On the Complexity of Graph Cuboidal Dual Problems for 3-D Floorplanning of Integrated Circuit Design

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1 On Complxiy o Grp Cuoil Dul Prolm or -D Floorplnnin o Inr Circui Din Rnn Wn Uniriy o Cliorni, Sn Dio L Joll, CA 99- rn@c.uc.u Cun-Kun Cn Uniriy o Cliorni, Sn Dio L Joll, CA 99- ckcn@uc.u ABSTRACT Ti ppr icu impc o mirin rom -D o -D on loorplnnin n plcmn. By lookin ic ormulion o rp cuoil ul prolm, o -D c n -lyr.-d c r unmnlly mor iicul n -D c in rm o compuionl complxiy. By comprion mon c, inrinic complxiy in -D loorpln rucur i rl in r-ciin rlion n opoloicl conncion n omricl conc. T rul o uur clln or pyicl in n CAD o -D inr circui. Cori n Sujc Dcripor J.6 [Compur Applicion]: Compur-i in Gnrl rm Alorim, ory Kyor -D inr circui, cuoil ul, compuionl complxiy. INTRODUCTION N cnoloi uc r-imnionl inrion r comin n orc kpin Moor l ill olin in nno r. By in imnion in currn -D VLSI circui, cn rly nnc inrion niy n ruc inrconncion ir ln, ic lp o impro ym prormnc n lor por conumpion. Mnil, xr imnion lo rin ir complxiy in in, CAD ool n ricion. To ully xploi n o ir imnion in -D inr circui, ir n o mur n unrn complxiy i rin, n c clln o nlin i complxiy. Plcmn o circui lock i n imporn p o in, ic lr complxiy incrmn mirin rom -D o -D. Currn lopin -D circui n ym-on-cip [8] r uully ci y i ckin, ic i ck o - D circui lyr i m ickn. Ti yp o plcmn i lo cll.-d plcmn [], [] inc i o no conin ull -D rucur. Full -D loorpln n plcmn rprnion r x- Prmiion o mk iil or r copi o ll or pr o i ork or pronl or clroom u i rn iou proi copi r no m or iriu or proi or commrcil n n copi r i noic n ull ciion on ir p. To copy ori, o rpuli, o po on rr or o ririu o li, rquir prior pciic prmiion n/or. GLSVLSI 9, My, 9, Boon, Mcu, USA. Copyri 9 ACM /9/ $.. plor in rl ork inc []. Full -D mn circui lock r cuoi plc in pc i no iinuil lyr. Tou y no -D cll lirry o uppor i cl o -D IC in, r r ull -D pplicion in rconiurl FPGA [9] r im i rr nor imnion. I i oun mo o loorpln rprnion ci in -D o no qully ci xnion in -D, uc Squnc-pir (-D) o Squnc-ripl (-D) in []. Sinc rprnion i irully rucur rom ic loorpln cn rcor, ry o xplor i complxiy rou nrl yp o rucur, rp. In i ork, icu complxiy o o cl o -D loorpln rou cuoil ul prolm in mo ic ormulion: Gin rp G = (V, E), cn in o cuoi V i conc rlion E? T prolm i imilr o rcnulr ul prolm in [6], xcp oluion in [6] mu rcnulr icion iou mpy pc. Opimizion on i prolm cn ppli on iniil loorplnnin o pyicl in. For xmpl, i pir o circui lock i, j r ily connc, l ( i, j ) E o mk m clor; or i o lock o i por niy, mk ( i, j ) / E or r iipion. A -D rcnulr ul cn ci y o coniion in [6] or [], n cn icinly nr in linr im. For cuoil ul, il -D c cn ol i imilr pproc, in -D c r unmnlly mor iicul in rm o compuionl complxiy. Lik -coloriliy prolm i y u -coloriliy i NP-r, on xr color or imnion rin ir ll o complxiy. In c pro -D cuoil ul prolm i NP-compl y rucin -coloriliy o i. For.-D cuoil ul, in i NP-compl n numr o lyr rc. T rul imply complxiy o IC pyicl in cn rly incr n xn circui on ir imnion, n or ju lyr o.-d circui. T r o i ppr i orniz ollo. Scion inrouc ic prolm ormulion. Scion pro nrl -D cuoil ul prolm i r n cion o.-d cuoil ul i lyr i r. Finlly cion mk comprion n concluion on rul.. PRELIMINARIES Triionl -D loorplnnin i o plc o rcnl in in r o m crin rquirmn. T ic conrin i no common r cn r y o or

2 mor rcnl. For -D, prolm com plcin o cuoi in pc iou common pc r y mulipl cuoi. A -D c cn rr -D c i c cuoi plc on loor. An jcncy rp cn conruc rom loorpln y inin rx o c cuoi n ( i, j ) n o corrponin cuoi r concin on urc. Wil i conrucion i y, rr conrucion rom rp o loorpln i no riil. In [6] r i o uicin n ncry coniion or rp o n jcncy rp o rcnulr icion. T icion i cll rcnulr ul o rp. For -D, in prolm on rp cuoil ul. () A rp n i cuoil ul () K no cuoil ul Fiur : Grp-loorpln rlion A nrl -D cuoil ul o n n-rx rp G = (V, E) i in o cuoi, c cuoi C i corrpon o rx i V. No o cuoi r common pr o pc. C i n C j r jcn (concin on urc y non-zro r) i n only i ( i, j ) E. Fiur o 6-rx rp n on o i cuoil ul, n -rx compl rp no cuoil ul. A.-D cuoil ul i in -D cuoil ul i n iionl conrin ry cuoi i inrl [l, l], r l i lyr inicin inr. A -D cuoil ul i in.-d cuoil ul i on lyr, i.. ry cuoi i plc in i inrl [,]. I i irn rom rcnulr ul [6] in o cuoi cn u o pc icion. Our ic prolm i o in cuoil ul o in rp G. For ny o -D,.-D or -D c, prolm i riilly in NP, cu i i y o riy i in o cuoi i oluion, i.. o cck r or c pir o i, j, ( i, j ) E C i n C j r concin on urc.. -D CUBOIDAL DUAL OF GENERAL GRAPHS To ci r rp -D cuoil ul i NP-r. W pro i y rucin ll knon NPcompl prolm, -coloriliy, o -D cuoil ul. W conruc G rom -coloriliy innc G C = (W, E ). Fir inrouc o 7 ric or c rx in W, on in iur. T 7 ric or i orm n ocron compo o 8 rron. Tr i cuoil ul o i rp, n conc urc n irn pir o cuoi r no inpnn. Lmm. In cuoil ul o 7-rx, cuoi o o oppoi ric on ocron (.., ) r on oppoi i o cnrl cuoi (o ). Proo. (Bri) Sinc n r jcn, cuoi r concin on common pln no p, n ir orlppin urc on p i rcnl R (iur ). Sinc our urrounin ric in loop r ll jcn o Fiur : 7-rx n i cuoil ul n, cuoi mu concin oulin o R n ror on i o R. An inc i lo jcn o urrounin ric, projcion o cuoi on p mu corin R. Tror n cuoi on oppoi i o. So or 7-rx N, concin ircion o n r m, no, (N). In m y, or o pir o ric (, ) n (, ) r on oppoi i o. Alo rom iur, cuoi o,,, cor ll 6 urc o cuoi. Rrin coorin xi, (N) i prlll o, i r poil ircion: x, y n z. T ircion cn u poil color in -coloriliy prolm, r N i color, (N). For (, ) E in G C, o ric cnno r m color. Ti conrin cn rliz icliqu n n o o N n N. A iur o, on xi prlll o,(n), occupi inrl [, ] n occupi inrl [, ]. I r i icliqu n {, } n {, }, n o n mu cor inrl [, ] on xi, o, (N ) cnno prlll o, (N). ( ) = ( ),, ( ), Fiur : Enorcin in irn ircion To compl rucion rom -coloriliy n o conruc G C on no. W 6 mor ric o 7-rx o urr rric concin ircion mon cuoi o,,. 6 Fiur : -rx n i cuoil ul Lmm. (iur ) Ain pir o ric o 7- rx, pir connc o {,, } n {,, }, pir connc o {,, } n {,, }, pir connc o {,, } n {,, }, n cuoi n m i cuoi (lon,6 ), 6 p

3 cuoi n m i cuoi (lon, ), cuoi n m ln cuoi (lon, ). In -rx r, oriinl 7-rx ini p, o cn ily lin mulipl ino m ircion i om iionl ric in G. A 6 B Fiur : To -rx i,(n) n,(n ) lin o m ircion (-linmn) A in iur (uin implii ocron o rprn -rx ), o ric A n B. Conir ir conncion i N on ri. Sinc A i imulnouly concin, n, y Lmm n iur, i mu on cornr orm y cuoi, n cuoi A i ror o. Similrly, B i concin, n, o i mu on cornr n cuoi B i lo. A rul, ircion A B i m, (N ). T m concluion cn oun on N, i.. ircion A B i m,(n). Tror i o iionl ric mk, (N) =, (N ). Bi linmn o,, lo n o lin o o ircion,,, n,6 o o r ll in prlll. () 6 () Fiur 6: -linmn n (compl)-linmn Fiur 6() i implii noion o linmn illur in iur, r only ircion o,(n) n, (N ) r prllliz. W cll i -linmn. In iur 6() r r iionl ric (cll -linmn), rul i, (N) =, (N ) n,6 (N) =,6 (N ), ic lo impli,(n) =,(N ). So in -linmn, o r complly lin in ry ircion. Alo noic ircion rom on o or in -linmn i lon, (N) or,6 (N), il i ircion in -linmn mu lon, (N). T o c nl linmn o pir o -rx N n N lon ny o x, i,(n) =,(N ). B on o linmn cn conruc conncion n -rx in G C = (W, E ) or rucion rom -coloriliy prolm. Torm. -coloriliy ruc o -D cuoil ul. Proo. Gin rp rom -coloriliy G C = (W, E ) i n ric no,,, n. For c rx i, conruc n -rx no in G, no 6 i,,, i,n, qunilly connc y -linmn. Tn or c no i,j, conruc uxiliry ollo: i,j -lin i,i,j,,i,j -lin i,i,j,,i,j -lin i,i,j, n inlly,i,j -lin i u i,j. For c ( i, j ) E, pick no u i,j n u j,i, connc { (u i,j ), (u i,j )} i { (u j,i ), (u j,i )} o o ric orm icliqu. In i y rp G cuoil ul i n only i G C i -colorl. I G C i no -colorl, n no mr o plc, r i l on pir o ric i n j uc ( i, j ) E, n, (u i,j ) =, (u j,i ). So (u i,j ) (u i,j ) n (u j,i ) (u j,i ) r on m ircion, n y iur i i impoil o orm icliqu n. y x k i, ( k, ) = y: i, ( k, ) = z: n,k, k, k, k,,k,,k, Fiur 7: Conrucion o cuoil ul rom G n G C i -colorl I G C i -colorl, cn conruc cuoil ul ccorin o iur 7. Vric,, n r plc on xy-pln n iur 7 i op i. Ec rx k color o {x, y, z}, ic ci ircion o no,( i,j). Ery in E i in uniqu i o conncin cuoi o no inrr. (i) I, ( i, ) i prlll o z, uxiliry {,i,j } cn plc lon lin, n y -linmn c,i,j i lr o i o ( i, j). (ii) Ori, ( i, ) i prlll o x or y, n c,i,j i lr o i o ( i, j ), n y - linmn o {,i,j } r y op i plc lon lin. In concluion, y lyou o iur 7, uxiliry {,i,j } cn plc lon lin y op i. For ny i, j uc ( i, j ) E n, ( i, ), ( j, ), cn ly conruc,i,j,i,j u i,j lon x,,j,i,j,i u j,i lon y, or ic r. So u i,j n u j,i cn m inrcin poin n orm icliqu o { (u i,j ), (u i,j )} n { (u j,i ), (u j,i )}. Tror cuoil ul o G i uccully conruc. Corollry. Finin rp -D cuoil ul i NP-compl.. LAYERED -D (.-D) CUBOIDAL DUAL OF LAYERED GRAPHS In l cion o nrl -D cuoil ul i r. No look.-d rion o prolm ic look l iicul.. -D Cuoil Dul o Plnr Grp T -D rcnulr ul prolm i ir ui in [6] n []. By uin -complion rp, impl rul o

4 ci i rp G rcnulr ul i Torm o [6]: A pln rp G i ll inrior c rinulr rcnulr ul i n only i r xi -complion o G. On our iniion o cuoil ul, ciin rul cn mor nrl n implii, iou uin -complion. Torm. A rp G -D cuoil ul i n only i G cn rn pln rp i no -rx cycl coninin inrior rx (ric). Ti cn pro y conrin in rp o - complion, ic i urn o rcnulr ul y [6], n cuoil ul cn n ily oin. T lo i on in iur 8. Conrucion lorim in linr im o rcnulr ul r inrouc in [] n []. c i j k l c n i j k l () Oroonl conc () Dimon Fiur : Bic in.-d cuoil ul W in n rp G lyr, ciin i.-d cuoil ul i no l iicul n Plnr -SAT, ic i pro o NP-compl in [7]. -SAT i ic NP-compl prolm inrouc in []. A Plnr -SAT innc m o ril U = {u,..., u n } n m o clu C = {c,..., c m } -SAT. Rrin c ril n clu rx, in (u i, c j) i clu c j conin u i, rulin rp G psat i plnr rp. p i n c i j k l c i j k l Fiur 8: From rp o i -D cuoil ul Corollry. Finin rp -D cuoil ul i in P...-D Cuoil Dul o Lyr Grp In prolm r, r in lyr rp G = (V, E, n, L : V {,, n}), i c rx in lyr n c ir in lyr or n o concui lyr, i.. ( i, j ) E L( i ) L( j ). T.-D cuoil ul i -D cuoil ul c cuoi i mu on lyr L( i). Fiur 9 o n xmpl. Lyr Lyr c c Fiur 9: A -lyr rp n.-d cuoil ul T rricion on cuoi n conc ruc rom o concin ircion. For ( i, j ), i i n j r on m lyr, concin ircion only coic. Y lo om ic inrouc complxiy. A iur () o, i o ric on lyr i n o ric on lyr i + r complly connc cliqu K, n in cuoil ul conc urc n o cuoi in lyr i mu oroonl o on in lyr i +. Bcu in iur, i o pir m ircion, compl conncion i impoil. T imon in iur () i imilr o 7-rx in lmm n iur, xcp i i in -D. p () Clu () 6-rx Fiur : -lyr urp o clu A or clu c i cn conruc in iur (), r i ric r on lyr n lck ric r on lyr. To pir o ric p n p on lyr r norc o oroonl conc urc y imon on lyr. Mnil o pir r lo connc rou r 6-rx ic olloin propri. Aum ircion o o uc i rmin, (iur ()) (i) i i on m ircion, i.. n r on m i o, 6-rx c imon, o mu on oppoi i o ; (ii) i i no on m i o, inc i on mor rx n imon, rom o in on on o o i o. No look iur (), rin rom ricl pir p. T ir ricl u o oroonliy norcmn rom p. By m norcmn, o i prlll o o, n o i prlll o o. Finlly o i orizonl rx pir p. Wi conncion, i ll 6-rx r ricl, n ll r lik imon, rulin in o, n mu ll ricl, ic l o conricion, i...-d cuoil ul o no xi. Ori i l on 6-rx i orizonl, n cn plc orizonl on i, n olloin lo orizonl. Rrl o ircion o on olloin, cn ly mk on oppoi i o, i.. orizonl. By

5 i propion, o i orizonl n.-d cuoil ul o iur () cn conruc. In ummry, -lyr urp o iur ().- D cuoil ul i n only i l on 6-rx orizonl. Ti mk rucion rom Plnr -SAT ri orr, inc in -SAT clu i ru i n only i l on o i ril i ru. Torm. Plnr -SAT ruc o.-d cuoil ul i lyr. Proo. W conruc -lyr rp G = (V, E,, L : V {,, }) rom G psat = (U C, E ) on in iur. Only o ric r on lyr, ic r u o lin m pir o ric on lyr, y ic ll m clu r lin in m ircion. Aum ricl ircion r i o pir on lyr, i i rn in iur. n imon r plc on lyr or n ril u,..., u n. u u u u n c c c m Fiur : Rucion rom Plnr -SAT o -lyr.-d cuoil ul For c u i pprin in c j, connc i imon D(u i) o 6-rx in j clu rou (m + n) imon on lyr, ic r connc i rx pir on lyr. T ir pir p comin ou o D(u i ) i in ircion, (D(u i )) i u i ppr u i in c j, ori in ircion, (D(u i )). In i y, rx pir o u i n u i r ly oroonl. T r cur in iur rprn conncion o rx pir p p p m+n+. Ec imon ( ) o nl n norc 9 urn, ir l or ri, n inl rx pir p m+n+ i oroonl o o 6-rx. Wn r i oluion o plnr -SAT ooln ormul, plc D(u i ) i, (D(u i )) ricl i u i i ru, or, (D(u i )) ricl i u i i l. Tn or ry clu l on 6-rx cn conruc i orizonl, o.-d cuoil ul o clu i conruc. Ec conncion cn plc rou r i (m + n) 9 urn roun m + n ocl. An inc rp G psat i plnr, conncion no inrcion. Alou y op i, conncion my inrc i cuoi on lyr, cn ly pick conncion cuoi on lyr o cro inrcion, n r i no conc n lyr n lyr. So.-D cuoil ul o G i ully conruc. Wn r i no oluion o plnr -SAT ormul, no mr o plc D(u ),, D(u n), r i ly clu i ll 6-rx lin in m ircion n ror inconrucil. Corollry. Finin lyr rp.-d cuoil ul i NP-compl i numr o lyr.. CONCLUSIONS W look r cuoil ul prolm o irn imnion, n com o rul o on icin lorim n o rn proo. Nurlly, iiculy o prolm mirin rom -D o -D i incrin. Dimnion Numr o lyr Hrn -D P.-D opn.-d NP-compl -D NP-compl A urpriin inin i ju lyr o - D c, ic cn ci y impl rul (Torm, or [6], []), in ck or, cn mk prolm o muc mor complx r i no ci lorim cn ci oluion, unl P=NP. T rlion n opoloicl conncion n omricl conc in -D loorpln i no inri n xn o -D rucur. Ti my lo xplin y -D pckin innc r mor iicul o nco or rprn n -D innc. Wi muc incr complxiy in -D rucur, xpc i clln or o inr n CAD ool lopr in uur -D IC in. Humn inllinc ill ply mor imporn rol in in lo n in iin uriic lorim in -D loorplnnin, plcmn n rouin ool. Furr rrc ill lpul o unrn nur o -D pyicl in prolm. 6. REFERENCES [] J. Bkr n S. Sni. A linr lorim o in rcnulr ul o plnr rinul rp. ACM/IEEE DAC, p 8, 986. [] S. A. Cook. T complxiy o orm-proin procur. r ACM STOC, p 8, 97. [] Y. Dn n W. Mly. Pyicl in o.d ck ym. IEEE ICCD, p 7,. [] B. Gopln n S. Spnkr. Plcmn o D IC i rml n inrlyr i conirion. ACM/IEEE DAC, p 66 6, 7. [] G. Kn n X. H. Rulr llin o -connc pln rp n i pplicion in rp rin prolm. Toricl Compur Scinc 7, p 7 9, 997. [6] K. Kozminki n E. Kinnn. An lorim or inin rcnulr ul o plnr rp or u in r plnnin or VLSI inr circui. ACM/IEEE DAC, p 6 66, 98. [7] D. Licnin. Plnr ormul n ir u. SIAM J. Compu. Volum, Iu, p 9, 98. [8] S. K. Lim. Pyicl in or D ym-on-pck: Clln n opporunii. IEEE DATE, p 6,. [9] G.-M. Wu, J.-M. Lin, n Y.-W. Cn. An lorim or ynmiclly rconiurl FPGA plcmn. IEEE ICCD, p,. [] H. Ymzki, K. Sknui, S. Nkk, n Y. Kjini. T D-pckin y m rucur n pckin uriic. IEICE Trn. Funmnl, p 69 6, Apr..

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