Using Graph-Grammar Parsing for Automatic Graph Drawing

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1 Using Grph-Grmmr Prsing for Automti Grph Drwing C.L. MCrery C.L. Coms D.H. Gill Dept. of Computer Siene n Engineering Equifx Inorporte The MITRE Corportion Auurn University Atlnt, Georgi MLen, Virgini.0 Introution Direte grphs, or igrphs, re n exellent mens of onveying the struture n opertion of mny types of systems. They re ple of representing not only the overll struture of suh system, ut lso the smllest etils in simple n effetive wy. However, rwing igrphs y hn n e teious n time onsuming, espeilly if the numer of verties n rs is lrge. In ition, muh time n e spent just trying to pln how the grph shoul e orgnie on the pge. These fts revel the nee for n utomte system ple of onverting textul esription of igrph into well orgnie n rele rwing of the igrph. Mny reserhers hve stuie this prolem n mny grph rwing systems hve een evelope [see 3 for omplete list]. The estheti riteri of the systems vry. The ojetives my inlue requirements of uniform ege length, minimum numer of ege rossings, stright eges, gri rwings (eges re either horiontl or vertil), miniml ens in the eges, minimum re overe, n isply of symmetries. Some limit the input grphs to prtiulr lss suh s plnr grphs, trees, grphs with mximum egree of four, or some pplition-speifi grphs suh s petri nets, network representtion, igitl system shemti igrms, PERT igrms, flowhrts, et. CG (Cln-se Grph Drwing Tool) is tool we use for our work with progrm epeneny grphs. Like ot [0] n its preeessor DAG [9], CG tkes textul esription of n ritrry irete grph n proues visul representtion of it. CG uses unique grph prsing metho to etermine intrinsi sustrutures in the grph n to proue prse tree. The

2 tree is given ttriutes tht speify the noe lyout. One exmple ttriute set is presente in this pper, ut the lyout n e tilore to suit ifferent estheti riteri. CG then uses tree properties n pts the work of Suigym [2], Wrfiel[ 3], Eens [7] n Frin [8] in routing the eges. The ojetives of the system re to provie n esthetilly plesing visul lyout for ritrry irete grphs. CG inorportes tehniques to reue ege rossings, gurntee few ens in long rs, n to shorten long rs whenever possile. CG is the first grph rwing tool to use grph grmmr s the funmentl struture tht esries the noe lyout. Brnenurg [] efines lyout grph grmmr s grph grmmr together with lyout speifition. The lyout speifition ssoites finite set of lyout onstrints with eh proution. Our pproh is to lssify the proutions of the grph prse n ssoite lyout ttriutes with eh proution type in the prse tree of the grph. 2.0 Noe Lyout The noe lyout is etermine y the omintion of () prsing of the grph into logilly ohesive sugrphs n (2) efining lyout ttriutes to pply to the resulting prse tree. The prse is se on simple grph grmmr, n the ttriutes tht re now progrmme into CG proue lyout whose noes re lne oth vertilly n horiontlly. 2. Grph Deomposition Cln-se grph eomposition (CGD) is prse of irete yli grph (DAG) into hierrhy of sugrphs. The sugrphs efine y our grph grmmr n prse re lle lns. Let G e DAG. A suset X G is ln iff for ll x, y " X n ll " G - X, () is n nestor of x iff is n nestor of y, n () is esennt of x iff is esennt of y. An lternte esription of ln epits it s suset of verties where every element not in the suset is relte in the sme wy (i.e. nestor, esennt or neither) to eh memer in the suset. Trivil lns inlue singleton sets n the entire grph. In Figure, sets {2,3}, {2,3,4,5,6},

3 {,2,3,4,5,6}, n {2,3,4,5,6,7} re the nontrivil lns. C={2,3} is ln sine vertex is n nestor of eh element of C n 5 n 7 re esennts of eh element of C. The set {2,3,4} is not ln sine 6 is esennt of only vertex Figure. Clns A simple ln C, with more thn three verties, is lssifie s one of three types. It is (i) primitive if the only lns in C re the trivil lns; (ii) inepenent if every sugrph of C is ln; or (iii) liner if for every pir of verties x n y in C, x is n nestor or esennt of y. Inepenent lns re sets of isolte verties. Liner lns re sequenes of one or more verties v i,v i+,...,v j-,v j where for i < k, v i is n nestor of v k. Any grph n e onstrute from these simple lns. Grph-grmmrs String grmmrs re speil se of grph-grmmrs. A string is isomorphi to liner grph, n in proution of the string grmmr, the replement string is onnete to the host string in nturl wy. In sequentil grph rewriting system or grph-grmmr, grphs re generte from some initil grph y proutions where the mother grph, sugrph of the host grph, is reple y nother grph, the ughter grph. The min prolem of grph-grmmrs is speifying the eges tht onnet the ughter grph to the host grph. The speifition of the eges onneting the

4 ughter grph to the host grph is lle the emeing rule. More formlly, grph-grmmr is system GG = (N, T,, P, H) where N n T re the nonterminl n terminl vertex lels, resp., is vertex lle the xiom or strt grph, P is set of proutions n H is n emeing rule. The prtiulr grph-grmmr of the work in this pper is lle the ln genertion system, CGS. All host n ughter grphs in CGS re Hsse grphs, i.e. irete grphs with no trnsitive eges. The xiom is single vertex. P is set of pirs (v,d) where v is ny grph vertex n D is primitive, inepenent or liner ln. For CGS, the reonnetion rule or emeing rule is hereity. All host n ughter grphs in CGS re Hsse grphs, i.e. irete grphs with no trnsitive eges. An emeing is lle hereitry when in-eges to the mother grph, single vertex, re reple with in-eges to the soures of the ughter grph n out-eges from the mother noe re reple with out-eges from the sinks of the ughter grph. More formlly, let the mother grph e vertex u. For eh vertex v in the set of soure verties in the ughter grph, (w,v) is n ege in the resultnt grph whenever (w,u) is n ege in the host grph. For sink verties, t, of the ughter grph,(t,w) is n ege in the resultnt grph whenever (u,w) is n ege in the host grph G m = 3 D () G G m = 3 D G () Figure 2. Proution exmples

5 Let us ll proution of this system CGS-proution. In pplitions of CGS-proutions, the ughter grph eomes ln in the resultnt grph. Figure 2 shows exmple proutions with the mother vertex ientifie s vertex 3. Quotient grph The onept of quotient grph is importnt euse it permits the lssifition of omplex lns into the tegories of liner, primitive or inepenent. Let C e ln n with {C,,...C k } prtition of C where eh C i is mximl proper suln of C. The quotient grph of C enote C/C...C k, is the grph with verties C,...,C k. Pir(C i,c j ) is n ege of C/C..C k iff x " C i, y " C j, n (x,y) is n ege of C. In Figure, the lns tht prtition the entire grph re: C = {}, C 2 = {2,3,4,5,6}, C 3 = {7}, n the quotient grph is liner. The lns tht prtition C 2 re {2,3}, {4}, {5}, n {6} n the quotient grph is primitive. Every ln n e ientifie s liner, primitive or inepenent oring to the lssifition of its quotient grph. Figure 3() shows the prse tree of the grph of Figure. Theorem : When the originl grph is Hsse grph, the grph generte y CGS is lso Hsse grph []. Theorem 2: Any Hsse grph n e generte y unique nonil sequene of proutions from CGS [6]. Theorem 3: For every Hsse grph there is unique eomposition into quotient grphs tht re ientifie s liner, primitive or inepenent [5,6]. Theorem 3 gurntees the existene of unique prse tree from the lns of grph. In the grph lyout lgorithm, the prse tree is use to speify the grph noe lotions. A ouning ox ttriute for eh tree vertex, x, speifies the length n with of retngle ontining the grph noes in x s sutree. The length n with re ompute for eh liner n eh inepenent ln from the ttriutes of the ln s hilren. The hilren of liner ln re isplye vertilly n the hilren of horiontl ln re isplye horiontlly in our exmple lyout.

6 Deomposing Primitives Primitive lns represent speil hllenge. They o not fll into the ler-ut tegories of verties tht shoul e li out horiontlly or li out vertilly. In ition, primitive lns n e ritrrily lrge, n there n e inefinitely mny of them. While this poses no speil hllenge to prsing, n every irete grph n e prse into lns from the three ln types, ritrry grph genertion is prolem sine list of ll primitive grphs must e inlue in the proution possiilities to rete ny speifi grph. Our pproh is to eliminte primitive lns y eomposing them further into hierrhy of pseuo-inepenent n liner lns. By ing eges from ll the soure noes of primitive to the union of the hilren of the soures, liner ln L is forme. L is ompose of ln of the soure noes (whih is ientifie s pseuo-inepenent) n the reminer of the primitive ln. The proeure then reursively prses the reminer of the primitive. This metho extrts series-prllel onstruts from primitive lns n represents the seril n prllel strutures in the prse tree. The pseuo-inepenent lns re trete s inepenent lns for lyout purposes sine their noes re not onnete. The prse tree of ny ompletely eompose grph is iprtite tree where the internl verties re lssifie s either liner or (pseuo-) inepenent. For onnete grph, the root vertex is lwys liner. In Figure, the primitive {2,3,4,5,6} eomposes into liner onnetion etween the inepenent lns {2,3,4} n {5,6}. The fully eompose prse tree is shown in Figure 3(). The iprtite prse tree n e eorte with lyout ttriutes tht n e synthesie to proue n esthetilly plesing rwing. 2.2 Extension to Cyli Grphs A simple trnsformtion is require to pply the grph eomposition metho to yli grphs. Cyles n e foun in epth-first grph trversl. To rek yle, the r tht ientifies the yle is given the reverse orienttion. When the lyout is rey, its orienttion will e orrete.

7 L = liner ln (L) P 7 L = primitive ln (P) = inepenent (I) or pseuo-inepenent (i) I i i () prse tree () prse tree with primitive remove Figure 3. Prse trees 2.3 Sptil Anlysis The prse tree of the grph n e given vriety of geometri interprettions. For exmple, retngle or ouning ox with known with n length n e ssoite with eh ln. Syntheti ttriutes n e ssoite with the prse tree hierrhy to show the emeing of the ouning oxes. For illustrtive purposes we hoose ttriutes we ll nturl tht give lne lyout, oth vertilly n horiontlly Bouning Boxes A simple two-imensionl lger efines the syntheti ttriutes tht speify the ouning oxes. The intrinsi ttriutes of singleton DAG noes (or equivlently prse tree leves) esrie unit squre ouning oxes. Liner lns require n re whose length is the sum of the lengths of the omponent lns n whose with is the mximum with of the omponent lns. Inepenent lns require n re whose with is the sum of the withs of the omponent lns n whose length is the mximum of the lengths of the omponent lns.

8 Definition: Denote the ouning ox ttriute of noe N in prse tree T y (L.N,W.N). We efine the nturl vlues of the ttriute to e: () (L.N, W.N) = (,), if N hs no hilren; (2) (L.N, W.N) <-- (L.C +...+L.C k, Mx(W.C,...,W.C k )), if N is liner noe with hilren C...C k; (3) (L.N, W.N) <--- ( Mx(L.C +...+L.C k,w.c,...,w.c k )), if N is n inepenent noe with hilren C...Ck. As n exmple onsier the prse tree in Figure 4. The ouning ox imensions re inite y the (i,j) pirs. 0 (5,2) Fig- v0 = liner ln = inepenent ln = vertex (,) (3,2) (,) v0 v6 (3,) (2,) 2 3 (,) (,) (,) (,) (,) v v3 v5 v2 v4 ure 4. Prse tree with lele ouning ox lels Noe Plement To hieve n esthetilly plesing lyout, the noes re entere within the ouning oxes. For hil C of liner noe N, the tul retngle in whih the noe is to e entere hs length L.C n with W.N. For hil D of inepenent noe I, the retngle in whih D is to e entere hs length L.I n with W.D. In Figure 4, for exmple, ln 3 is entere in (3,) ouning ox n vertex v0 is entere in (,2) ouning ox. Figure 5 shows the noe lyout for the prse tree of Figure 4.

9 v0 v v2 v3 v4 v5 v6 Figure 5. Bouning oxes Cln Expnsion n Contrtion Sine lns re efine s groups of noes with ientil onnetions to the rest of the grph, lns n esily e ontrte to single noe. Any noe not in the ln tht ws onnete to ln soure or sink will e onnete to the ontrte noe. By llowing segments of the grph to e ontte, the user n simplify grphs for viewing y ontrting those prts whih re not relevnt to the investigtion. Contrte noes n e expne to show the originl ln onfigurtion. Similrly, it is possile to isply only the ln, ignoring the rest of the grph. One of the mjor ifferenes etween the grphs rwn y the nturlly ttriute CG n hierrhil systems [2][2] is tht the noes in CG's grphs re spe in lne wy oth vertilly n horiontlly. Hierrhil rwings prtition noes into levels, n ll noes of the sme level re ple on the sme horiontl xis. Noes ten to unh towr the top of the grph is these systems. By using our grph eomposition tehnique, CG is le to etermine lne vertil sping s well s lne horiontl sping. 3.0 Drwing Ars An importnt onsiertion tht went into the formultion of the r rwing lgorithm ws the question of whih pirs of verties n hve rs etween them. The nswer lies in the efinition of liner n inepenent lns. The verties of liner ln n hve rs etween

10 them even if the verties re not onseutive in the lns. Within inepenent lns no two verties my e onnete y n r. For these resons, the lgorithm nee only onsier routing rs etween verties within ommon liner ln. There re two mjor prolems to e solve in routing rs: reuing ege rossings n long r prolems. Prouing optiml ege rossings is NP-hr, even for noes in only two levels [4]. CG moifies the Bryentri orering tehnique of Wrfiel [3] to e pplie to lns, s well s iniviul noes. The long r prolems inlue reuing r lengths whenever possile n the proper routing of rs tht onnet noes not ple on jent levels. To voi visul onflits, long rs my hve to e route roun noes from intermeite levels. Bens in the rs re often neessry for this routing, ut my use visul onfusion. Bens shoul e minimie. 3. Routing Long Ars Before noes ssume their finl position in the lyout, their lotion is etermine y onsiering plement tht will shorten long rs when possile or strighten rs for those tht nnot e shortene. To esrie the metho, severl efinitions re require. The level of noe x, l(x), in DAG G is the length of the longest pth from soure to x. The height of noe x, h(x), is the length of the longest pth from sink of the grph to x. The height of DAG G, H(G), is the mximum of the noe heights or, equivlently, the mximum of the noe levels. The length of r (x,y), L(x,y) = l(y) - l(x). An r, (x,y),is lle long if L(x,y) >. Otherwise, it is lle short. Theorem 4: In DAG G, if l(x) + h(x) = H(G), then there exist pth from soure to sink k,,...x= i,... k, suh tht L( j, j+ )= for # j < k. We sy noe x is speifie if l(x) + h(x) = H(G). The vertil positioning of speifie noe is etermine sine it is ontine in pth whose rs re ll of length. Short eges re

11 route iretly to verties while long rs re route through flse verties emee in lns etween the two verties. Sine ens re hr to follow n istrting to the eye, our strtegy gurntees tht trnsitive rs will hve t most two ens. Furthermore, no eges will ross long rs exept possily t either en of the r. 3.. Reogniing Long Ars A long r is inient to noe x iff there exists noe y suh tht (x,y) "E or (y,x) "E n $l(y)-l(x)$>. If the the long r, (x,y), is trnsitive r, it is reple y ummy noe n rs (x,) n (,y). When the ugmente grph is prse, is omponent in n inepenent suln of the ln ontining x n y. The exmple grph in Figure 6 illustrtes this sitution. The only trnsitive r is (,). Here, noe n rs (,) n (,) re e, n r (,) is elete. The ugmente prse inlues new inepenent ln, {,, } with omponents {,} n {}. The ugmente prse tree is shown in figure 6. Sine the only role of is tht of ple holer for the ummy noe, resonle moifition of the ttriute grmmr woul e to give the ummy noe smller with. The lyout in 6(e) illustrtes the se where the ouning ox of is ssigne the imensions (, 0.5). () originl grph () prse tree () grph with ugmenting noe x () new prse tree (e) lyout Figure 6. Routing trnsitive r

12 3..2 Positioning Noes Inient with Long Ars For non-trnsitive long rs inient t x, if l(x) + h(x) < H(G), then some flexiility exists for justing the vertil positioning of x. The vertil positioning will e moifie in one of three wys: (i) x will e entere etween preeessor n suessor, (ii) x will e ple one level elow its preeessor, or (iii) x will e ple one level ove its suessor. If there is suessor of x n preeessor of x tht re oth speifie, then x will e ple miwy etween, seprting two long rs. If only suessor or preeessor, y, is speifie, x will e move s lose s possile to y. These moifitions re me y inserting ummy noes into the prse tree, ompleting the norml lyout, routing the eges through the ummy noes n then removing the unneessry noes. The ummy noes re inlue with x in liner ln. The ition of the liner lns insures the noes with e ple in vertilly jent positions. Speifilly, the prse tree trnsformtions will e: (i) if ll preeessors, w, n ll suessors,, of x re stisfie, let = min %[l() - l(w) - 2] / 2&, where the minimum is tken over ll w, jent to x. Noe x is reple in the prse tree y liner with l() - l(w) - noes: ummy noes re ple to the left of x n l() - l(w) ummy noes re ple to the right of x. Figure 7 illustrtes the proess. In the figure, r (,) is trnsitive r. The grph is ugmente with noe x to remove the trnsitive ege. Now rs (x,) n (e,) re long. Furthermore ll preeessors n suessors of x n e re stisfie. Noe x in the prse tree is reple y liner ln with = (5-0-2)/2 = ummy noe to the left of x n l() - l() = 2 ummy noes to the right of x. Similrly, e is reple y liner ln with 2 noes. (ii) if there exists y suh tht y is speifie, (x,y) "E, n l(y) - l(x) >, let = min[l(y)-l(x)-], where the min is tken over ll y jent to x. Reple noe x in the prse tree y liner noe with + hilren: ummy noes to the left of x. Figure 8 illustrtes this

13 solution. Here (,) is long r tht is not trnsitive. = 3, n prse tree noe is reple y liner ln with 3 ummy noes followe y x. (See Figure 8()). Note tht neither the ummy noes nor the potentil eges etween them re require for the lyout. (5,3) = (length, with) % = level = height 2 %0 5 %0 5 e # x f g $ 3 e f g () prse tree # # 4 4 & e () originl grph (6,3) & ' 2 f g () ugmente prse tree (4,3) (4,2) (4,) " (,2) (2,) (2,2) (2,) " x " " " e f " " g " e f g " () ugmente prse tree with ummy liner Figure 7. Lyout for long trnsitive ege (e) lyout (iii) if there exists noe y suh tht y is speifie, (y,x) "E, n l(x) - l(y) >, let = min[(l(y)- l(x)-2]. Reple noe x in the prse tree y liner noe with + hilren: noe x is the left-most hil n ummy noes re to its right. Though the prse tree hnges n lrger ouning oxes re ssigne to these liner noes, it is importnt to leve the ouning oxes of the nestors unhnge. No itionl spe is require for the ummy noes. They re ple in the grph only for the purpose of routing eges. One ll noes n ummy noes re ple, ll rs re short n their plement follows.

14 (0,) (0,4) (5,4) e (,3) (,3) f (,2) (4,4) (,0) (,0) g (2,2) (2,2) h %0 # # %0 (4,2) () originl grph i (3,) () Prse Tree (,2) (,2) (,) (,) & ' i 0 (4,0) # # 3 e 3 f g h # $ 2 2 (5,4) (,2) (4,4) ( ( ( 2 (4,) " " " (4,2) i ( ( ( ( e f g h i () trnsforme prse tree e f g h () lyout Figure 8. Lyout to shorten long ege 3.2 The Bryentri Metho n its Apttion to Clns The Bryentri metho is heuristi for reuing the numer of ege rossings in two onseutive levels of grph. Two vlues lle the row ryenter n the olumn ryenter quntify the horiontl istne etween jent verties. Reuing these istnes proues fewer ege rossings. The metho trverses the levels of the grph n reursively omputes the ryenter for jent levels strting with level ero. With eh omputtion, the noes on one level re permute to reue the row ryenter vlue. The proess is repete etween top n ottom lyers until no reorering ours or limit on the numer of omputtions hs een rehe.

15 Apting ryenters to lns involves repling the jent levels of hierrhy y jent inepenent ln omponents of liner lns. The proess first inspets the smllest liner lns in the prse tree (those losest to the leves). The ryenter metho is use on non-trivil jent inepenent sulns of these liner lns. One the proper orering hs een estlishe for the noes within tht liner ln, tht orering remins fixe. The proess ontinues with the quotient grph where the liner noe hs een ontrte. Ientifying n using only the liner lns results in signifint reution in the mount of omputtion require to perform the ryentri metho. In the originl ryenter metho, the ryenters re ompute on mtries of sie r i *r i+ where r i is the numer of verties in level i n the omputtion requires r i *r i+ multiplitions n (r i -)*r i+ itions t eh level for eh itertion etween the top n ottom levels. With the pplition to the prse tree, fter the proper plement of the inepenent lns within liner, there is only one noe tht represents the entire struture of the liner ln. 3.3 Smoothing Ars. The CG system elimintes shrp orners y using B-splines. When n r onnets n tul vertex with ummy vertex, the spline tehnique esrie y Enns [7] n Frin [8] is use. See Figures 9, 0 n The Implementtion The CG system hs een implemente on Sun Mirosystems SPARCsttion running the SunOS 4..2 operting system. It ws written in C++ n uses the InterViews toolkit evelope t Stnfor University. This toolkit provies n ojet-oriente interfe to the X Winow System. Using this interfe, the system genertes rwing of irete grph from textul esription. The winows in Figures 9, 0, n ontin the igrph isplys generte y the

16 CG system. The verties of the igrph re represente y irles, eh of whih ontins the numer of the vertex. Verties n e selete y liking on them with the left mouse utton. This tion highlights the vertex y reversing its foregroun n kgroun olors. When ll the omponents of ln hve een selete, the user my then ontrt the ln. Figure 9 shows lyout n the results of ontrting lns {, 2} ' {3, 4, 5}, n {6, 7, 8, 9, 0}. The ontrte lns n e selete extly like the verties. The expn utton n then e use to expn selete lns to revel their ontents. This expnsion n ontrtion of lns llows the user to onentrte on speifi sugrphs while rowsing the igrph. Figure 9. Grph with 5 noes n ontrtions

17 Figure 0. Rnom DAG with 33 Noes

18 Figure. Rnom DAG with 3 noes.

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