Data structures and algorithms to support interactive spatial analysis using dynamic Voronoi diagrams. Abstract

Transcription

1 Data structures and algorthms to support nteractve spatal analyss usng dynamc Vorono dagrams Mark Gahegan *a and Ickja Lee b a Department of Geography, The Pennsylvana State Unversty, Walker Buldng, Unversty Park, PA 80, USA. b Department of Computer Scence and Software Engneerng, Unversty of Newcastle, NSW 308, Australa. Abstract To support the need for nteractve spatal analyss, t s often necessary to rethnk the data structures and algorthms underpnnng applcatons. Ths paper descrbes the development of an nteractve envronment n whch a number of dfferent Vorono models of space can be manpulated together n real tme, to ) study ther behavour, ) select approprate models for specfc analyss tasks and 3) to examne how choce of one model over another wll affect the nterpretaton of data. The paper studes sx specfc Vorono dagram varants: the Ordnary Vorono Dagram, the Farthest-pont Vorono Dagram, the Order-k Vorono Dagram, the Ordered Order-k Vorono Dagram, the k th Nearest-pont Vorono Dagram and the Multplcatvely Weghted Vorono Dagram, and develops algorthms and data structures to store, rebuld and query these varants. From ths, a generalsed Vorono data structure s proposed, from whch specfc Vorono varants can be reconstructed dynamcally as requred. Algorthms for dagram reconstructon and for queryng neghbourhood topology or adjacency relatons) of generator ponts and Vorono regons are presented. An applcaton program, developed on these deas, s used to generate example results as proof of concept. It may be downloaded from a supportng webste. Keywords: Vorono Dagrams, Interactve Spatal Analyss, Algorthms, Data Structures. Correspondng author. Tel ; fax E-mal address: mark@geog.psu.edu

2 Introducton There s an ncreasng trend towards nteractve spatal analyss, where models and data are manpulated n real-tme, and often va a vsual means e.g. Baley and Gatrell, 99; Anseln, 999; Dykes, 997). Supportng these knds of dynamc modellng and analyss efforts places addtonal demands on the computatonal methods and data structures used n that they must be capable of supportng teratve updates, rather than beng executed once as a batch process. Ths paper descrbes algorthms and data structures to support adaptable modellng and analyss usng a number of Vorono dagram varants Vorono dagrams are also known as Thessen polygons, Drchlet tessellatons roxmal polygons). A dynamc and vsual approach to Vorono dagrams can enable drect nteract wth the data, to study cause and effect of varous changes and updates graphcally and to learn about the varous types of dagrams and ther effect on space tessellaton. A number of geometrc models have been suggested to represent geographc space. These models dffer n ther powers and capabltes to guarantee the formalsaton of spatal concepts. Two fundamental data models have been heavly used n desgnng geographcal databases; namely the vector object-based or feature-based) and raster tessellaton-based or mage-based) models Egenhofer and Herrng, 99; Goodchld, 99; Gatrell, 99). However, there has been much debate over the past decade or so concernng the relatve merts that these models offer. Many researchers e.g. Goodchld, 99a; Okabe et al., 994) have ponted out problems and dffcultes caused by these tradtonal representatons and ther lmtatons for varous types of spatal analyss, ncludng for dscrete pont datasets for whch vector and raster representatons do not normally mantan spatal adjacency neghbourhood relatonshp) for dscrete, unconnected objects Gold, 99). For useful examples of the use of adjacency n the analyss of pont datasets see Baley and Gatrell 99)). It s not the purpose of ths paper, however, to argue that one representaton s nherently superor to another n all cases, t s suffcent to conclude that alternatve models may be more approprate for certan classes of applcaton. The Ordnary Vorono Dagram or OVD has been suggested as an alternatve to overcome some of the lmtatons of conventonal geographc data models Gold, 99a; Okabe et al., 99). The man strength of ths approach s that t s, to some extent, an ntegraton of both vector and raster models. It explctly encodes topology spatal adjacency: n the form of shared vertces) lke the vector approach, but also provdes a space-fllng model lke the raster approach. In other words, all space s fully occuped, and fragmented nto tles usually equvalent to zones of nfluence) around each dscrete map object. Therefore, every locaton n the space can be assgned to at least one of the members n any underlyng pont dataset. Another strength of the Vorono approach s that t permts many operatons to be

3 performed n a local fashon, rather than global, usng the explct spatal adjacency relatonshps Gold, 994). As a result, the tessellaton can be mantaned dynamcally usng local updates followng from any changes. Snce topology constructon s computatonally expensve, ths s an appealng property and s especally useful when the objectve s to allow users to make reasonable decsons quckly, rather than provdng a globally optmum soluton Gold, 993). Fnally, Vorono constructons form an ntegral part of many useful dscrete nterpolaton methods e.g. Watson, 99). Vorono dagrams offer some nterestng and useful ways to model a geographc space accordng to relatonshps between the ponts contaned. Ponts can represent facltes such as hosptals and fre statons, retal outlets such as dstrbutors or shoppng malls, or ndeed any other object for whch a pont s a sutable surrogate. Space s parttoned by regons formed from the ponts; n the smplest case the OVD) ths s acheved by defnng a zone of nfluence around each pont, from whch all other ponts are further away. Ths mmcs the proxmty, catchment or attractveness of the pont when compared to other ponts or when vewed from dfferent locatons n the space. By changng the defntons of the underlyng model the space can be parttoned dfferently, to model dfferent condtons, goals or constrants. Readers who are unfamlar wth these concepts may want to look ahead to Fgures 3-, whch show Vorono dagrams and neghbourhoods of a pontset represented n sx dfferent Vorono models concurrently. Changes made to a specfc pont, such as addton, deleton and relocaton affect all dagrams smultaneously. The applcaton software shown n these fgures mplements the data structure and algorthms as descrbed here. Although there are some examples of dynamc dagram creaton n the lterature e.g. Gold, 998; Ickng et al., 999) there has been lttle research drected at the supportng algorthms and data structures, partcularly for hgher order varants. Notable exceptons for the OVD only) are the work of Green and Sbson 978), who proposed an edge-swappng procedure for node nserton and Heller 990), who consders the dynamc deleton of ponts. However the OVD s lmted to a collecton of nearest neghbourhood operatons, such as generatng a buffer zone and searchng for the nearest faclty from a gven locaton Okabe et al., 994). Furthermore, the OVD s only useful when a pont dataset has constant propertes, such as weght nfluence), growth rate and neghbourhood. In many practcal cases, a pont dataset can have dfferent weghts e.g. as attracton values), growth rates and even varous dstance concepts such as the second-closest, the farthest, and so on. Okabe et al. 99) provde comprehensve descrptons of a wde range of Vorono varants. By usng such an extended set, analyss and modellng capabltes can be enlarged so that data can be analysed accordng to a number of dfferent spatal neghbourhoods. For nstance, fndng the next nearest hosptal from a partcular locaton when the nearest hosptal s fully occuped or closed, or the modellng of catchment areas when two retal outlets 3

4 have a dfferent attracton value. These examples requre the nd Nearest-pont Vorono Dagram NVD) and the Weghted Vorono Dagram WVD), respectvely. Unfortunately, these Vorono varants do not always mesh well wth the data structures used for the OVD, but requre a more generalsed form. Whlst there has been a good deal of research conducted nto data structures and algorthms for the OVD e.g. Aurenhammer, 99; Gold, 99a; 998), and assocated Delaunay trangulaton e.g. Gubas and Stolf, 98), stll lttle s known of the data structure and algorthmc requrements of a generalsed form of the Vorono dagram. As a consequence, mplementaton s more dffcult, and may act as a barrer to the uptake of these potentally useful concepts. Certanly, each dstnct Vorono model has specfc requrements n terms of the descrpton of neghbourhood that must be supported by the data structure, and the varous constrants that apply to ts formaton and update. The noton of Who s my neghbour? s extended by these varants, complcatng the topologcal relatonshps that must be represented Gold, 99b). By addressng these ssues, t should become easer to embed a varety of Vorono models nto specfc applcaton packages and generc GIS, thus mprovng the representatonal and modellng capabltes for certan analyss tasks. The research reported here s part of a larger project to nvestgate the commonaltes and dfferences across a range of dfferent Vorono varants, from the perspectve of supportng data structures and dynamc update methods. The overall am s to propose an ntegrated data structure and mplementaton framework that all types can share, to facltate dynamc modellng and analyss. To ths end, a supportng applcaton has been developed, downloadable from or The applcaton allows a user to buld any of sx possble Vorono dagram varants: the Ordnary Vorono Dagram OVD), the Farthest-pont Vorono Dagram FVD), the Order-k Vorono Dagram, OKVD), the Ordered Order-k Vorono Dagram OOKVD), the k th Nearest-pont Vorono Dagram KNVD) and the Multplcatvely Weghted Vorono Dagram MWVD), These can be concurrently vsualsed and the applcaton supports nteracton between any number of lnked vews. It then becomes possble to study how these models dffer n ther representaton of neghbourhood and ther relatonshp to the underlyng pont set generators and therefore to select approprate choces for specfc analyss problems. Thus, the objectve of ths paper s: To nvestgate data structures and methods classes) for a generalsed form of the Vorono dagram for modellng dscrete pont sets. The data structure must be capable of supportng a number of The applcaton s mplemented n the C++ programmng language usng Mcrosoft Developer Studo 4.0 runnng under the MS-Wndows 9/98 operatng system on IBM compatble PC s. Vsual C++, Developer Studo 4.0, MS- Wndows and Wndows NT are all trademarks of the Mcrosoft corporaton.) 4

5 dfferent Vorono dagrams concurrently, rovde for the constructon of approprate topologc nformaton for Vorono regons and ther generatng ponts as requred. It s our ntenton that algorthms for the dynamc update of Vorono varants also mplemented n the applcaton) wll be reported n a later paper, leavng us free to concentrate here on data structure requrements and constructon of topology dynamc update s mplemented n the applcaton referenced above). Intutvely, t s recognsed that any type of Vorono dagram can be constructed f all the vertces and ther topology are known. A complete dagram can then be obtaned by connectng the neghbourng vertces wth a lne OVD, OVD, OOVD and NVD), a crcular arc MWVD), a hyperbolc arc addtve weghted Vorono dagram) or even hgher order polynomal curve compound weghted Vorono dagram). To determne generc structure requrements, the defnton ropertes of each Vorono varant were carefully nvestgated based on mplementaton of the defntons presented by Okabe et al. 99) and evdenced n the accompanyng software applcaton. The followng sectons each consder a dfferent Vorono varant, descrbng the data structure needs and the algorthms from whch the underlyng Vorono dagram can be reconstructed and Vorono neghbours located. Followng from ths, the fndngs are then fused together n Secton to produce a generc data structure encompassng all uncovered needs. Due to space consderatons, detals of the Ordered Order-k Vorono Dagram OOKVD) are omtted from the text. From the perspectve taken here, t s very smlar to the Order-k Vorono Dagram, OKVD) except that the k generators of a regon are ordered. Also, the k th Nearest-pont Vorono Dagram KNVD) s only covered brefly n the descrpton of the Farthest-pont Vorono Dagram FVD) Secton 3). Ordnary Vorono Dagram OVD) The OVD s normally defned as the regon of nfluence. Every locaton n the plane s assgned to the closest member n the pont set usng a Eucldean dstance metrc). If a locaton s equally close to two or more members of the pont set, the locaton s shared by those members. As a result, locatons assgned to one member n the pont set form the nteror of a Vorono regon, whlst those assgned to more than one regon form the regon boundares Vorono edges). Snce every locaton n the plane s assgned at least once, the Vorono regons are collectvely exhaustve and the resultng tessellaton s referred to as a planar OVD Okabe et al., 99). Underlyng the Vorono tessellaton s a Delaunay tessellaton the dual of the Vorono dagram)

6 constructed by jonng all generators that share a Vorono edge, as Fgure shows. The Delaunay trangulaton conssts of non-overlappng trangles where no ponts fall wthn the crcumscrbng crcle crcumcrcle) of any trangle. Fgure. The relatonshp between the Ordnary Vorono Dagram OVD) the sold lne) and ts dual Delaunay Trangulaton the dotted lne).. Data Structure There are two establshed and common data structures used to encode OVDs. These are referred to as the Wnged-Edge structure, based on the Vorono regons, and the Delaunay structure, based on the dual trangulaton. Okabe et al. 99) adopt the rcher Wnged-Edge structure from whch a varety of useful nformaton, necessary for many applcatons, can be drectly extracted. Ths data structure conveys explct nformaton about local ncdence relatons among Vorono vertces, Vorono edges and Vorono regons, so that t can be retreved easly and wthout computaton.) The alternatve Delaunay structure does not explctly store Vorono regons, but snce the Vorono edges can be reconstructed from the trangulaton Gold, 99a) there s no loss of nformaton. One of the advantages of preservng trangulaton rather than polygon sets n the orgnal form s that the number of vertces and edges s constant three), gvng a fxed-length record structure that smplfes mplementaton and mproves effcency of storage and retreval. The cost s that computaton must occur to reconstruct the Vorono structure as and when t s requred. Rather than beng a problem, ths can be advantageous snce t enables any form of Vorono dagram to be reconstructed from the pontset, dependng on the user s current need. Another advantage of the trangulaton approach s that, by makng use of neghbourhood nformaton, the structure can be updated dynamcally and locally, so the scope of changes s reduced. Small changes are more dffcult to solate usng the Wnged-Edge data structure. As Fgure shows, the supportng data structure for Delaunay trangulaton can tself be mplemented n

7 three ways dependng on whether the vertces ponts), edges, or trangles are stored. B 4 T A 3 T D C Pont # x, y Pont # Trangle # Pont # x, y Edge # Pont Next Twn A : A T, T A : A 3 0 B : B T B : D 0 C : C T, T C : 3 C D : D T D : 4 B 0 a) Pont based data structure C 0 A 4 3 b) Edge based data structure Pont # x, y A : B : C : D : Trangle Frst Second Thrd # edge edge edge T 3 4 Edge Frst Second Frst Second # Pont Pont T 0 A D T 0 C D 3 T T A C 4 T 0 A B T 0 B C c ) Trangle based data structure Fgure. Three data structures for descrbng Delaunay trangulaton, based on the pont, edge and trangle: a) pont based data structure; b) edge based data structure modfed from Heller, 990); c) trangle based data modfed from Bjorke, 988). The lst of propertes for each element s shown n Table. The pont-based structure Fgure a)) contans nformaton about how neghbourng ponts form a trangle. It s the smplest structure, but topologcal nformaton concernng adjacency between the elements s rather mnmal. Also, the length of the neghbour lsts s varable the number of trangles concdent wth a generator vares from pont to 7

8 pont) whch gves rse to operatonal neffcences. In the edge-based data structure Fgure b)), all nformaton about the trangle s mplct n the edge table. Therefore, no trangle table s explctly requred, although there may be a trade-off here of storage effcency versus computatonal burden. Ths data structure stores more nformaton than the pont-based data structure, but also requres more storage resources. The trangle-based structure Fgure c)) contans redundant data; as well as ncreasng the storage overhead, the edges of a trangle appear n both the edge table and trangle table. Ths property makes the trangle-based structure less sutable for dynamc trangulaton, because there are more edges to update. Table. Geometrc and topologc propertes of data structure elements appearng n Fgure. Geometrc Informaton Topologc Informaton Pont x and y coordnate. ponters to adjacent edges and ponters to adjacent trangles. Edge Trangle par of coordnates for each endpont. coordnates of the three vertces of each trangle. ponters to adjacent edges and ponters to adjacent trangles. ponters to adjacent trangles. The three types of data structures descrbed above do not deally ft the purposes of ths study, where the emphass s on supportng many dfferent types of Vorono dagram wthn a dynamc modellng envronment, and where the underlyng dataset s subject to regular localsed changes. Also, some of the hgher Vorono methods use two or more ponts as a sngle generator e.g. Order- and Order-k methods), complcatng ssues of neghbourhood. Fnally, the generalsed Vorono edges are not restrcted to a lne; for nstance the MWVD uses two knds of dfferent Vorono edges lne and crcular arc). For now, we make the observaton that a sutable arrangement for the OVD would appear to be a hybrd of those shown n Fgure, where only ponts and Delaunay trangles are stored. Such a structure s shown n Fgure 3. Ths s bascally the same as Fgure c), but wth the redundant edges table removed and the orgn of each crcumcrcle specfcally added. Ths addton s an optmsaton compromse, snce the crcumcentre can be computed as requred from ts trangle vertces. Notce also that each pont contans a weght varable, allowng the attractveness of ponts to vary. The reasons for these choces should become evdent as other Vorono types are studed n the followng sectons. 8

9 a) b) Pont table Pont # X Y Weght c) Delaunay trangle table Trangle # Pont Pont Pont 3 Center 3 0, -87) 3 0,-4) ,-3) Fgure 3. The hybrd data structure of the OVD: a) Dagram; b) Pont table; c) Delaunay trangle table.. Regeneraton of ordnary Vorono regons Snce the data structure explctly holds crcumcentres for all trangles, the OVD can be produced by smply connectng the crcumcentres of neghbourng trangles.e. those that share a Delaunay edge). A full descrpton of an approprate reconstructon algorthm s provded by Gubas and Stolf 98) and Okabe et al. 99)..3 Retreval of spatal adjacency nformaton Neghbours to a partcular pont can be dentfed straghtforwardly. Fgure 4 shows an example where the retreved nformaton ncludes all generator ponts p, p, p 7 ) adjacent to the chosen pont p ), a correspondng Vorono regon the shaded area) and the neghbourng Vorono regons v p ), v p ), v p ) and v p7 ) ) ncdent to that pont. The steps for retreval of adjacency nformaton concernng a partcular pont are:. Fnd any trangle that has the selected pont as a vertex n the Delaunay trangle table Fgure 3). For nstance, p p p s chosen frst among four Delaunay trangles p p p, p p p7, p p p7 and p p p ) whch have the pont p as a vertex as shown n Fgure 4b).. Arrange the three vertces of the chosen trangle n a clockwse order startng from the partcular 9

10 pont. For nstance, the trangle p p p ) s re-arranged to p p p. 3. Fnd the next trangle p p p ) whch has the frst two ponts p ) of the arranged trangle p p p ) n step. Then, draw the ordnary Vorono edge a lne connectng two centres of crcumcrcles of trangles p p p and p p p )) and mark the mddle pont p ) of the arranged trangle p p p ) n step as a neghbour by connectng a lne between the two common ponts p ). 4. Repeat steps ~ 3 for the trangle selected n step 3 untl the frst trangle p p p ) s agan found. a) b) Fgure 4. Ordnary Vorono Dagram and underlyng Delaunay Trangulaton showng adjacency nformaton for a pont; a) Delaunay Trangulaton n = 7), b) Vorono regon surroundng a pont. The shaded area, vρ) represents the Vorono regon of generator ρ, the sold lnes connect ρ to all neghbourng ponts. In the OVD, the Vorono regon of a generator pont always contans the generator pont, smplfyng reconstructon of topology. Ths s not the case for some other forms of Vorono dagram, as wll be seen n the followng sectons. 3 Farthest-Pont Vorono Dagram FVD) Conceptually, the FVD s smlar to the OVD but nstead of descrbng the closest regon t descrbes the farthest. Gven a set of dstnct ponts n the plane, all locatons are assgned to the farthest member of the pont set. The result s agan a tessellaton of the plane nto a set of regons assocated the members n the pont set. The tessellaton s mathematcally defned after Okabe et al., 99) as follows. 0

11 Let P = { p,..., pn } polygon, v p ) defned as: fp R n < ) be a set of dstnct ponts, then the farthest-pont Vorono v p ) = p d p, p ) d p, p ), p P /{ p } fp { } = p d p, p ) max{ d p, p )}, p p { p } The FVD s, n fact, an order-k Vorono dagram Lee, 98). If j j. j j / p j p s the farthest pont from p, then the set of ponts P \{ p } s the set of the frst, the second,..., the n-) th nearest pont from p. Thus the v p ) s the same as the order-n-) Vorono polygon assocated wth P \{ p } from the order-n-) fp Vorono dagram generated by P. 3. Data structure Trangles defne the Vorono vertces n both OVD and FVD. The crcumcrcles of trangles n a FVD nclude all other members n a pont dataset, and the crcumcentres are the farthest-pont Vorono vertces. An example of ths appears n Fgure, where the farthest-pont Vorono vertces q and q represent the centres of crcumcrcles of two trangles p p3 p4 and p p p4. The Vorono vertex q s the crcumcentre of trangle p p3 p4, also t s an ntersecton pont of the Vorono regons v p ), v p ) and v p ). 4 3 Fgure. Dervaton of FVD the dotted lne) from ts data structure the sold lne) n = 4.) These Vorono vertces must all be stored n the data structure of the FVD. As can be seen, all these vertces have exactly three Vorono edges accordng to the non-cocrcularty assumpton. Therefore, a complete FVD can be obtaned by drawng three edges of every vertex from the data structure. Ths s exactly the same requrement as already presented for the OVD, so the structure shown n Fgure 3 can

12 be appled wthout modfcaton to ts form. The only change s one of nomenclature; the Delaunay trangle table now refers to the farthest not the nearest) trangles. 3. Regeneraton of the farthest-pont Vorono regons The FVD s formed from the data structure by the followng steps llustrated n Fgure ):. Read a farthest trangle from the farthest trangle table. For nstance, farthest trangle p p3 p4 ) s chosen frst n Fgure.. For all three edges p p, p p and p p 4 ) of the trangle do step If an edge s shared by another trangle n the farthest trangle table, then draw a Vorono edge a lne between two centres of crcumcrcles). For nstance, the edge p p 4 s shared by the trangle p p4 p, a lne between two centres of crcumcrcles gves Vorono edge q q. If an edge s not shared by any other trangle n the farthest trangle table, then draw a perpendcular lne to the centre of the crcumcrcle of the trangle n the opposte drecton of the edge. The edge p p 3 of trangle p p3 p4 s not shared by any other trangles, so a perpendcular lne s constructed from q headng away from the edge p p 3. Ths lne bsects the Vorono regons v p ) and v p3 ). 4. Repeat steps -3 untl no trangle remans n the farthest trangle table, by whch tme a complete FVD s constructed. 3.3 Retreval of spatal adjacency nformaton Once the data structure for FVD s constructed the topologcal nformaton can be retreved from the data structure,.e. the farthest Vorono edges ncdent wth a generator pont, the farthest neghbourng ponts and the farthest-pont Vorono regon adjacent to a generator. To explan how ths s acheved, consder the pont set shown n Fgure. Retreval s acheved by the followng steps:. Locate a trangle that ncludes the chosen pont as a vertex from the farthest trangle table. In the example, f pont p s chosen, then two trangles p p4 p and p p p could be located. Assume here that the trangle p p4 p s read frst.. Mark the two other ponts n ths trangle p 4 ) as neghbours of p.

13 3. For the two edges that nclude p as one of ther end ponts do step 4. In ths case, edges p p and 4 p p of the p p4 p have p as an end pont. 4. If the edge s shared by another trangle then draw a Vorono edge lne) between the two centres of crcumcrcles of these trangles. If an edge s not shared by any other trangle n the farthest trangle table, then draw a perpendcular lne to the crcumcentre of the trangle n the opposte drecton of the edge. Ths s the same as step 3 n the prevous algorthm).. Repeat steps ~ 4 untl no further trangles contan the chosen pont, producng a farthest-pont Vorono regon and Farthest-pont Vorono neghbours of p. a) b) Fgure. Farthest-pont Vorono Dagram a) nformaton ncdent/adjacent to a generator pont: b) A farthestpont Vorono regon and neghbours n = ). As seen from Fgure a), the Vorono regon v p ) shares Vorono edges wth regons v p ), v p ) and v p4 ). Therefore, ponts p, p 4 are neghbours of p. Ths nformaton s easly obtaned from the two trangles p p4 p and p p p n the farthest trangle table that have p as a vertex. An example of retrevng adjacency nformaton relatng to a farthest-pont Vorono regon s shown n Fgure b). In ths case an ntal step s requred to dentfy the pont farthest from the selected locaton, after whch the prevous procedure can be appled. Ths nformaton can be retreved by selectng any locaton n the shaded area usng the Vorono applcaton referenced prevously. 3.4 Dscusson The FVD has a couple of dfferent characterstc from the OVD. Frstly, there s no one-to-one 3

14 correspondence between the generator pont and the Vorono regon; ponts lyng wthn the convex hull of a dataset wll never form a farthest-pont Vorono regon. These ponts have no neghbours and therefore no spatal adjacency nformaton. For nstance, the pont p n Fgure does not have any topology n the sense of the farthest-pont Vorono dagram. Secondly, the Vorono regon does not contan ts generator pont e.g. the regon v p ) does not contan the generatng pont p ). These two characterstcs cause some algorthmc complcaton but do not mpose any new data structure needs over those of the OVD. 4 Hgher Order Vorono Dagrams OVD - OKVD) In the hgher order Vorono dagrams the number of generator ponts for each regon ncreases from the smple case as n the OVD) to a set. Order- dagrams contan a par of generators per regon and are descrbed here n detal. Extenson to order-k s straghtforward and s covered at the end of ths secton. In general, f a set of the frst nearest and second nearest ponts to a locaton s { p, p j } assgned to v { p, }),.e. the order- Vorono polygon assocated wth { p, } p j p j, the locaton s. Note that no dstncton s made as to whch pont s the nearest the set s unordered. Followng ths assgnment rule, all locatons n the plane are assgned to at least one of the sets of two ponts. A collecton of the resultng order- Vorono polygons form a tessellaton OVD) generated by the set of ponts. Let P = p,..., p } { n R n < ) be a set of dstnct ponts, then the order- Vorono polygon, ) v ) s defned Okabe et al., 99) as: p ) v ) = { { p d p, p ), ), ), ), / ) d p p j p d p p j p j P p }, p ) ) = { } { } p max d p, p pk k ) p k P Under the non-cocrcularty assumpton, for every vertex unque crcle centred at of P n ts nteror. mn d p, p p j j ) p j P / P q of an order-k Vorono polygon, there exsts a q whch passes through three ponts of P and contans zero or,..., k- ponts. 4. OVD data structure Bascally, the data structure must here manage two dstnct types of trangles and assocated crcumcrcles). The crcles contanng no ponts n ther nteror can be treated n the same way as vertces of the OVD. These vertces are called old vertces by Lee and Schachter 98). The relatonshp 4

15 between the OVD and the OVD s shown n Fgure 7; the two Vorono dagrams do not share edges but share three vertces q, q and q 3 the flled crcles). Fgure 7. The relatonshp between the OVD the dotted lnes) and the OVD the sold lnes) generated by the same set of generator ponts n = ). Table. The data structure for OVD; a) Pont table; b) Delaunay trangle table; c) Order- trangle table n =, the locatons of the generator are the same as those n Fgure 7). Pont # X Y Weght a) Pont table Trangle # Pont Pont Pont 3 Centre 4-4,-00) 3 4,-3) ,-00) 4, 04) 9, -38) 3 8,-7) b) Delaunay trangle table Order-# Pont Pont Pont 3 InPont Centre, -8) 4 4,-00) ,-00) ,-8) 3 4,-8) ,-403) c) Order- trangle table

16 All vertces of the OVD are also vertces of the OVD, but some vertces of OVD.e. centres of crcumcrcles that contan a pont n ther nteror) are not part of the OVD e.g. vertex q4 n Fgure 7). The old vertces can be stored n the Delaunay Trangulaton table Fgure 3), leavng only the vertces not n the OVD requrng specal treatment. Ths addtonal need s catered for here by constructng a new table, so the OVD data structure conssts of three components. Table shows the data structure for the OVD depcted by Fgure 7. The frst parts of ths, a) and b), reman unaltered wth an Order- trangle table added c), contanng only those trangles whose crcumcrcles nclude a pont. The Order- trangle table needs to store the three vertces formng a trangle and the pont lyng wthn the trangle. Smlar to the data structures of OVD or FVD, edges are not stored, but nstead are mplctly defned n the two trangle tables. 4. Regeneraton of the Order- Vorono dagram The complcatng factor here s that vertces exst n one of two tables. Consder the example shown n Fgure 8. Vertces q and q3 are the centres of crcumcrcles of Delaunay trangles p p p and p 3 p p ) are found n the Delaunay trangle table n Table b) as trangle ID and. On the other hand, the vertces q, q, q 4 and q appear n the Order- trangle table shown n Table c), as Order- ID, 3, and, respectvely. These two tables need to be processed together to derve an OVD. q 9, ), ), ), ), 3) 4, ) 3, ) 3, ) Fgure 8. Dervaton of OVD from ts data structure n =, generated by the same dataset of ponts used n q 8 Table, p represents a pont, q represents OVD vertex and, j) represents a Vorono regon generated by two ponts p and p j respectvely).

17 The converson process s acheved by the steps below:. Read an Order- trangle e.g. Order- ID 3: p p3 p contanng InPont p n Table c)).. Make three new trangles by combnng two vertces from the trangle read and ts InPont vertex. Followng the example these would be: p p3 p, p3 p p and p p p. 3. For every new trangle do step Search for the trangle n the trangle table Table b)). If t s found, then draw a lne between the centre of the trangle and the centre of the Order- trangle and go to step. The trangle p3 p p exsts n the trangle table so a lne s drawn between ther two centres, creatng the Vorono edge q q 3 n Fgure 8. If the trangle s not found, then do step.. Search for the trangle n the Order- trangle table. In ths example, the thrd vertex p ) of the trangle p p p ) should be the InPont feld and the frst two vertces p ) should be the vertces of Order- trangle n the trangle table. If the trangle s found, then draw a lne between the two centres. For example, the trangle p p p s found n Table c), the frst two ponts p and p are the vertces of the Order- trangle p p p ) s the InPont. So, draw a lne between the two centres of the Order- trangles ID and 3). Ths results n a Vorono edge q q n Fgure 8. If the trangle s not found, then draw a perpendcular lne from the Centre feld of the orgnal Order- trangle n the opposte drecton of the edge between the frst and second vertex.. Repeat steps - untl no Order- trangle remans n the Order- trangle table. 7. On completon, an OVD s obtaned from the data structure. However, there s an excepton that needs specal care. By the converson algorthm descrbed above, some Vorono edges e.g. q q and 9 q q 3 8 n the example) are not reachable from the centres of Order- trangles. Snce the algorthm reads the entre Order- trangle table, any edges not drectly connected to Order- trangles wll be mssed. The problematc vertces are reachable, however, from the vertces used n Step 4 above), so ths s modfed to nclude a procedure to draw such edges. In the example gven, the vertex q s reachable from the vertces q or q that are processed n Step Retreval of spatal adjacency nformaton Calculatng the OVD neghbourhood of a pont requres a dfferent approach. Fgure 9 shows an 7

18 example used throughout ths descrpton, obtaned by selectng the pont p n the Vorono program. As defned, the neghbours of p are p, p, p 7. Therefore, the order- Vorono regon of p conssts of four regons: v p, p ), v p, p ), v p, p ) and v p, p ) 7. The regons are generated by the combnaton of p and one of ts neghbours. The shaded area n Fgure 9a) s the zone of nfluence of p n the sense of OVD. For nstance, f p were a hosptal, people lvng n the regon v p, p ) wll use the hosptal p, f the hosptal p s closed or fully occuped. So, the number of patents attendng hosptal p wll be affected by the neghbourng hosptals p, p, p 7 ). Fgure 9. Order- nformaton a) ncdent/adjacent to a generator pont p) and b) to a Vorono regon: n = 7). The retreval procedure s acheved by the steps below:. Fnd neghbourng ponts of a partcular pont n the Delaunay table. The neghbours n the OVD are those n the OVD. Therefore, read the Delaunay table and fnd the neghbours p, p, p 7 ) of the pont p ).. For every neghbour, draw ts Vorono regon wth the pont p and draw a lne from the neghbour to p. For nstance, for the neghbour p draw a Vorono regon v p, p ) and a lne p p to show the neghbourhood relatonshp. 3. On completon, adjacency nformaton ncdent/adjacent to a partcular pont s obtaned. Calculatng the OVD regon of two generators requres the followng method:. Fnd the frst and second nearest generators from a gven locaton. Two ponts p 7 are the 8

19 frst and second nearest ponts to any locaton n the shaded area n Fgure 9b).. Once two generators are obtaned, then draw the Vorono regon v p, p ) of two ponts, and show a 7 par of generator ponts by drawng a lne between two ponts p Dscusson The OVD data structure has two propertes that make t dffcult to update locally and dynamcally wth comparson to the data structure of OVD. Frstly, the Order- trangles are overlappng they do not tessellate unquely). We note n passng that ths also causes complcatons for nserton and deleton algorthms whch we do not have room to descrbe here. Secondly, the OVD data structure s a combnaton of a Delaunay trangle table and an Order- trangle table, due to the addtonal feld needed to specfy the InPont. So when an OVD s derved or spatal adjacency nformaton s retreved, these two tables must be processed together. Generalsng the OVD to hgher order neghbourhoods can be acheved by addng addtonal trangle tables. From the defnton gven at the start of Secton 4, the Vorono vertces of order-3 Vorono dagram are defned by three dfferent trangle tables Delaunay trangle table, Order- trangle and Order-3 trangle ). In general, the data structure for OKVD wll requre k trangle tables: Delaunay trangle table, Order- trangle table,..., Order-k trangle table, each wth progressvely more InPonts. Extendng these modfcatons to address the OOVD and OOKVD are also straghtforward. The unon of the ordnary Vorono edges and the OVD or OKVD) produce the OOVD OOKVD) edges, so they requre no addtonal supportng structures. Multplcatvely Weghted Vorono Dagram MWVD) So far t has been mplctly assumed that each generator pont has the same weght. However, ths assumpton may not be approprate n many settngs e.g. Boots, 980; Aurenhammer and Edelsbrunner, 984). Rather, we would lke the ablty to adjust weghts reflectng the varable propertes of the generator ponts for nstance, to reflect the attractveness of some object such as a shoppng centre or a cty. The basc propertes of an MWVD are agan descrbed by Okabe et al. 99) and brefly mentoned here snce the followng descrptons draw on them. In a nutshell, the defnton of the MWVD s exactly the same as the OVD except the Eucldean dstance s replaced wth the Multplcatvely Weghted MW) dstance, gven by: 9

20 d mv p, p ) = x x, w > 0. w where, w s weght n) and n s the number of members n the pont dataset. A MW Vorono regon s then gven by: v p ) = { p d p, p ) d p, p ) }, j. MWVD I n = mw Because of the dstance-decay effect of the weghtng functon, the bsector between two generator ponts s a crcle; known as the Apollonus crcle n classc geometry. In the specal case of weghts beng equal w = w j ), the bsector becomes a straght lne.e. a crcle wth an nfnte radus). So, an edge s a crcular arc f and only f the weghts of the MW-Vorono regons sharng the edge are dfferent, otherwse, edges are straght. A MW-Vorono regon s a non-empty set; t need not to be convex, or connected; and t may contan holes. In fact, a MW-Vorono regon v p ) s only convex f the mw j MWVP weghts of adjacent MW-Vorono regons are greater than or equal to) may share dsconnected edges. w. Two MW-Vorono regons Fgure 0 shows an example of an MWVD. Notce that the Vorono regon v MWVD p ) ndcated by the shaded area s nether convex nor connected. Even worse, t has a hole p ). Interestngly, the v MWVD hole has another hole v MWVD p ) n t. As a result, the MWVD s not only more complex than the GVDs 7 mentoned earler, but also more dffcult to mplement. 3) 3) 3) ) 3) ) ) Fgure 0. A Multplcatvely weghted Vorono Dagram MWVD) generated by 7 ponts the numbers n parentheses represent weghts assocated wth the generators). 0

21 From the non-cocrcularty assumpton, all the Vorono vertces wll be equally MW-dstant from exactly three members n the pont dataset. In Fgure 0, the MW-Vorono regon v MWVD p ) has two Vorono vertces q and q. These two vertces are equally MW-dstant from the three ponts p, p. Thus, the two Vorono vertces q and q are the ntersecton ponts of exactly three MW-Vorono regons v MWVD p ), v MWVD p ) ). For each Vorono vertex, these three neghbourng v MWVD ponts must be stored n the data structure for MWVD. However, the holes do not have any ntersecton pont, so must be treated dfferently.. MWVD data structure The MWVD data structure requres two tables a pont table and an MW-trangle table) as shown by Table 3. Also, a weght for each generator must be stored n the pont table, explanng why ths feld appears n Fgure 3, despte not beng used by the OVD, the FVD, the OVD, the OOVD or the NVD. Structurally, the MW-trangle table s exactly the same as the Delaunay trangle table shown earler. The only dfference s caused by the holes) mentoned above. Table 3. The data structure for the multplcatvely weghted Vorono dagram MWVD) shown n Fgure 0 n = 7): a) pont table; b) MW-trangle table. Pont # X Y Weght : : : : : : : : n a) Trangle # Pont Pont Pont 3 Centre : : 3 4 : : 3 : 3 4 : 7 0 : : b) The MWVD shown n Fgure 0 has sx Vorono vertces q ~ q ) and two holes. These vertces are stored n Table 3b) as trangle IDs,,, and the two holes n a smlar manner as trangle ID 7 and 8, respectvely. For nstance, the vertex q s an ntersecton pont of Vorono regons v MWVD p ), v MWVD p ) and v MWVD p ), so these three neghbourng ponts p, p are stored together as

22 trangle ID. By contrast, the holes v MWVD p ) ) ) do not have any ntersecton pont, v MWVD 7 but are created by the generator and the frst nearest MW-dstance pont from the generator. For example, the hole v MWVD p ) s created by ts generator p ) and the nearest MW-dstance pont p ). In other words, the hole s an Apollonus crcle generated by the two ponts p ). Accordngly, the hole requres two ponts rather than three, so s stored n the data structure wth the second pont set to zero to ndcate t s a hole. Note also that the frst and second records trangle ID and ) n Table 3b) have the same ponts p, p ) as ther neghbours. Ths s because the two vertces q and q are generated by the same MW-Vorono regons: v MWVD p ), v MWVD p ) ). A separate v MWVD method s requred to dfferentate the two vertces when rebuldng the MWVD or retrevng adjacency nformaton and wll be descrbed n the followng secton.. Regeneraton of the multplcatvely weghted Vorono dagram The followng method regenerates the MWVD from the data structure.. For each generator, retreve the set of neghbourng Vorono vertces. From the dagram shown n Fgure the MW-regon v MWVD p ) conssts of sx Vorono vertces q ~ q ) and sx Vorono edges q q, q q, q q, q q, q q and q q ) Sort these Vertces by polar angles wth respect to the generator. Vertces of p wll be n order q, q, q, q, q 3 and q 4 after sortng. 3. Once sorted, every par of consecutve Vorono vertces q q, q q, q q, q q, q q and q q 4 n the example) must be tested to see f they form a vald edge. Generally, each par has at least two ponts n common. For nstance, the vertex q 3 has p, p 4 as ts members and vertex q 4 has p, p 4. Therefore, the Vorono edge q q 3 4 s a part of an Apollonus crcle generated by the two common ponts p Once these two ponts are dentfed, then the requred porton of the Apollonus crcle can be drawn by constructng an arc n a counterclockwse drecton from the frst vertex to the second q 3 to q 4 n ths example). To avod drawng arcs more than once a smple test can be nserted here. An arc s only drawn when the weght of the generator pont s smaller than that of the adjacent generator. In the above example,

23 arc q q would not be drawn because the weght 8) of generator 3 4 p ) s greater than the weght 3) of adjacent generator p 4 ). The arc q q 3 4 has to wat untl the MW-Vorono regon v MWVD p4 ) s drawn, because the weght 8) of adjacent generator p ) s greater than the weght 3) of generator p ). As a result, only two Vorono edges arcs q q and q q 3 ) wll be drawn wth respect to the 4 MW-Vorono regon v MWVD p ).. Upon completon, the MWVD s obtaned from the data structure. ) ) 8) 8) 0) ) 3) ) 0) ) 3) ) a) b) Trangle # Pont Pont Pont 3 Centre : : : : : 3 : 3 : 4 : 4 : : : : : : : : c) Fgure. Dervaton of MWVD from ts data structure: a) the MWVD; b) the MWVD wth two bsectors; c) the assocated vertex table. However, there s a complcaton that needs specal care. A regon may be comprsed of two arcs between the same par of vertces. For example, the MW-Vorono regon v MWVD p ) enclosed by two Vorono edges q q and q q ) s formed entrely by two Vorono vertces q and q. These two vertces have three ponts p, p 3 ) n common and are generated by the ntersecton of two Apollonus crcles 3 3

24 b p, p ) and b p, p ). As Fgure b) shows, ths produces four arcs two crcles) lnkng these 3 3 vertces, wth the actual edges that should be drawn shown as sold arcs and the others as dotted lnes. The correct arcs can be dentfed by comparng lengths and selectng the smaller of each par. The length of an arc s ) s gven by: s = r θ, where r s the radus of an Apollonus crcle and θ s the radan of the arc.).3 Retreval of spatal adjacency nformaton Adjacency nformaton for the MWVD data structure comprses the neghbourhood of a MW generator pont and regon. Mostly, retreval s straghtforward here, due to the smlarty of MWVD and OVD. Calculatng the neghbourhood of a pont nvolves determnng whch ponts share a vertex, as explaned n the prevous secton. Fgure a) shows the MWVD generated by generator ponts and Fgure b) shows the nformaton retreved by choosng the pont p n the Vorono program. The adjacency nformaton ncludes the correspondng MW-Vorono regon the shaded area) and the Vorono edges ncdent to the pont p. The neghbours lnked to the pont by a lne) adjacent to the pont p are retreved as well. Two ponts p 3 are not connected to p. Ths s because they are not neghbours of p ; the MW-Vorono regon v MWVD p ) shares no Vorono edge wth ether v MWVD p ) or v MWVD p ). 3 ) ) ) 3) ) ) ) ) 3) ) 3) ) ) ) ) a) b) Fgure. Weghted nformaton ncdent/adjacent to a generator: a) A MWVD; b) Retreved adjacency nformaton by clckng over the pont n =, the numbers n parentheses represent weghts assocated wth the generators). Snce a MW-Vorono regon always contans ts generator, the nformaton retreved by selectng any 4

25 locaton n the MW-Vorono regon must be exactly the same as that retreved by selectng ts generator pont. The nearest MW-dstance generator from any locaton n the shaded area n Fgure b) s ts generator p. If any locaton s selected n the MW-Vorono regon v MWVD p ) then the adjacency nformaton shown wll be retreved..4 Dscusson The MWVD s smlar to the OVD descrbed frst ndeed, f the weghts of all generator ponts are same, then the MWVD degenerates to the OVD. However, the weghtng of ponts enables the MWVD to model a wder-range of stuatons, wth the prce for ths beng a more complex reconstructon algorthm. The format of the supportng data structure s exactly the same as that of OVD, wth the caveat that holes as well as trangles must be represented. Results and Summary Before presentng a general data structure, we make the followng key observatons as a summary of our fndngs: ) The Vorono regons of OVD and OOVD also OKVD and OOKVD) have multple ponts as ther generator. ) The Vorono regons of FVD and NVD do not contan ther generator pont. 3) In the MWVD, members of pont dataset may have dfferent weghts attached to them. As a result, a weght feld s added to the pont table. 4) The Vorono regon may contan one or more holes n the MWVD. ) A Vorono regon may be dsjont n the MWVD and s always dsjont n the NVD. ) A Vorono edge can be a crcular arc n the MWVD. 7) There s no one-to-one correspondence between generator ponts and Vorono regons n the FVD, NVD and MWVD.. Proposed data structure A revew of the prevous descrptons reveals that the data structure needs for dynamc, generalsed Vorono dagrams can be met by two tables of the form shown n Table 4 as follows. The pont and trangle table are shared by all varants. An addtonal Order- trangle table s also needed by the OVD, OOVD and NVD. The hybrd trangle-based data structure shown n Table 4 merges the trangle table

26 and Order- table to gve two dstnct components: a pont table contanng x, y coordnates and weght, and a vertex table detalng the Vorono type GVD#), the vertex dentfer vertex#), three vertces pont, pont, pont 3) the InPont Secton 4.) and crcumcentre. Despte all of the dfferences shown above, the methods developed ndcate every type studed s adequately catered for by ths structure. Table 4. A hybrd data structure for all Vorono types studed: a) A pont table; b) A vertex table. a) Pont # X Y Weght : : : : : : : : : : : : : : : : b) GVD# Vertex # Pont Pont Pont 3 InPont Centre : : : : : : : OVD : OVD 3 : MWVD : If any two trangles n the trangle table have two vertces n common they are consdered as neghbours. The three neghbours two for the holes n the MWVD) wth respect to the Vorono vertces are always defned whatever the dagram type. As prevously mentoned, the InPont feld n the vertex table s only used for the OVD, t takes the value zero for all other types. Also, the hole n the MWVD does not have any ntersecton pont, ndcated by zero stored n the second feld Pont ) n the vertex table. Importantly, ths data structure s amenable to the dynamc nserton and deleton of ponts. Ths s achevable because the data structure solates the scope of change, so that only drectly-affected regons of any dagram need to be reprocessed followng an update. The am, of course, s to ensure that computatonal overheads are mnmsed, and consequently, that the algorthms are scaleable to large numbers of data ponts. For ordnary Vorono dagrams OVDs), localsng the update procedure s reasonably straghtforward due to the relatvely smple structure of the neghbourhood. However, hgher Vorono types possess more elaborate neghbourhood topology, resultng n more complex update algorthms that often affect more of the underlyng ponts set. The consequence s that the algorthmc effcency of ncremental update decreases. However, when compared to the cost of a complete dagram rebuld, the benefts of an ncremental approach are stll clearly justfed. Detals wll be presented n the forthcomng companon paper.. Example results Fgure 3 shows a range of Vorono dagrams generated from the same set of ponts n = ) so they can be compared drectly n truth the FVD s generated by generator ponts wthout four boundary ponts).

27 Fgure 3. An OVD, OVD, OOVD, MWVD and NVD generated by the same set of generator ponts n =, the FVD s generated wthout the four boundary ponts). 7

28 Fgure 4. Adjacency nformaton ncdent/adjacent to a generator pont n =, the locatons of the generator pont are the same as those n Fgure 3). 8

29 Fgure. Adjacency nformaton ncdent/adjacent to a Vorono regon n =, the locatons of the generator pont are the same as those n Fgure 3 and Fgure 4). 9

30 In the applcaton program, all dagrams are lnked, so that any changes affect all dagrams smultaneously. Fgure 4 uses the same data ponts to contrast the adjacency of a generator pont between the sx GVDs. As a result, t s possble to gan some nsght nto partcular types that mght be sutable for a gven task. If each shoppng centre has dfferent facltes that attract customers, then the MWVD adjacency nformaton can be used. If a factory s to be sted as far as possble from a partcular cty, then the FVD can be used. If a shoppng centre s plannng to merge wth one of ts neghbours, then the OVD can be used the optmal merger mght be wth the neghbourng shoppng centre that promses the bggest ncrease of catchment area). Fgure shows the neghbourhood of a Vorono regon, agan based on the same data ponts. The neghbourhood of a partcular locaton the shaded area) can be compared among the varants. Usng the OVD and the MWVD, ths regon represents a zone of nfluence. In the FVD ths zone can be used to fnd a ste farthest from the bggest market rval. The second nearest competng company to the partcular locaton can be examned usng the NVD..3 Observatons, comments and future work A data structure, and assocated reconstructon methods are presented for a range of Vorono varants OVD, MWVD, FVD, OVD, NVD). In the OVD, MWVD, NVD and FVD, dscrete objects are defned as adjacent neghbours only f ther correspondng Vorono regons share a common boundary Vorono edge). Snce a generator s a pont n these dagrams, t s easy to defne the neghbours. It s more dffcult to defne neghbours n the hgher order Vorono dagrams OVD and OOVD) snce a generator s not lmted to a pont, but rather s defned by a set of ponts a par of ponts n order examples gven n ths paper). In ths case, all members of the set of ponts that generate a Vorono regon are consdered as adjacent neghbours. The Vorono Applcaton program developed from the descrbed structures and algorthms was used to produce Fgures 4-, wth annotaton added afterwards where approprate to support explanaton n the text. The applcaton appears to be useful for exploratory vsual analyss, for contrastng models and for educatonal purposes. Although not specfcally descrbed here, the applcaton appears well-suted to dynamc modellng tasks, where the data are subject to change. Ths s because all dagrams and ther assocated topologcal relatonshps are reconstructed 'on the fly' as they are requred. A later paper wll gve detals of the dynamc nserton and deleton algorthms evdent n the applcaton program for the descrbed Vorono varants. Vorono spaces requre the followng two geometrc constrants, so they have been coded nto the applcaton to avod conflct. Ponts are all dstnct n the sense that no ponts may concde n the plane. If two ponts concde, 30