U be a neighbohood of w J(, ) accoding to Theoem 4, small enough to intesect only the segments F,F + and G,G + of J(, ) and J(, ). Let H denote the segment of U F adjacent to w; see Figue 8 (3). Since H G + F G + =, segment H, togethe with ats of G + and U, fom a simle Jodan cuve encicling a domain D, which is shaded gey in Figue 8 (3). Bounday oint w is accessible via some ac α, whee α D. Ou assumtion F D(, ) imlies that H cannot ente D(, ); hence, D belongs to U D(, ), imlying α D(, ). On the othe hand, at least a segment of α stating fom w must also belong to D(, ). This is because α can leave D(, ) only though oints on J(, ) to the left of H. Theefoe, at of α lies in VR(, S) and is an access ath fo w. 4.2 Many sites By Lemma 3, the Voonoi diagam V (S) consists of all oints in the lane that ae contained in the closues of at least two Voonoi egions. Now we show that only finitely many oints ae contained in the bounday of thee o moe egions. Lemma 9 Let B be the set of all oints on the boundaies of at least thee Voonoi egions in some Voonoi diagam V (S ), whee S S. If S = n 3 then B is of size at most 2 ( n 3). Poof. Fist, we conside the case whee S = { 1, 2, 3 }. If one of the thee Voonoi egions is emty, V (S) euals a single bisecting cuve, and B is emty. Othewise, we aly the following elegant agument of Thomassen s [34]. In each Voonoi egion VR( j,s) we choose a oint 11

12 G G F b G + c d a w (1) F + F G + d c b a w (2) F + F α U H D w G F + G + (3) Figue 8: Discussion of case BC deicted in Figue 7. denoted by a j. Now suose, by way of contadiction, that B contains thee diffeent oints, v 1,v 2,v 3. By Lemma 8 and axiom A 2, we can find acs α i,j connecting v i to a j, such that α i,j \{v i } is fully contained in VR( j,s). We may even assume that fo each j, the thee acs α 1,j,α 2,j,α 3,j contained in the egion of j fom a lane tee T j ooted at a j. 10 Since the Voonoi egions of 1, 2, 3 ae disjoint, the tees T 1,T 2,T 3 ae ealizing a lane embedding of the biatite gah K 3,3, which is imossible. Hence, B 2 holds. Now let S > 3, and let v B be a oint on the bounday of the Voonoi egions of,, S S. Since the Voonoi egions of these indices in V (S ), whee S := {,, }, can only be lage than those in V (S ), oint v belongs to the egion boundaies of,, in V (S ), too. ( We have just shown that thee ae at most two such oints in V (S ). Since thee ae only n ) 3 subsets S of S of size 3, the claim follows. We ecall that a finite lane gah is an embedding in the lane of a finite abstact gah, that mas each edge e onto an ac whose endoints ae the embedded vetices adjacent to e. Two acs do not intesect excet at a common endoint. If the two endoints of an edge coincide, the edge is a loo, maed onto a closed Jodan cuve. Owing to the stuctue of Voonoi diagams, we allow fo a secial vetex, the invese image of the noth ole unde steeogahic ojection to the shee. 11 Now we can ove the main esult of this section. Theoem 10 The abstact Voonoi diagam V (S), whee S = n, is a finite lane gah of O(n) edges and vetices. Poof. If n = 2 then V (S) consists of a single bisecting cuve with both endoints at, and we ae done. Let us assume n 3. By Lemma 3, the set V (S) consists of all oints contained in the closues of two o moe Voonoi egions. The oints on the bounday of thee o moe egions of V (S) fom a subset B of B, which is finite by Lemma 9. The est of V (S) is decomosed into sets B, consisting of all oints on the bounday of exactly the Voonoi egions VR(, S) 10 This can be achieved as follows. Fist, we choose α 1,j. Then we tace α 2,j fom v 2 to a j, and cut it at the fist oint of α 1,j it meets. Finally we tace α 3,j fom v 3 to a j, and cut it at the fist oint of the atial tee aleady constucted. 11 Altenatively, one could comactify the Voonoi diagam in the way suggested in [21], by cliing the unbounded ieces of V (S) at a sufficiently lage cicle. 12

13 and VR(, S). Such a set B,, if non-emty, consists of disjoint segments of the bisecting cuve J(, ). We claim that each endoint, v, of such a segment e belongs to the set B { }. This can be seen as follows. Suose v. Clealy, v belongs to the closues of the egions of both, and, but the extension of e beyond v on J(, ) does not. Thus, we can eithe find oints of some Voonoi egion VR(, S), whee / {, }, abitaily close to v on J(, ). O a segment e of J(, ) extending e beyond v belongs to V (S). By Lemma 3, segment e is contained in the closues of the Voonoi egions of two sites, at least one of which,, is not in {, }. In eithe case, v in VR(, S). Since J(, ) is simle, at most two segments e of B, can shae a oint of B. Thus, the numbe of these segments is finite. We conclude that V (S) { } = B, B { } S is a finite lane gah. The elements of B ae the finite Voonoi vetices, while the edges ae the segments of the sets B,. Because B, and B, ae disjoint if {, } {, }, edges can intesect only at thei endoints. The O(n) bound follows fom the Eule fomula, as usual. Once V (S) is known to be a finite lane gah, the following iece of ie lemma can be shown; fo a oof see, e. g., in Rinow [31], o comae Lemma in [21]. Lemma 11 Fo each oint v in the lane thee exists an abitaily small neighbohood U of v, whose bounday is a simle closed cuve, such that the following holds fo each subset S of S. Let v V (S ). If v is inteio oint of some Voonoi edge e B, of V (S ) then U is divided by e in exactly two domains, one contained in VR(, S ), the othe in VR(, S ). Othewise v is a Voonoi vetex of V (S ), of degee k 3. Afte suitably enumbeing S, the Voonoi edges e i adjacent to v belong to B i, i+1 in counteclockwise ode, whee 0 i k 1 is counted mod k. The edges e i 1 and e i, togethe with U, bound a iece of ie contained in VR( i,s ); these ieces ae domains with Jodan cuve boundaies. The sites 0, 1,..., k 1 ae aiwise diffeent. The last fact can be seen as follows. Point v does not belong to any Voonoi egion and can, theefoe, not fom a connection between diffeent ieces of the same Voonoi egion. On the othe hand, no connecting ath can un aound some othe iece, because abstact Voonoi egions ae connected and thei closues ae simly-connected. This follows fom Lemma in [21]. Fo convenience, we include this statement and its simle oof. Lemma 12 Let C VR(, S) be a closed cuve. Then each bounded connected comonent of R 2 \ C is contained in VR(, S). Poof. The comlement of C consists of disjoint connected comonents exactly one of which, Z, is unbounded. Let Z be a bounded connected comonent, and assume that some oint z Z does not belong to the Voonoi egion of. Then z D(, ), fo some. We may even assume z D(, ) because a small enough neighbohood U of z is contained in the oen set Z, and has nonemty intesection with D(, ), so that we could ick a suitable z fom U. Since D(, ) is unbounded, it contains a oint y Z. Because D(, ) is ath-connected thee is an ac α D(, ) unning fom z to y. It must meet the cuve C, which contadicts C D(, ). One should obseve that fom the axioms stated in Definition 1, only A1 was used in the oof of Lemma

14 4.3 Chaacteizing admissible cuve systems In this subsection we show that only the subsets S S of size 3 need to be checked, in ode to ensue that a given cuve system J = {J(, ); S} is admissible in the sense of Definition 1. Fo the old AVD model, this fact has been stated in [22], and been oven fo all bisecting cuve systems whose (finitely many) aiwise intesections ae oe cossings only. Hee we give a diffeent, moe geneal oof based on Lemmata 13 and 14 below. We assume that J = {J(, ); S} is a system of cuves such that each J(, ) fulfills axiom A 1 of Definition 1. Lemma 13 With the above assumtions, we have R 2 = S VR(, S),, S : D(, ) D(, ) D(, ). Poof. The diection fom left to ight has been shown in Lemma 5 without using axiom A 2. To ove the convese diection, let z R 2. Fo an abitay ɛ> 0 let U := U ɛ (z) be an ɛ-neighbohood of z. As long as thee exists a set D = D(, ) such that U D but U D we elace U by its oen subset U D. This ocess teminates afte at most ( n 2) many stes. Fo the final set U, and fo each ai of oints fom S we have The elation U D(, ) o U D(, ). (3) < : U D(, ) is anti-symmetic, tansitive by the ight hand side of ou lemma, and, by fact (3), eithe < o < must hold. Thus, < defines a total ode on S. Let ɛ denote the minimum element with esect to < in S. Then, fo each oint ɛ in S we have U D( ɛ,) which imlies U VR( ɛ,s). Thus, U ɛ (z) contains oints of VR( ɛ,s). As ɛ tends to 0, the index ɛ may vay, but since S is finite thee must be a subseuence of ɛ tending to 0 fo which all ɛ ae the same. Conseuently, z VR(, S). Lemma 14 With the assumtions fom above the following holds. If each set VR(, S ), whee S S and S =3, is ath-connected then each Voonoi egion with esect to some T S is ath-connected, too. Poof. If T = {, } then VR(, T )=D(, ) is ath-connected. Fo T = 3 the claim follows by assumtion. Let T S be of size 4, and conside two oints x, y VR(, T ). Let t t T be diffeent fom. By induction, thee exist an ac π that connects x, y in VR(, T \{t}), and a connecting ac π in VR(, T \{t }). All oints contained in both π and π belong to VR(, T ). Suose that some oint z π is not contained in VR(, T ), and let f, g be the fist oints of π one meets when tavesing π in both diections away fom z. The two segments π f,g and π f,g of π, π between f and g fom a domain, D; see Figue 9 (i). We claim that thee exists an ac α f,g D VR(, T ) fom f to g. Then, by simultaneously elacing all (countably many) segments π f,g of π with α f,g, a ath connecting x, y in VR(, T ) will esult, thus oving ou lemma. This claim is shown as follows. Acs π f,g and π f,g togethe fom a loo in VR(, T \{t, t }). This egion is simly-connected by Lemma 12; one should obseve that axioms A 2,A 3 have not been used in its oof. Hence, domain D too is subset of VR(, T \{t, t }). Now we distinguish two cases. Fist, we assume that the union R := VR(t, T ) VR(t,T) 14

15 does not seaate f fom g in D. Then D \ R belongs to VR(, T ) and contains the desied ac. In the second case, R does seaate f fom g in D. We obseve that VR(t,T) cannot intesect π, which is contained in VR(, T \{t}), and VR(t, T ) cannot intesect π. Thus, both Voonoi egions have non-emty intesections with D. Now we conside the index set S := {, t, t }. By assumtion, thee exists an ac ρ connecting f and g in VR(, S ). Ac ρ must avoid the union, R, of the Voonoi egions of t and t with esect to S, which includes R. In the esence of D R, ac ρ can be homotoic with eithe π f,g o π f,g. In the fist case, which is deicted in Figue 9 (ii), ρ and π f,g togethe encicle oints of VR(t, S ) D(t, ) although they ae both contained in D(, t), which is simly-connected. In the second case, ρ and π f,g encicle oints of VR(t,S ) D(t,), but ae themselves contained in D(, t ) again a contadiction. f D R f π f,g α f,g VR(t, T ) π f,g VR(t, S ) VR(t,T) D π f,g VR(t,S ) ρ π f,g g (i) g (ii) Figue 9: In (i), f and g can be connected by an ac in VR(, T ) D. In (ii), a ath ρ connecting f and g in VR(, {, t, t }) must go aound the Voonoi egion of t o t. Now we can state the esult of this subsection. Theoem 15 Let J = {J(, ); S} be a system of cuves each of which ojects onto a closed Jodan cuve though the noth-ole of the shee. Suose that fo each subset S S of size 3 the Voonoi egions VR(, S ) ae ath-connected, and that thei closues cove the lane. Then J is admissible in the sense of Definition 1. 5 Constuction of AVDs In this section we demonstate that both algoithms develoed fo constucting abstact Voonoi diagams, divide & conue [21] and andomized incemental constuction [24], also wok fo the AVD model esented in this ae. Since eithe algoithm was based on its own set of axioms, denoted by DC-AVD fo divide & conue and by RIC-AVD fo andomized incemental constuction, we also discuss how these sets diffe fom the new axioms intoduced in Definition 1. Since the bisecting cuves ou algoithms ae dealing with can be athe comlex objects, some cae is euied in establishing the cost of building an AVD. If all cuves in J wee algebaic of bounded degee, they could be descibed in constant sace. Also, whateve elementay oeations on bisecting cuves ae tyically necessay, could be caied out in constant time. In geneal, even an elementay task, like testing two cuves fo intesection, may be undecidable. Theefoe it seems easonable to seaate these issues fom the task of constucting V (S). To this end, we shall assume that a set of elementay oeations on bisecting cuves, that may deend on the algoithm consideed, is casuled in a basic module. Only this module can access the cuves diectly. Each call to the basic module efomed by ou algoithm is consideed one ste, just like a standad RAM oeation. The algoithm is not chaged fo the basic module s 15

16 eal time and sace consumtion, which deends on the comlexity of the bisecting cuves J(, ). 5.1 Randomized incemental constuction A andomized incemental algoithm fo abstact Voonoi diagams was fist esented by Mehlhon et al. [27] and then genealized in [24]. The latte ae is based on the following assumtions. As in Definition 1, a set S of n indices is given, and fo each ai of indices in S a cuve J(, ) =J(, ) that slits the lane into two unbounded domains, D(, ) and D(, ). In addition, a total ode on S is assumed to be at of the inut. Now the extended Voonoi egion of with esect to S is defined by EVR(, S) := R(, ), whee if, and S\{} R(, ) := D(, ) R(, ) := D(, ) J(, ), if. In addition, a egula Voonoi egion was defined as V (, S) := EVR (, S), whee E denotes the inteio of E. Finally, the Voonoi diagam is defined as the union of the boundaies of all extended Voonoi egions EVR(, S). Then the following oeties ae euied. Definition 16 The ai (J, ), whee J := {J(, ); S}, is called admissible fo RIC- AVDs if the following axioms ae fulfilled. R 1 ) Each cuve J(, ), whee, S, is maed to a closed Jodan cuve though the noth ole by steeogahic ojection to the shee. R 2 ) Fo any,,, s in S, the intesection J(, ) J(, s) consists of at most finitely many connected comonents. R 3 ) Fo each subset S S and fo each S, if EVR(,S ) is non-emty then V (,S ) is non-emty, too, and both sets ae ath-connected. R 4 ) Fo each subset S S, we have R 2 = S EVR(,S ). The following lemma shows a close connection between RIC-AVDs and ou new AVD concet. Lemma 17 If, fo some ode on S, cuve system J is admissible fo RIC-AVDs then J is admissible fo AVDs in the sense of Definition 1, and fo each S S and S we have VR(,S )=V (,S ). 16

17 Poof. Fist, we ove the euality of Voonoi egions. Let z V (,S ). By definition, z is an inteio oint of the extended egion of, so thee is a neighbohood U of z contained in EVR(,S ). Clealy, U cannot intesect a bisecting cuve J(, ); othewise, U would contain oints of D(, ), but such oints do not belong to EVR(,S ). Theefoe, z U lies in all sets D(, ), S, hence in VR(,S ). The convese inclusion is obvious. Now assume that cuve system J is admissible fo RIC-AVDs. By the above, each Voonoi egion VR(,S )=V(,S ) is ath-connected, thus oving axiom A 2. A oint z R 2 not contained in any egion VR(s,S ) belongs to some set EVR(,S ) \ V (,S ), by R 4. By the iece of ie Fact 1,. 163 [24], z is contained in the closue of some egion V (,S ) = VR(,S ), which oves A 3. Hence, J is admissible fo AVDs. In Definition 2,. 163 [24], Voonoi vetices and Voonoi edges wee defined in the same way as fo ou Voonoi diagam, namely as (maximal) sets contained in the closues of two (es.: of at least thee) egula Voonoi egions. 12 By Lemma 17, egula Voonoi egions ae the same as ous. Thus, the two definitions yield identical gahs as Voonoi diagams. The only diffeence lies in the fact that in RIC-AVDs, Voonoi edges ae, as oint sets, distibuted among the Voonoi egions. Fo examle, Figue 10 (i) shows a cuve system admissible fo each set of axioms consideed in this ae. In (iii) the extended Voonoi egion EVR(, S) of consists of the closue of its inteio, V (, S), lus segment σ of the adjacent Voonoi edge. In fact, we have σ J(, ) J(, ) R(, ) R(, ) = EVR(, S) because of,. The est, τ, of this edge belongs to the extended egion of since. Point v is a Voonoi vetex of the RIC-AVD but the oint w, whee σ and τ meet, is not, because it lies in the closue of only two egula Voonoi egions. (i) R(, S) (ii) V(, S) = VR(, S) (iii) R(, S) σ w R(, S) V(, S) = VR(, S) σ v τ V(, S) = VR(, S) Figue 10: (i) A cuve system fo S := {,, } that is admissible unde all definitions. (ii) Voonoi egions in the DC model, assuming. Segment σ is contained in the egion of. (iii) The egula egions in the RIC model eual the Voonoi egions in ou new model. The RIC algoithm fom [24] iteatively adds a andom index s to the set R aleady consideed, and udates both the Voonoi diagam V (R) and a histoy gah, H(R). This incemental ste deends only on the stuctue of the intesection V (s, R {s}) V (R); 12 Fo the definition of Voonoi edges and vetices in ou model, comae the end of the oof of ou Theoem

18 see the analysis stating on. 165 [24]. This set, in tun, does not deend on the ode on S. Theefoe, we can un the incemental algoithm on a system of cuves, admissible accoding to ou Definition 1, without sulying an ode, and the Voonoi diagam V (S) will be constucted. No ham can come fom the fact that the finite intesection oety R 2 is missing, fo the following easons. All geometic oeations, in aticula those on bisecting cuves, ae casuled in a basic module, as was exlained in the intoduction to this section. With RIC-AVDs, this module takes as inut a subset S of five indices of S, and oututs a combinatoial descition of V (S ); see. 169 [24]. The incemental algoithm itself woks in a uely combinatoial way on the oututs of this module. Its coectness does deend on the fact that V (S) is a finite lane gah and on the iece of ie fact, but these ae guaanteed by ou Theoem 10 and by Lemma 11. We have thus obtained the following counteat of Theoem 2,. 181 [24]. Theoem 18 Let J be a cuve system admissible in the sense of Definition 1 fo index set S, whee S = n. Then the abstact Voonoi diagam V (S) can be constucted in an exected numbe of O(n log n) stes and in exected O(n) sace, by andomized incemental constuction. A single ste may involve a call to a basic module that etuns a combinatoial descition of a diagam of size five. Lemma 17 shows that cuve systems admissible fo RIC-AVDs fom a subclass of cuve systems admissible fo new AVDs. This inclusion is stict, fo two easons. Obviously, bisecting cuves ae now allowed to intesect infinitely often. Also, abandoning the total ode on S makes moe cuve systems admissible, as the following lemma shows. Lemma 19 Thee ae admissible cuve systems enjoying the finite intesection oety, which ae not admissible fo RIC-AVDs unde any ode on index set S. Poof. Figue 11 shows a cuve system admissible unde Definition 1, with Voonoi edges dawn bold. Fo each emutation ijk of 123 thee exists a oint z ijk J( i, j ) J( j, k ) D( k, i ). Unde ode i j k we would have z ijk R( i, j ) R( j, k ) R( k, i ). This events z ijk fom being contained in any extended Voonoi egion, thus violating condition R 4. VR( 1,S) z z z 231 z 213 VR( 2,S) 1 2 VR( 3,S) z 132 z Figue 11: An admissible cuve system that cannot be made admissible fo RIC- o DC-AVDs by any ode on S. Summaizing, we have seen that unde the new AVD axioms the incemental algoithm becomes both moe natual and moe oweful. 18

19 5.2 Divide & Conue The divide & conue algoithm of [21] was designed fo the oiginal definition of abstact Voonoi diagams that will now be eviewed. As in Subsection 5.1, the inut cuves J(, ) and the ode on S wee used to define Voonoi egions R(, S) = EVR(, S) that ae eual to the extended Voonoi egions fo RIC-AVDs. Again, the Voonoi diagam V (S) was defined as the union of all egion boundaies. The imotant diffeence is that fo DC-AVDs the inteios of the Voonoi egions wee not euied to be ath-connected. Moe ecisely, only the following oeties wee stated. Definition 20 The ai (J, ), whee J := {J(, ); S}, is called admissible fo DC- AVDs if the following axioms ae fulfilled. D 1 ) Each cuve J(, ), whee, S, is maed to a closed Jodan cuve though the noth ole by steeogahic ojection to the shee. D 2 ) Fo any,,, s in S, the intesection J(, ) J(, s) consists of at most finitely many connected comonents. D 3 ) Fo each subset S S and fo each S, the set R(,S ) is ath-connected and has a non-emty inteio. D 4 ) Fo each subset S S, we have R 2 = S R(,S ). Due to D 4, each oint of the lane belonged to a Voonoi egion. Since only the Voonoi egions, but not necessaily thei inteios, need be ath-connected, Voonoi egions could contain cut-oints. A simle examle is shown in Figue 12. Because of,, we have J(, ) R(, ) and J(, ) R(, ), so that segment σ J(, ) J(, ) belongs to R(, ) R(, ) = R(, S) whee S = {,, }. All oints of segment σ ae cut-oints of the Voonoi egion R(, S), whose emoval disconnects this egion. In eaation fo a ecise definition of Voonoi edges, a segment σ of J(, ) was called a {, }-bodeline if and σ R(, S) R(, S), o vice vesa; comae Definition 2.3.4,. 40 [21]. In Figue 12, σ is both, a {, } and a {, } bodeline. In a way, the bounday of R(, S) has been sueezed togethe along σ. But the DC-AVD algoithm, that will be sketched below, oeated as if the two bodelines foming σ wee disjoint. Moe comlicated situations could aise at such oints whee moe than two Voonoi egions met. Let us take a look at the ieces of ie aound the oint v deicted in Figue 13. Since Voonoi egions ae connected, by D 3, and because thei closues ae simly-connected (see Lemma 12 and the discussion eceding it), only the Voonoi egion to which oint v belongs could contibute seveal ieces of ie to the neighbohood of v. These ieces wee connected via v. In Figue 13, fo examle, oint v belongs to R(, S) and connects the thee ieces of this Voonoi egion. At oint v, the Voonoi egion of was concetually thickened, thus slitting v into as many induced oints v i as thee ae ieces of R(, S); comae Definition 2.5.1,. 46 [21]. In Figue 13 this thickening esults in thee induced oints, v 1,v 2,v 3. Of the bodelines adjacent to v, each v i inheits those contained in the wedge between two consecutive -ieces. Each 19