Treemaps for Directed Acyclic Graphs

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1 Trmps or Dirt Ayli Grphs Vssilis Tsirs, Soi Trintilou, n Ionnis G. Tollis Institut o Computr Sin, Fountion or Rsrh n Thnology-Hlls, Vssilik Vouton, P.O. Box 1385, Hrklion, GR Gr {tsirs,strint,tollis}@is.orth.gr n Dprtmnt o Computr Sin, Univrsity o Crt, P.O. Box 2208, Hrklion, Crt, GR Gr {tsirs,strint,tollis}@s.uo.gr Astrt. Gn Ontology inormtion rlt to th iologil rol o gns is orgniz in hirrhil mnnr tht n rprsnt y irt yli grph (DAG). Trmps grphilly rprsnt hirrhil inormtion vi two-imnsionl rtngulr mp. Thy iintly isply lrg trs in limit srn sp. Trmps hv n us to visuliz th Gn Ontology y irst trnsorming th DAG into tr. Howvr this trnsormtion hs svrl unsirl ts suh s prouing trs with lrg numr o nos n sttring th rtngls ssoit with th uplits o no roun th srn. In this ppr w introu th prolm o visulizing DAG s trmp, w prsnt two spil ss, n w isuss omplxity rsults. Kywors: Trmp, Dirt Ayli Grph (DAG) Visuliztion, Gn Ontology. 1 Introution Th Gn Ontology Consortium (GO) [14] tss stor thousns o trms tht sri inormtion rlt to th iologil rol o gns. Th inormtion in GO is orgniz in hirrhil mnnr whr th trms r pl in lyrs tht go rom gnrl to spii. Th GO orgniztion n rprsnt y irt yli grph (DAG) whr th st o vrtis V is th st o trms n n g is us to lr th is or prt o rltionship twn two trms. Trmps grphilly rprsnt hirrhil inormtion vi two-imnsionl rtngulr mp, proviing ompt visul rprsnttions o lrg trs through r, olor n shing ts [3,5,11]. Trmps, hv lso n us to visuliz ompoun grphs tht ontin oth hirrhil rltions n jny rltions [7]. This work ws support in prt y INFOBIOMED o: IST n th Grk Gnrl Srtrit or Rsrh n Thnology unr Progrm ARIS- TEIA, Co 1308/B1/3.3.1/317/ S.-H. Hong, T. Nishizki, n W. Qun (Es.): GD 2007, LNCS 4875, pp , Springr-Vrlg Brlin Hilrg 2007

2 378 V. Tsirs, S. Trintilou, n I.G. Tollis In th ontxt o GO, trmps hv n us to visuliz mirorry t, whr h gn trnsript is ssign ll possil pths tht strt rom it n trmint to th root (th ll trm) o GO [1]. Symoniis t l. in [12] propos to ompos th omplt GO DAG into tr y upliting th nos with mny in-oming gs, n thn to us trmp lgorithm to visuliz th tr, s Figur 1. Th uplition o no howvr triggrs th uplition o ll o its snnts. Thror th trnsormtion o DAG into tr ls to trs with (potntilly xponntilly) mny mor nos thn th originl DAG. Symoniis t l. in [12] rport tht th initil GO DAG h trms, whil th prou quivlnt tr h trms. Anothr rwk o upliting th nos is tht th rtngls ssoit with th multipl rplis o no r sttr roun th srn. In this ppr w introu th Fig. 1. Exmpl o trnsorming DAG into tr n thn rwing it s trmp prolm o rwing DAG s trmp without onvrting it to tr irst. W onsir svrl vritions o th prolm, w prsnt som hrtriztions o simpl milis o DAGs tht mit suh rwing, n provi omplxity rsults or th gnrl prolm. 2 Prolm Dinition 2.1 Nottions Suppos tht G = (V,E) is lyr irt yli grph (DAG) with prtition o th no st V into susts L 1,L 2,...,L h, suh tht i (u, v) E, whr u L i n v L j,thni>j. Without loss o gnrlity w ssum tht th lyring is propr, sin th long gs tht spn mor thn two lyrs my rpl y pths hving ummy vrtis in th intrnl lyrs [2]. Lt R v not th isply rgion o no v V. Evry irt g = (u, v) rom no u to no v orrspons to rwing rgion R = R v R u whih is th prt o th hil s rwing rgion R v tht is rwn within th prnt s rwing rgion R u. Givn vrtx v V w not th st o its in-oming n out-going gs y Γ (v) ={ E/stintion() =v} n Γ + (v) ={ E/origin() =v} rsptivly.

3 2.2 Trmp Drwing Constrints Trmps or Dirt Ayli Grphs 379 Trmps hv th invrint tht th rwing rtngl o ny no (irnt rom th root) is ontin within th rwing rtngl o its prnt. Whn th grph is tr this invrint n sily stisi sin vry no hs on prnt. Whn th grph is DAG, th ov invrint shoul rpl y th invrint tht th rwing rtngl o ny no is ontin within th union o th rtngls o its prnt nos. Aprt rom this invrint it is plusil to ssum tht th rwing rtngls o siling nos o not ovrlp n tht th rwing rtngl o prnt no is ovr y th rwing rtngls o its hilrn nos. Th ov invrint n ssumptions r summriz in th ollowing inition. Dinition 1 (Trmp si rwing onstrints). Th rwing is onstrin y th ollowing ruls. B1. Th isply r o th DAG (srn) is rtngl. B2. Evry no is rwn s rtngl (R v is rtngl or vry v V ). B3. I two istint nos u, v V r ssign to th sm lyr thir rtngls o not ovrlp (r(r u R v )=0). B4. Th rtngl o hil no oupis non-zro r in h on o its prnt no rtngls. (r(r )=r(r u R v ) 0i =(u, v) E). B5. Th rtngl o hil no is ontin in th union o rtngls o its prnt nos (R v (u,v) E R u ). B6. Th rtngl o prnt no is ovr y th rtngls o its hilrn nos (R u (u,v) E R v ). Th rwing ruls o th ov inition r quit gnrl sin thy o not onstrin th r o th l nos, n th proportion o hil s no r tht is rwn on h on o its prnt rtngls. To simpliy th nlysis o th prolm w onstrint ths two prmtrs y mking th ollowing ssumptions. Dinition 2 (Trmp itionl rwing onstrints) A1. Th l nos r rwn in qul r ( srn r numr o l nos )rtngls. A2. Th rwing rtngl o hil no oupis th sm r on h on o its prnt rtngls (For vry non sour no v, r(r )= r(rv) Γ (v), or vry Γ (v)). In th ollowing w will us th trm trmp rwing to hrtriz rwing oring to th si n itionl rwing onstrints. Hving in th rwing ruls, w n in th ollowing prolms: 1. Givn DAG G 1,osG 1 mit trmp rwing? 2. In s tht th nswr to th irst prolm is ngtiv, wht is th minimum numr o no uplitions tht r n to trnsorm G 1 intodag G 2 tht mits trmp rwing?

4 380 V. Tsirs, S. Trintilou, n I.G. Tollis 2.3 Exmpls n Countr-Exmpls o DAGs Tht Amit Trmp Drwing Exmpls o DAGs tht mit trmp rwing ppr in Figur 2. From th () Th K 33 DAG () A irulr DAG Fig. 2. Exmpls o DAGs tht n rwn s trmps g h i j () Consutiv onitions violtion () Ar imln Fig. 3. Exmpls o DAGs tht o not mit trmp rwing ountr-xmpls o Figur 3 w s tht thr r DAGs tht nnot rwn s trmps. Th DAG in Figur 3() nnot rwn u to jny onstrint violtion. Th l nos,, g, h, i, j onstrin th prnt nos,,, to rwn in jnt rtngls. Howvr w nnot hv onigurtion whr ll th pirs {, },{, },{, },{, },{, },{, } o rtngls r jnt. In this s in orr to rw th DAG w n ithr uplit on o th nos,, g, h, i, j or rw on o ths nos using two isjoint rtngls. Th two oprtions r similr n whn ppli to hil no rmov th orrsponing jny onstrint. In gnrl, u to th our olor (mp oloring) thorm thr xists ountrxmpl involving iv prnt nos n tn hilrn nos (on hil no or vry pir o prnt nos), vn in th s tht w rlx onstrint B2, llowing rwings to simply onnt rgions o th pln. Th xmpl o Figur 3() shows DAG tht os not mit trmp rwing, u to r imln mong th irst lyr nos,,. Assuming

5 Trmps or Dirt Ayli Grphs 381 tht th l nos hv unit r, thn nos n hv r 1/2+1/3 whil no hs r 1 + 1/3. Howvr, i w rlx onstrint A2, thn this DAG mits trmp rwing. 2.4 No Duplition Usully, DAG nountr in prti os not mit trmp rwing. In this s w shoul rlx on or mor o onstrints B1-B6, A1-A2 or hng th orm o th DAG. Symoniis t l. in [12] hos to trnsorm th DAG into orst o trs y multipl no uplitions. An xmpl o no uplition is shown in Figur 3(), whr tr th rtion o two rplis o no, on with two prnts n on with on prnt, th DAG is trnsorm into nw DAG whih mits rwing, s Figur Fig. 4. Atr th uplition o no th DAG o Figur 3 is trnsorm into DAG tht hs trmp rwing Spil Css W will ontinu y onsiring two spil ss. Th irst s is s on rstrit orm o DAGs, th son on rstrit orm o trmps. 3.1 Two Trminl Sris Prlll Digrphs A Two Trminl Sris Prlll (TTSP) igrph is rursivly in s ollows [2,13]. An g joining two vrtis is TTSP igrph. Lt G 1 n G 2 two TTSP igrphs. Thir sris n prlll ompositions, in low, r lso TTSP igrphs. Th sris omposition o G 1 n G 2 is th igrph otin y intiying th sink o G 1 with th sour o G 2. Th prlll omposition o G 1 n G 2 is th igrph otin y intiying th sour o G 1 with th sour o G 2 n th sink o G 1 with th sink o G 2. Du to its rursiv strutur TTSP igrph lwys mits trmp rwing. Th s TTSP igrph is rwn s rtngl. In sris omposition th rtngl o grph G 2 is rwn on th top o th rtngl o grph G 1.In prlll omposition th rtngl o th omposit grph is sli into th rtngls o G 1 n G 2.

6 382 V. Tsirs, S. Trintilou, n I.G. Tollis Algorithm 1. Construt th omposition tr o G [13] n mrg th jnt P -nos. In th rsulting tr th P -nos my hv two or mor hilrn. 2. Using th omposition tr lult th siz o th omponnts. () In sris omposition siz(g) =siz(g 1 )=siz(g 2 ). () In prlll omposition siz(g) =siz(g 1 )+siz(g 2 )+...+siz(g k ). 3. Using th omposition tr rursivly rw th omponnt rtngls. () Th rtngls hv r proportionl to th siz o th orrsponing omponnt. () In sris omposition, th rtngls o th two omponnts oini. () In prlll omposition, us ny o th xisting trmp lgorithms to ly out th omponnt rtngls. G 1 G 1 () Th s TTSP igrph G 2 G 2 G 2 is rwn on top o G 1 () Sris omposition G 1 G 2 G 1 G2 () Prlll omposition Fig. 5. Rursiv inition o TTSP igrph n th orrsponing rursiv trmp rwings () A TTSP () Sli n Di lyout () Squrii lyout Fig. 6. Exmpl o TTSP igrph trmp rwing 3.2 On Dimnsionl Trmps Dinition 3. A trmp is ll on imnsionl i th rtngl rprsnting no is ivi with vrtil (or horizontl) lins into smllr rtngls rprsnting its hilrn n th orinttion o th lins is th sm or ll th nos o hirrhy.

7 Trmps or Dirt Ayli Grphs 383 Sin th hight (rsp. with) o ll th rtngls is onstnt n qul to th hight (rsp. with) o th srn, th prolm is on imnsionl n th rtngls R q n rprsnt y intrvls I q. Also only th orring n not lngth o th intrvls I = I u I v, =(u, v) E onstrin th prolm. For this rson w onsir th out-going gs o vry non-l no u V s sunos insi th no. Th in-oming gs o no n onsir s in-oming gs o vry on o its sunos. A rwing n L k I n o th sunos n L k o lyr k {2,...,h} orrspons to n orring o th sunos n L k. Fig. 7. A on imnsionl trmp xmpl. A suno is rt or h out-going g. Dinition 4 (ONE DIMENSIONAL TREEMAP FOR DAG). Th rognition prolm INSTANCE: ADAGG. QUESTION: Cn G rwn s on imnsionl trmp? Suppos tht u is no in lyr L k. W will giv th nssry n suiint onitions tht th nstor sunos o u must stisy in orr to l to rw I u s n intrvl. With th trm nstor sunos w mn th sunos rhl rom no u i th irtion o th gs is rvrs. Lt P u,i not th st o nstor sunos o no u L k in lyr i {k +1,...,h}. Thorm 1 (Nssry onitions). Suppos tht in on imnsionl trmp rwing o grph G =(V,E) nou L k n rwn s n intrvl I u. Thn th union o th rwings o th nstor sunos o u in lyr i, Pu,i I, is n intrvl or vry lyr i {k +1,...,h}. Proo. By inution on th lyrs L k+j, j =0,...,h k. For j =0,I u is n intrvl y th hypothsis. Now, suppos tht or j = 0,...,i < h k, thr is n orring o th sunos in vry on o th lyrs k + j,...h, suh tht Pu,k+j I to n intrvl, ut Pu,k+j+1 I nnot n intrvl. Thn thr is t lst on no v L k+j+1, v / P u,k+j+1, whih is twn two nos α, β P u,k+j+1.thn th intrvl Pu,k+j I intrsts th intrvls I α n I β ut not th intrvl I v, ontrition. lyr k+j+1 I I v I lyr k+j I Pu, k j

8 384 V. Tsirs, S. Trintilou, n I.G. Tollis Thorm 2 (Suiint onition). I thr is n orring o th sunos in L k+1 suh tht th prnt sunos P u,k+1,onou L k, k<hr onsutiv in this orring thn u n rwn s n intrvl. Proo. Sin P u,k+1 r onsutiv Pu,k+1 I is n intrvl. Thn simply rw I u in this intrvl. Algorithm From th ov thorms vry non-sour no u L k ins onstrints on th missil suno prmuttions in h on o th lyrs L i, i {k,...,h}. Thror th ision prolm is trnsorm to h 1 onsutiv ons ision prolms [4,9]. On onsutiv ons prolm or h lyr L i, i {2,...,h}. Thr is on list list i o onstrints or h lyr L i, i {2,...,h}. Initilly th lists r mpty. Thn w th onstrints s ollows. or i =2toh o or v L i o to th list i th onstrint tht th sunos o v r onsutiv. or i =1toh 1o or v L i o or j = i +1toh o to th list j th onstrint tht th sunos P v,j r onsutiv. Complxity nlysis Without loss o gnrlity w ssum tht thr r no l nos in lyrs 2,...,h. Suppos tht n i = L i, i {1,...,h} n tht m i gs go rom lyr i to lyr i 1, i {2,...,h}. For i {2,...,h} th list list i hs t most n i trivil onstrints n t most n j onstrints u to nos t lyr L j, j < i. In totl it hs n i n 1 onstrints. Eh ostrint hs siz t most m i. Thror th totl tim is: h h O(m i (n i n 1 )) = O(m i n) =O(m n) i=2 whih is polynomil on th input siz m = L i=1 m i n n = L i=1 n i 4 Th Gnrl Cs 4.1 Th Rognition Prolm Tking th nos o lyr L k isolt rom th rst o th DAG, th prolm is similr to loorpln prolm whr th isply r is isst into n k = L k i=2

9 Trmps or Dirt Ayli Grphs 385 sot rtngls, i.., rtngls whos r is ix ut thir imnsions my vry. Th numr o possil isstions (th solutions sp) is oun low y Ω(n k!2 3n /n 4 k )novyo(n k!2 5n /n 4.5 k )[10]. Consiring two onsutiv lyrs L k+1 n L k o DAG, th lyouts o th two lyrs r onstrin y th gs mong th two lyrs, oring to th rwing ruls. Th omin solution sp my mpty or ontin numr o solutions. W will show tht iing whthr th solution sp is mpty or not is NP-omplt. Dinition 5 (TREEMAP FOR DAG). Th rognition prolm INSTANCE: ADAGG. QUESTION: Cn G rwn s trmp? Thorm 3. Th TREEMAP FOR DAG ision prolm is NP-omplt vn i w rstrit it to two lyr wkly onnt DAGs. Proo. Givn isstion o th isply r (srn) into L k rtngls or h lyr L k, k {1,...,h} o DAG G, w n hk in polynomil tim i ths isstions orrspon to trmp rwing o G. Thror th prolm longs to NP. Nxt w will show tht th prolm TREEMAP FOR DAG is NP-hr. Th proo will on y ruing th 3-PARTITION prolm to rstrit vrsion o th TREEMAP FOR DAG prolm. Nmly, s input w onsir only two lyr DAGs. For simpliity o th proo w will llow n input DAG to ompos o svrl wkly onnt omponnts. Dinition 6 (3-PARTITION). INSTANCE: AmultistA o 3m positiv intgrs A = {α 1,α 2,...,α 3m } whr th α i s r oun ov y polynomil in m n Σ 4 < α i < Σ 2,whr Σ = 1 m (α 1 + α α 3m ). QUESTION: Cn A prtition into m tripls A 1,A 2,..., A m suh tht h tripl hs th sm sum. Spiilly h tripl must sum to Σ. Th onition Σ 4 <α i < Σ 2 ors vry st o α i s summing to Σ, tohv siz xtly 3. Th 3-PARTITION is strongly NP-omplt sin it rmins NPomplt vn whn rprsnting th numrs in th input instn in unry [6]. Th rution is on y lol rplmnt n using n norr. Enorr: Th DAG us s norr hs 2m+2 nos t th son lyr. Th nos β n 1, 2,...,2m + 1. At th irst lyr thr r (m +1)Σ +6m +4 nos. Eh o th nos 1, 2,...,(2m +1) hs two prnts. On is no β n th othr is th orrsponing numr no in th irst lyr. Th β-no rtngl is rwn in on si o th norr n prlus ny othr rtngl to rwn long this si. Also th β-no togthr with th γ-no or th nos 1, 2,...,2m + 1 to rwn s onsutiv rtngls. For vry pir o (j, j + 1) o son lyr nos thr xists irst lyr no whih hs thm s prnts n onstrins thm to onsutiv. Th son no whih hs s prnts th nos j n j + 1 is us or omplting th rwing (grg

10 386 V. Tsirs, S. Trintilou, n I.G. Tollis us th othr si & lignmnt norr m 2m m 2m+1 2m+1 nos lignmnt norr grg olltion jny norr & grg olltion Fig. 8. Th norr us in th proo nos sum norr Lyr 2 nos m 2m+1 Lyr 1 nos m 2m+1 Fig. 9. On possil rwing o th norr. Th rtngls 1, 2,...,2m + 1 r or y th rtngl γ to hv th sm with. olltion). Also or grg olltion on no is onnt to no 1 n on no to no 2m +1. Finlly, vry o numr no o th son lyr hs Σ hilrn nos. Lol rplmnt: For h α i A w onsir two lyr DAG whih hs on no t lyr two (prnt no) n α i nos t lyr on (hilrn nos). lyr 2 i lyr 2 lyr 1 i nos Th rwing o no i is rtngl with r α i, ut without ny onstrint on th spt rtio o its sis. In orr or th inl rwing to rtngl th 3m rtngls shoul ill th hols o th norr. Th rution rom th 3-PARTITION to TREEMAP FOR DAG uss polynomil numr o rsours sin th numrs involv in 3-PARTITION r oun y polynomil in m. Th 3-PARTITION instn hs solution i n only th TREEMAP FOR DAG instn hs solution sin in vry hol n it xtly thr rtngls.

11 4.2 Minimiztion o No Duplition Trmps or Dirt Ayli Grphs 387 Dinition 7 (MINIMUM DUPLICATION OF NODES). INSTANCE: ADAGG 1 n n intgr K. QUESTION: Cn G 1 trnsormintodagg 2 tht mits trmp rwing y upliting t most K nos. Commnt: Th prolm MINIMUM DUPLICATION OF NODES is NPomplt sin its rstrition or K = 0 is th prolm TREEMAP FOR DAG, whih is NP-omplt. 5 Disussion In this ppr w introu th prolm o rwing DAG s trmp. W in th rognition n minimiztion prolms n w show tht in th gnrl s thy r NP-omplt n NP-hr rsptivly. W lso onsir two spil ss y rstriting th orm o th DAG n o th trmp rsptivly. W r urrntly invstigting rwing huristis s on rlxtions o th rwing onstrints n\or rstritions on th orm o DAGs. Th rsults o this rsrh will pulish in susqunt ppr onrning th pplition o ths thniqus to hirrhilly orgniz ontologis. Rrns 1. Bhrk, E., Dng, N., Bri, K., Shnirmn, B.: Visuliztion n nlysis o mirorry n gn ontology t with trmps. BMC Bioinormtis 5(84) (2004) 2. Di Bttist, G., Es, P., Tmssi, R., Tollis, I.G.: Grph Drwing: Algorithms or th Visuliztion o grphs. Prnti - Hll, Nw Jrsy, U.S.A. (1998) 3. Brson, B., Shnirmn, B., Wttnrg, M.: Orr n quntum trmps: Mking tiv us o 2D sp to isply hirrhis. ACM Trnstions on Grphis 21(4), 833 (2002) 4. Booth, S., Lukr, S.: Tsting or th onsutiv ons proprty, intrvl grphs, n grph plnrity using PQ-tr lgorithms. Journl o Computr n Systm Sins 13, 335 (1976) 5. Bruls, M., Huizing, K., vn Wijk, J.J.: Squrii trmps. In: Proings o Joint Eurogrphis n IEEE TCVG Symposium on Visuliztion, p. 33. Springr, Hilrg (2000) 6. Gry, M., Johnson, D.: Complxity rsults or multiprossor shuling unr rsour onstrints. SIAM Journl on Computing 4(4), 397 (1975) 7. Holtn, D.: Hirrhil Eg Bunls: Visuliztion o Ajny Rltions in Hirrhil Dt. IEEE Trnstions on Visuliztion n Computr Grphis 12(5), 741 (2006) 8. Hsu, W.-L.: PC-Trs vs. PQ-Trs. In: Wng, J. (.) COCOON LNCS, vol. 2108, p Springr, Hilrg (2001) 9. Minis, J., Porto, O., Tlls, G.: On th onsutiv ons proprty. Disrt Appli Mthmtis 88, 325 (1998)

12 388 V. Tsirs, S. Trintilou, n I.G. Tollis 10. Shn, Z.C., Chu, C.: Bouns on th Numr o Sliing, Mosi, n Gnrl Floorplns. IEEE Trnstions on Computr-Ai Dsign o Intgrt Ciruits n Systms 22(10), 1354 (2003) 11. Shnirmn, B.: Tr visuliztion with tr-mps: 2- sp-illing pproh. ACM Trnstions on Grphis 11(1), 92 (1992) 12. Symoniis, A., Tollis, I., Rzko, M.: Visuliztion o Funtionl Aspts o mirorna Rgultory Ntworks Using th Gn Ontology. In: Mglvrs, N., Chouvr, I., Koutkis, V., Brus, R. (s.) ISBMDA LNCS (LNBI), vol. 4345, pp Springr, Hilrg (2006) 13. Vls, J., Trjn, R., Lwlr, E.L.: Th rognition o Sris Prlll igrphs. SIAM Journl on Computing 11, (1982) 14.