维普资讯 http://www.cqvip.com Jo urnal o fShanghai Uni ver si t y (Engl i sh Edit i on),2005,9( 5):407—41 0 Ar t i cl e I D:1 007— 6417( 2OO5) O5— 0407— 04 Dr awi ng We i ght ed Di re ct e d Gr aph f r om I t’ s Adj acency Mat ri x MAO Guo— yong(毛 国勇), ZHANG Wu(张武 ) School f Comput o r e Engi nee ri ng and Sci ence,Sh a ngh ai Uni ve rs it y,Shangh ai 20007 2,P. R.Chi na Abst rac t Thi s paper pr opo ses an al gor i t hm f or buUdmg wei ght ed di rect ed gr aph, de f m es t he w ei ght ed di rect ed rel at i on sh ip m at r i x of t he gra ph,a n d desc ibe r s lgor a it hm i mpl em ent at i on usi ng t hi s ma tr i x.Based on t hi s lgor a it m h ,a n ef f ect i ve way or f bui l di ng n d a dr aw i ng wei ght ed di rect ed g ra p hs i s pr esent ed,f or mi ng a f ounda ion t or f v i s ual i mpl ement at ion of t he a lgor it m h n i t he gr aph t heor y. Key words wei ght ed di rect ed gr aph,adj acency mat ri x,r el a ion t sh ip mat r i x. m at ix r show n i n Fi g.1 i s stud i ed f ir s t. F i g.2 shows a 1 Int roduct i on Wi th rapi d devel opm ent of com put er technol ogy,r e— i rect d ed g ra p h cor respond i ng to t he rel ati on sh i p m atr ix. search on the graph t heor y has pr ovi ded a great deal of advanced resul t s.How ever,one can not f ind m any r e— s e r che a s on v isua l bu i l d of gr a phs ba sed on adj ace nc y m at ri x or rel ati onshi p m atri x of t he graph , a lt hough such r esea r ch i s usef ul i n t he t eachi ng of gr a ph theor y and ot her pract i ca l appl i cat i ons.Take wei ghted di rect— ed graph a s an exampl e, onl y af ter bui l di ng a graph si u ng adj acency or r el a t i on sh ip mat r i x,c an one isual v — Fi g.1 Relati on sh i p nmtri x l y n d ef a fect ivel y i m pl em ent al gor ithm s such a s shor t— est path, connected com ponent s , and m i i m um span— n ni ng t r ee.As bui l ing d g r aph si u ng adj acency mat r i x or r el at i onsh i p mat ri x i s d iff erent f rom graph drawi ng, popu lar d r aw i ng sof t wa r e i s not desi rabl e.Researchers have devel oped dynam i c gr a ph ed it or speci f i cal l y u s ed i n d r aw i ng graph. I t on l y faci l i t ates d r aw i ng of g r aph com pa r ed w i t h ot her d r aw i ng sof t wa r e. So f r no ef a - f ect i ve al gor it m s have been devel h oped for bu il i ng d gr aph si u ng a ̄ac ency ma t r i x or r el a t i onsh ip mat r i x. Fi g. 2 Di rect ed g ra p h 2 M odel and Approach One can observe f rom the m at ix and the core- r Usuall y d i rect ed gr aph i s r epr esented by a rel ati on— sh i p m at ri x. Each rel at i on sh i p m at ri x corresponds t o a i rect d ed graph. I n or der to bu il d a graph ba s ed on a rel at ion s h i p m at ri x , the charact er isti c of r el at i on s h i p spond i ng gr aph t hat t he sum of each col umn i n h e m a— t tr ix i s 0, that i s, posi t i ve num bers equa l to negat i ve nu m ber s i n each col m n.A u posi t i ve nu m ber repr esent s n edge that a i nci dent s f rom th i s ver t ex ,a n d a negat i ve Recei ved Feb.23.2004: Revi sed Aug. 1 8,200 4 nu m ber represent s an edge that i nci dent s to th i s ver- Pr oj ect suppor t ed by Sci ence Foundat ion of Sha ng hai Muni ci pa l tex.Connect i ng the two vert i ces cor espond ing to neg— Conmi ssi on of Educat i on (Gr a nt No.03A203) ati ve a n d ver ti ces posi t i ve nu m ber s , i.e.,con ecti ng MAO Guo— yong. Ph.D .Ca n di dat e. E— m ar l :g ynl ao @ m ai l.shu. edu. cn; ANG W u, Ph. D . , Prof., E— m ai l:zhang @ s t af f. shu.edu.cn Vl corespondi ng to a posi ti ve nu m ber and core— spond i ng t o a negati ve nu m ber i n t he 5t h col m n i u n 维普资讯 http://www.cqvip.com 408 Journa /o fShanghai Uni ver sit y Fi g.1,we get e5 connect ing Vl and .Connect ing ver ti ces i n each col umn one by one, one can get 7 edges i n t 0ta1.equal to t he number of col m ns i u n t he m at ri x. The edge di rect ion can al so be det ermi ned by the above approach.The d irect ed edge i s obt ai ned si mpl y by usi ng the ver tex cor respond ing t o a posi ive nu t m ber as a source vert ex a n d a ver tex cor espond i ng t o a negat ive nu m Fig.4 W DRM bera s a dest inat ion ver tex .The arrow i— d rect ion i s det er mi ned by t he edge di rect ion.From this irect d i on,t he l ength of arrow a n d the heml i ne,a n ar— row ca n be drawn a s shown i n Fi g.2. The arr ow shoul d be sl i m a s t here may be m ore tha n one edges that i nci dent to a cert ai n vert ex.Fi g.3 i l l st u rat e s t h e Mgonthm of arrow , supposi ng t hat M l a n d M 2 are WDRM can be ach i eved i f we change t he val ue 1 i n t he rel at i on sh i p mat ix r i nt o t he rea l wei ght.The nega— ti ve va lue i n a rel at ion sh i p m a t ri x rem ai s t n he sam e becau s e i t i s onl y u s ed f or det er m i ni ng d ir ect ion. There m ay exi st a l oop bet w een w o vert t i ces i n pract i— ca l appl i cat ion s .Theref ore, an a r c shoul d be u s ed i n— sou r ce dest i nat i on and ver ti ces of a d ir ect e d edge,r e— st ead of a l i ne t o con ect these t w o vert i ces.W e st il l spect i vel y. use l i ne to con ect t w o vert i ces i f no l oop ex i st s be— w een t them ,as shown i n Fi g. 5. Fi g. 3 Ar r ow al gor i t hm Fi g. 3 s hows t hat d ir ect i on,l eng th a n d wi dt h of t he arrow a r e M l M 2, IM 2, and 2 × IM 3, respect ivel y. Coord i nat es of M l a n d M 2 det er mi ne the angl e be— Fi g. 5 W ei ght ed di r ec t ed gr aph t ween M l M2 n d a t he hor i zont al l i ne ( we cal l t hi s angl e s l oppy”).Coor inat d es of I a re de t er m i ned by t is h r om t F he propert y 0f W DRM .each edge i n Fi g. 4 i s a ngl e a n d t he l eng th of ar r ow .Coor inat d es of M3 a re s soci a at e d w i th t w o ver ti ces.Thu s ,t her e a r e onl y w o t “ det er mined by t he coord i nat e s of I, t h e angl e of M3 I K , wh ich e qual s 90一 s l oppy, a n d t he l eng th of IM 3.Coord i nat e s of M 4 ca n be obtained i n a si mi l r a non- zero el em ent s i n each col m n .I u f si u ng l a n d —l to represent respecti vel y a posi t i ve wei ght and a nega— t i ve wei ght i n the m a tr i x, a loop ex i st s bet ween t w o vert i ces cor espond i ng t o t w o non— zer o el ement s when w a_ y. Gi ven a wei ght e d d i rect e d graph G Def t ni t i on i thout w paral l el e dges,G =(V,E), V= {vl,v2, … , = v },E ={ el,e2,… ,e },P X q ma t r i x M (G) (m ),a n d f w, 1 fl 0re.a n a r c shou l d be u s ed to con ect t hese t w o vert i CeS . Det er mi nat ion of t he a r c’s di rect i on i s the sam e a s of a l i ne. I f i s the sou r ce ver tex of ej i n G , w i s t h e we i g h t of ,w>0, 1 【 一l ,Ⅱ i s t h e d e s t i n a t i o n v e r t e x o f 0, el em ent s of w o col t um ns a r e exact l y t he sam e.Ther e. I f Vi d0es n’ t i nci dent on . e st il l u s e t he ver tex cor espond i ng to a posi ti ve el em ent i n t he m at r i x a s a source ver t ex, n d a negat a ive el em ent s a a dest inat ion vert ex.The di — rect i on of ar row i s cl ockw i se.An r c i a s determ i ned by at l ea s t 3 vert i ces, so we m u s t f ind at l ea s t one add it — i ona l vert ex. Accord in g to the arrow al gor it hm , we Then M (G)i s r ef er red t o s a t he wei ght ed di r ect ed r e— l at i on s hi p mat r i x( WDRM)of G,a s s how n i n Fi g. 4. ext end t he l eng t h of ar r ow a n d I M 4 a ppr opr ia el t y, 维普资讯 http://www.cqvip.com Vo1. 9 No. 5 Oct. 2005 MAO G Y,et a1.: Dra wi ng Wei ght ed Di rec t ed Gr aph f r om I t’s Adj acenc y Mat r i x 409 dr aw a new arrow of t he sam e si ze i n M 1, t hen a vert ex.t he cent er of el — der t o get coord inates of each cl ockw i se arc f rom M l to M 2 i s det ermi ned by M 1, l i pse i s f ixed,and ll a ver ti ces a r e draw n on ci rcumfer— M 6, M 4 and M 2 as show n i n Fi g.6.W e ca n l abel t he ence of t h e el l i pse.The angl e of ci rcumf erence i s d i— w ei ght of a m at ri x besi de t h e di rect ed edge or d ir ect ed i ded equal V l y am ong a ll ver ti ces.Suppose the rat io be— t w een h e l t ong axi s a a n d t he shor t axi s b i s R ,f rom £ urc . he t equa t i on x / (Rb) + /6 =l a n d ig. F 8,one ca n get t h e l engt h of OP wh i ch con ect s t h e el l i pse cent er n d t a h e vert ex d ivi i ng t d h e angl e of c ̄cum f erence equal l y: 0P[ si n z r at i o CO S e , ∈[ 0, 2 )' e wher e OP[0]equa ls b,and i r anges f rom 0 t o t he Fi g.6 Arc algori thm 3 Com puter Impl em entati on num ber ofver texes m i nu s 1.The coord i nat es of P ca n be der ived by he t va lues of OP[i]a nd t he a ngl e臼: P[i]. x=0. x+OP[i]* s i n(i*臼) /( 180 /  ̄)), Al m ost al l sof tw a r e packages associ at ed w i h t t h e P[i]. Y=O. Y一0P[i]* COS(i*臼) /(1 80 /  ̄)) gr a ph t heor y use ad jacency mat r i x a s i n put.So i t i s ne ces s ar y t o se u ad jacenc y mat r i x f or dat a i nput,con— vert i t t o a wei ght ed d ir ect ed rel ati onshi p m atri x, and h e code segm ent u T s ed for com puti ng t he coord i— nates of ver ti ces i s l i sted a s f ol l ows. t hen u s e t h e al gor it hms a s descr ibed above t o draw a graph . . The a d j acency mat r i x cor r es pond ing t o a we i ght ed 。 i rect d ed m at ri x i n F i g. 4 i s gi ven i n F i g.7. Fig.8 Comput at ion of the l ength of OP and h e coordi t nates of P on ell i pse Program 1 angl e =2 pi/number o fve rt e x:/ /The ci r cum f er ence of el — l i pse i s di ded equa ll y by verti ces f or(i=0;i<numbe ro fve rt e x;{ ++ ) Fi g. 7 Ad j ace ncy mat r i x cor r espondi ng t o WDRM i n Fi g. 4 I t i s ea s y t o conver t a n ad jacency mat r i x i nt o 1 center. =420: ce n te r .Y = 260;//the coordinates 0f the el li pse’s center W DRM .One thi ng should be rem em bered i s that a val — ue — l i s used t o determ i ne t he d irect i on of edge i n WDRM .wh il e t he act ua l wei ght i n n a ad jacency mat r i x 印 [i]=r at i o*b/sq ̄(( si n (angl e) si n(angl e) +r at i o COS (angl e)*rat i o COS (angl e )) P [i].x =ce? t t r[ e 0].x +i nt(印 [i]*si n (i m ay i ndeed be — 1. Ther ef ore t i s w ei h ght cannot be put i nto W DRM d i rect l y i n order to avoi d am bi gui t y .I f angl e)); / /X coor di na t es of each ver t ex P [i].Y =ce nt r 0]. e 一i nt(印 [i]*cos(i* her t e a re negat i ve wei ght s i n t he ad ja cency mat r i x, one can si m pl y f ind t h e nf ini m um wei ght f irst, a n d cha n ge a ll wei ghts i n t h e m at ri x i nt o posi ti ve val ues by add i ng a posi t i ve val ue to t h e ori gi na l w ei ght s. Not e agai n t hat t h e ch nged va l ues i n t h e WDRM i s onl y s eful u i n det er m i m ng t he d irect ion of edge.The or igi — nal wei ght s m u st be u s ed when i m pl em ent ing a l go— ri t hm s i n t h e gr aph t heory or l abel i ng wei ght s. Thi s sof tw a r e i s i m pl em ent ed u si ng VC + + 6.I n or— angl e )); / / Y coor di nat es of each ve  ̄ex I I f M .i s t he source ver tex and M 。 t he desti nat ion vert ex t hen t h e code segm ent .u s ed for com puti ng the coor ina d t e s of t he ot her t wo ver t exe s(s ay M 3,M4)of h e arr t ow i s a s f ol l ow s. Program 2 s p =a t an2((M l Y —M 2.Y ),(M 1.x — M 2.x)); / / cor n— 维普资讯 http://www.cqvip.com 4l O Journal o fShanghai Uni ver si t y put e t he sl ope of edge f rom M l t o M 2; cos Y = co s ( sl opy ); Usi ng t he a l gor i t hms and t he above programs,we s i n Y = si n ( s泖 ); ca n get gr aphs ba s ed on t he adi acenc y ma t r i x a s M3.x=M2. x +i nt(Le ngt ho farr ow * C OS Y) 一 M l a s t he desti nat i on. show n i n F i g. 5. nt(Lengt i ho farrow /rat e * si n Y ); 3.Y=M2.Y + i nt(Le ngt ho farr ow si n Y) 4 Summ ary +i nt(Le ngt ho farro w /rat e * COS Y )); M4.x=M2. x +i nt(Lengt ho farr ow * co s Y) +i nt (Lengaofarr ow /rat e si n Y ); 4. Y=M2.Y — i nt(Lengt ho farr ow /r at e * COS Y) 一 nt(Le i ngt hof arr ow * s n Y ). i I n t hi s c ode s egment,l e ngt ho farr o w r e pr es ent s t he l e ng t h of ar r ow, a nd l e ngt ho farr o w/ rat e r e pr es ent s hal f widt h of ar row , as IM 4 or IM 3 shown i n Fi g. 3. An arr ow ca n be construct ed by connect i ng M 2, M 3, n d a M 4.Di f f er ent l eng th a n d i dt w h l ead t o ar r ows of I he proposed m ethod i s superi or t o ot her draw i ng s of t wa re s a i t ca n bui l d gr aph s di rec t l y f rom a n adj a— cency m at ri x,m aki ng i t possi bl e t o visua ll y i mpl em ent som e a l gor ithm s i n t he graph t heory such a s m i ni m um spanni ng t ree, short est pat h, a n d con ect e d com po— nent s .Therefore,t hi s sof tw a r e i s pr act i ca ll y useful i n i sua v li zed t e ach i ng of the graph t heor y a n d i n t he i m— pl em ent at i on of t h e a l gor it hm s i n the g r aph t heor y, wh i ch i s t he f oundat i on of m any engi neer ing appl i ca— t i ORS. vari ous shapes. The val ue of l engt ho farr o w ca n var y bet we en 1 / 4 t o 1 / 3 of t he l eng th of M l, M 2.The coor di nat e s of M6 i s Ref efences l l J Dong Li —mi n.Desi gn and real i zat i on of dynami c obj ect - cal cul ated usi ng t he fol l owi ng code segm ent to draw or i ent e d gr aph edi t or [ J]. Comput er Engi nee ri ng and an ar c .det er mi ned by M l,M 6,M 4,and M 2(see Fi g. Appl i cat i ons,2001,37( 3):l12— 11 5 (i n Chi nese). 6). 1 2 J Hua n g Ji ng— wei.Bi nar y t ree drawi ng al gor it hm based on genet i c a lgor it m lJ h J. Jour na l o f So tware,2000,11 f Program 3 (8):l 11 2一 l 117 (i n Chi nese). M6.x:M1.x + i nt(一Le ngt ho farrow * COS Y一 (Le ngt ho farr ow /r at e si n Y)); 6.Y:M1.Y + i nt(一Le ngt ho farro w * si n Y+(Le ngt ho— farr ow /r at e COS Y)). The coord i nat es of M 4 ca n be ca l cu l at e d by usi ng a ne w va lue of l engt ho farr o w i n Pr ogr am 2.Now we ge t coord i nates of 4 vert ices, w hich ca n be used to det er— mine an a r c f rom M 、t 0 M . The ar row at t he endpoi nt of a r c ca n be draw n by execut i ng P r ogr am 2 usi ng M 4 a s t h e source vert ex n d a 1 3 J Mao Guo— yong. Al gor it hms a n d i t s i mpl ement at i on i n buddi ng wei ght ed di r ec t ed gr aph 【 J].Jour na l f o Uni ve r- s o f Sh a ngh ai for Sci ence and Tec hnol ogy,2003,25 (3):263—266 (i n Chi nese). [4] Wang Shu— he.Gr aph Theor y n d a lgor A it m h [ M].Uni ver — si 毋 of Sci ence a n d Technol ogy of Chi na Pr ess, Hef ei , 1 990 (i n Chi nese). 1 5] Davi d J K,S c ot W ,Geor ge S.P r ogr an ̄ni ng Vi sua l C++ lM J.Fi f t h ed it i on,Bei j ng i Hope r es P s,Be i j i ng,1999(i n Chi nese). (Edi t or YA0 Yue - yuan)
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