Monotonic Prllel nd Orthogonl Routing for Single-Lyer Bll Grid Arry Pckges Yoichi Tomiok Atsushi Tkhshi Deprtment of Communictions nd Integrted Systems, Tokyo Institute of Technology 2 12 1 S3 58 Ookym, Meguro-ku, Tokyo, 152 8550 Jpn Tel: +81-3-5734-2665 Fx: +81-3-5734-2902 e-mil: {yoichi,tsushi}@lb.ss.titech.c.jp Abstrct In this pper, we give the necessry nd sufficient condition tht ll nets cn be connected by monotonic routes when net consists of finger nd bll nd fingers re on the two prllel boundries of the Bll Grid Arry pckge, nd propose monotonic routing method bsed on this condition. Moreover, we give necessry condition nd sufficient condition when fingers re on the two orthogonl boundries, nd propose monotonic routing method bsed on the necessry condition. I. Introduction Bll Grid Arry (BGA) pckges s shown in Fig. 1, in which I/O pins re plced in grid rry pttern, relize number of connections between chips nd the printed circuit bord (PCB). Bonding fingers re connected to chips, nd solder blls re I/O pins of the pckge in grid rry pttern. Since the structure of BGA pckges is simple, mny routes cn be relized in few lyers in the pckges if connection requirements nd routing ptterns re suitble for the structure. In current pckge routing design, the designer genertes stisfctory routing ptterns by using the properties of connection requirements effectively. But it tkes much time for lrge pckges since the huge number of routes needs to be relized. So, the demnd for utomtion of pckge routing is incresing. In this pper, we consider routing for single-lyer BGA pckge s the first step for BGA pckges routing. In the literture on for plnr routing, there re lot of problem formultions nd pproches. For exmple, problem formultions for single-row nd double-row routing, where terminls re plced on single-row nd doublerow, re proposed in [1] nd [2], respectively. Though these problem formultions re similr to problems for single-lyer BGA pckges, pproches for them re not enough to obtin stisfctory routes for BGA pckges. Actully, mny prts of the routing process for BGA pckges re relized mnully with support tools. In order to obtin stisfctory routing pttern, the nlysis of mnul routing ptterns is necessry. In the routing pttern by mnul, though routes my snke, most of them do not go bck. The route which do not go bck re sid to be monotonic. In monotonic routing ptterns, it is expected tht the totl wire length tends to be smll, nd it is esy to decide the route of ech net. But there exists netlist tht cnnot be relized by monotonic routes in one lyer with ny design rule. There lso exists netlist in which design rule my be stisfied if non-monotonic routes re llowed. In these cses, nonmonotonic routes re needed. Though we im to relize nets in one lyer under certin design rule, in this pper we propose n pproch in which ll nets re relized by monotonic routes. The obtined monotonic routes will be n initil solution in itertive improvement to stisfy the design rule. In litertures for BGA pckge, severl pproches focusing on monotonic routes were proposed. The first pproch for single-lyer BGA pckges ws proposed in [3] nd it ws improved in [4]. Their pproch genertes optiml uniform distribution of wire by generting connection requirements. An pproch for 2-lyer BGA pckges ws proposed in [5]. It is given connection requirements, nd optimizes the totl wire length nd the wire congestion by improving vi ssignment. Also, severl pproches considering non-monotonic routes were proposed. The pproch for multilyer Pin Grid Arry (PGA) nd Bll Grid Arry pckges were proposed in [6] nd [7], respectively. They ssign ech net to lyer, nd relize nets in respective lyer. All of them divide the pckge into severl sectors, nd nets re relized within ech sector. Bsiclly, ech sector consists of bonding fingers on the sme boundry of the pckge nd solder blls, nd net in ech sector consists of bonding finger nd solder bll. Nmely, their pproch cnnot be pplied if it is impossible to divide the pckge into such sectors. So, we propose n pproch for the region consisting of solder blls nd bonding fingers on two boundries of the pckge s shown in Fig. 2. Section II introduces routing model, nd gives some definitions for nlysis. Connection requirements re the set of nets nd re clled netlist. A prllel netlist, in which bonding fingers re on two prllel boundries of the pckge, is shown in Fig. 2(). In section III, we give
bonding wire Orthogonl chip bonding finger Prllel S bv bs b H b b bh S b V b S b Fig. 3. Reltionship between nd b solder bll Fig. 1. A Bll Grid Arry pckge () Prllel Netlist Fig. 2. Monotonic Netlists Decision Problem (b) Orthogonl Netlist the necessry nd sufficient condition tht ll nets in the prllel netlist cn be relized by monotonic routes, nd propose monotonic routing method bsed on this condition. An orthogonl netlist, in which bonding fingers re on two orthogonl boundries of the pckge, is shown in Fig. 2(b). In section IV, we give necessry condition nd sufficient condition tht ll nets in the orthogonl netlist cn be relized by monotonic routes, nd propose monotonic routing method bsed on the proposed necessry condition. There my exist more thn one monotonic routing pttern tht corresponds to prllel netlist nd n orthogonl netlist, respectively. How to select one mong them tht meets the design rule, if it exists, is in our future works. Since it is not gurnteed tht our routing method for orthogonl netlists completes routing, we implement our method for orthogonl netlists with C++ lnguge, nd pplied it to orthogonl netlists in section V. Section VI concludes this pper. A. Definitions II. Preliminry In this pper, we ssume tht the BGA pckge hs connection requirements between bounding fingers plced on boundries of the pckge nd solder blls plced in grid rry pttern. A solder bll, which we will refer to s bll, is n I/O pin of the pckge nd is connected to the PCB. A bonding finger, which we will refer to s finger, is connected to the chip by bonding wire. In this pper, we ssume tht ll nets re two-terminl nets connecting finger to bll. A netlist is the set of such nets nd is represented by N. We refer to finger plced on bottom boundry of the pckge s bottom finger, nd refer to net which consists of bottom finger nd bllsbottomnet. Similrly,topnetndleft net re defined. Bottom nets nd top nets re lbeled ccording to the order of fingers from the left to the right s,,,...nd,,,..., respectively. Left nets re lbeled ccording to the order of fingers from the bottom to the top s,,,....letb, T nd L be the sets of bottom, top nd left nets, respectively. We define the reltion between nets ccording to their bll positions s shown in Fig. 3. Let (x,y )nd(x b,y b ) be the coordintes of the blls of nets nd b, respectively. The reltion between nd b is defined s follows. If x <x b,y = y b then is sid to be to the left of b nd the reltion is represented by H b. If x = x b,y <y b then is sid to be below b nd the reltion is represented by V b. If x <x b,y <y b then is sid to be to the lowerleft of b nd the reltion is represented by S b. If x <x b,y >y b then is sid to be to the upperleft of b nd the reltion is represented by S b. b H, b V, b S nd b S re defined symmetriclly. B. Order Grphs Weusesomeordergrphswherevertexv corresponds to net v N. The edge from vertex u to vertex v is represented by the ordered pir (u, v). In this pper, every order grph hs edges corresponding to the order of fingers in ech boundry. Ef b,et f nd El f re the sets of edges corresponding to the order in bottom, top nd left boundries, respectively. Formlly, they re given s follows: Ef b = {(b i,b j ) b i,b j B,i<j}, Ef t = {(t i,t j ) t i,t j T,i<j}, Ef l = {(l i,l j ) l i,l j L,i<j}. C. Monotonic Routes A boundry, where the finger of net is plced, is clled the finger boundry of the net. For exmple the finger
b 7 b 7 b 6 b 9 b 9 b 6 b 8 b 8 t4 b 6 b 7 b 8 b 9 b 6 b 7 b 8 b 9 () All routes re monotonic (b) R(b 6) is non-monotonic Fig. 5. MPN Decision Problem Fig. 4. Monotonic nd non-monotonic routes boundry of in Fig. 2() is the bottom boundry. In this pper, monotonic route nd non-monotonic route re defined s follows: Definition 1 If the route from finger to bll intersects ny stright lines running prllel with the finger boundry t most once, then the route is sid to be monotonic. Otherwise the route is sid to be non-monotonic. Let R(v) be the route of net v N. All routes re monotonic in Fig. 4(), but R(b 6 ) is non-monotonic in Fig. 4(b). If ll nets in netlist cn be relized by monotonic routes without intersecting ech other, the netlist is sid to be monotonic. A netlist is sid to be single if the fingers of nets in the netlist re plced on the sme boundry. Consider tht nets consist of bottom fingers nd blls s shown in Fig. 4. A single netlist is monotonic if nd only if nets on ech row re in incresing order. Since netlist in Fig. 4() stisfies this condition, it is monotonic. On the other hnd, either R(b 6 )orr(b 9 ) is non-monotonic in Fig. 4(b) since b 6 nd b 9 re in decresing order. A monotonic routing pttern for monotonic single netlist, in which ll routes re monotonic, is unique. Similr observtions re found in [3, 4, 5]. Whether single netlist is monotonic is decided by the order grph G S. E(G S ) consists of Ef b nd edges tht correspond to the order of nets on ech row. For exmple in Fig. 4(), G S hs edges (, ), (,b 8 )nd(,b 8 )corresponding to the bottom row. Similrly, G S hs edges for other rows. Clerly, G S is cyclic if nd only if there exist nets on row which re in decresing order, such s b 6 nd b 9 in Fig. 4 (b). III. Prllel Netlists A prllel netlist is netlist in which fingers re plced on the two prllel boundries of the pckge. In this section, we nlyze prllel netlists. A. Monotonic Prllel Netlists The Monotonic Prllel Netlist (MPN) Decision Problem is defined s follows: Definition 2 MPN Decision Problem Input: A prllel netlist. Question: Is it possible to relize ll connection requirements by monotonic routes? An exmple of MPN Decision Problem is given in Fig. 2(). In this cse, N = B T. The necessry nd sufficient condition for being monotonic is tht nets on ech row re in incresing order without distinguishing bottom nd top nets. This condition is represented by the order grph G P. E(G P ) consists of Ef b, Et f nd edges corresponding to the order of nets on ech row. In Fig. 5, G P hs edges (, ), (, )nd(, ) corresponding to the bottom row. Similrly, G P hs edges for other rows. In Fig. 5, the trnsitive edges like (, ) re omitted. A prllel netlist is monotonic if nd only if G P is cyclic. Theorem 1 A prllel netlist is monotonic if nd only if the order grph G P is cyclic, where E(G P )=E b f Et f E p nd E p = {(x, y) x, y N, x H y }. Proof. If the order grph G P is cyclic, then n order cn be obtined by G P. A monotonic routing pttern cn be relized ccording to the order s shown in section III.B. Conversely, consider tht G P hs cycle C. If C consists of only bottom nets, then non-monotonic routes re needed since it mens tht bottom nets re in decresing order on row. The sme discussion is possible for top nets. So we ssume tht C consists of bottom nets nd top nets. Without loss of generlity, we ssume tht C hs (b j,t p )nd(t q,b i ), where b i,b j B (i <j)nd t p,t q T (p <q). Since bj H tp nd tq H bi, non-monotonic routes re needed by t lest one of them s shown in Fig. 6. B. A Prllel Routing Method A prtil order is defined by G P, nd some orders re obtined by the prtil order. This order corresponds to n order such tht the sources in G P re removed one by one. For exmple, the following order
t p t q b j t p t q b i b i b j Fig. 6. bj H tp tq H bi Fig. 7. An exmple of monotonic routing pttern for MPN Fig. 8. Exmples of MON Decision Problem nd it s monotonic routing pttern O is obtined for Fig. 5. According to the obtined order, we put virtul fingers on bottom boundry of the pckge for top nets nd put virtul fingers on top boundry for bottom nets. All nets cn be relized becuse we cn connect finger to its virtul finger vi its bll by monotonic route one by one from the left. An exmple of solution re given in Fig. 7. For monotonic prllel netlist, there re severl monotonic routing ptterns since the order obtined by G P is not unique in generl. The selection of the order tht reduces the density is in our future work. IV. Orthogonl Netlists An orthogonl netlist is netlist in which fingers re plced on the two orthogonl boundries of the pckge. In this section, we nlyze orthogonl netlists. A. Monotonic Orthogonl Netlists The Monotonic Orthogonl Netlist (MON) Decision Problem is defined s follows: Definition 3 MON Decision Problem Input: An orthogonl netlist. Question: Is it possible to relize ll connection requirements by monotonic routes? An exmple of MON Decision Problem is given in Fig. 2(b). In this cse, N = B L. A.1 A Sufficient Condition An orthogonl netlist is monotonic if the order grph G s, which hs edges corresponding to the order of nets on ech row nd column without distinguishing bottom nd left nets, is cyclic. According to the order given by G s,we put virtul fingers. Nets re relized by connecting ech finger to its virtul finger vi its bll from the lower left. Exmples of MON Decision Problem nd it s monotonic routing pttern re given in Fig. 8. I l p b i l q b j Fig. 9. A directed grph G R b l bh l Fig. 10. A constrint between two nets Theorem 2 An orthogonl netlist is monotonic if the order grph G s is cyclic, where E(G s )=E b f El f E s nd E s = {(x, y) x, y N, x H y x V y }. If in the second column in Fig. 8 is swpped for, G s becomes cyclic. But, the netlist is monotonic since monotonic routing pttern for the netlist is given in Fig. 2(b). So, this condition is not necessry condition. A.2 A Necessry Condition A route cn be regrded s the set of points. Let b be bottom net nd v be net. If there exists point (x b,y b ) on R(b) ndpoint(x v,y v )onr(v) such tht x b <x v nd y b = y v,thenr(v) issidtobetotherightofr(b). In other words, R(v) is sid to be to the right of R(b) if there exists point on R(v) which is to the right of R(b). Similrly, R(v) fornetv is sid to be bove R(l) forleft nets if there exists point on R(v) whichisbover(l). Theorem 3 Let G R be the directed grph constructed for routing pttern, where the vertices correspond to the nets, nd edge set is defined s follows: An edge (b, v) (b B,v N) exists if nd only if R(v) istotherightofr(b). An edge (l, v) (l L,v N) exists if nd only if R(v) is bove R(l). If the routing pttern is monotonic, then G R is cyclic. Proof. Consider tht G R is cyclic. Let C be cycle in G R. The cycle cnnot consist of only bottom nets or only left nets since there is no non-monotonic route. So, there is n edge from bottom net to left net nd n
ls l l H l S l l V b2 l H b1 l S Sl Fig. 12. An exmple of lterntive constrints (, )+ (, ) () b2 S b1 (b) b2 S b1 Fig. 11. Constrints between three nets edge from left net to bottom net. Without loss of generlity, we ssume tht C includes (b j,l p )nd(l q,b i ), nd tht R(b i )isbover(l q ), where b i,b j B (i j) nd l p,l q L (p q). See Fig. 9. Since ll routes re monotonic, the bll of b j nd R(b j ) need to exist in region O shown in Fig. 9. Similrly, the bll of l p nd R(l p ) need to exist in region I. Therefore, R(l p ) is not to the right of R(b j ), since x i <x o if y i = y o where (x i,y i )nd(x o,y o ) re points in region I nd O, respectively. However, G R hs the edge (b j,l p ). It contrdicts definition of E(G R ). So, G R is cyclic if ll routes re monotonic. G R is not defined when monotonic routing pttern is not given. However, depending on the reltionship between bottom net b nd left net l, there re cses such tht it is decided tht R(l) is to the right of R(b) or R(b) is bove R(l) in ny monotonic routing pttern. In such cses, (b, l) or(l, b) existing R for ny monotonic routing pttern. A grph where only such edges exist is the grph obtined from G R by removing some of edges. Therefore, we consider such order grph G n. Clerly, n orthogonl netlist is not monotonic if G n is cyclic. For exmple, if R(b) ndr(l) re monotonic nd b H l s shown in Fig. 10, then n edge (b, l) ising n since R(l) is lwys to the right of R(b). Similrly, R( )islwys bove R(l) if three blls of nets, nd l re plced s shown in Fig. 11(). In Fig. 11(b), R( )ndr( )re lwys bove R( ). A necessry condition is given focusing on two or three nets. G n cn be constructed by necessry conditions for being monotonic. Theorem 4 An orthogonl netlist is not monotonic if the order grph G n is cyclic, where E(G n )=E b f El f E h E v E E E E, E h = {(b, x) b H x }, E v = {(l, x) l V x }, E = {(l, b j) bj S bi ( l H bi l S bi ) bj S l }, E = {(b, l j) li S lj ( b V li b S li ) b S lj }, E = {(l, b i ) bj S bi l S bi ( bj H l bj S l l V bj )}, E = {(b, l i) lj S li b S li ( b H lj b S lj lj V b )}, where x N, b, b i,b j B, l, l i,l j L nd i<j. In ddition, there re lterntive constrints. When three blls of nets, nd re plced s shown in Fig. 12, R( ) is non-monotonic if R( )istotheright of R( )ndbelowr( ). So R( )iseitherbeloworto the right of R( )ndr( ). Therefore, either (, )or (, ) should exist in G n. These constrint is represented by (, )+ (, ). There exists n lterntive constrint (l, b i )+ (b j,l)if ( bj V bi bj S bi ) bi S l bj S l,whereb i,b j B (i <j)nd l L. Similrly, lterntive constrints for two left nets nd bottom net re defined. Assume tht there is n lterntive constrint (l, b i )+ (b j,l). If G n becomes cyclic when (l, b i ) is dded in G n,(b j,l) should be selected. But if G n does not become cyclic for either edge, then it is not esy to decide which is to be selected. When there re some lterntive constrints, n orthogonl netlist is not monotonic if G n re cyclic for ll combintions of lterntive constrins. This constrints should be nlyzed thoroughly in our future work since the number of combintions is exponentil for the number of lterntive constrints. B. An Orthogonl Routing Method An initil order grph is constructed by necessry conditions without lterntive constrints. If decision is possible for n lterntive constrint, the corresponding edge is dded. The order of nets is determined by G n by removing the source one by one. The order grph is updted if removl of the source forces n lterntive constrint nd the decision is possible for it. In this method, the updted order grph might become cyclic depending on the selection of source. According to the obtined order, fingers re connected to blls by monotonic routes one by one from the lower left. Formlly, routing of bottom net is defined s follows: A bll is sid to be connected if its route is completed. Otherwise, bll is sid to be unconnected. Let b be bottom net. R(b) psses s the left s possible on condition tht R(b) psses to the right of the unconnected left net blls in the lower-left region of b nd connected blls. For exmple, consider R( ) in Fig. 13. R( )nd R( ) become non-monotonic if R( ) psses to the left of them. Therefore, R( ) needs to void blls of nets nd s shown in Fig. 13(b). Similrly, routes of left nets cn be decided. V. Experiments nd Results We implemented our method for orthogonl netlists with C++ lnguge nd pplied it to monotonic orthogonl netlists since it is not gurnteed tht our method completes routing. Monotonic orthogonl netlists re
l 6 l 5 l 6 l 5 () Before is relized. (b) After is relized. Fig. 13. An exmple of mking routes 10 time [sec] 1 0.1 0.01 0.001 0.0001 10 100 1000 10000 the number of nets Fig. 15. An exmple of output (100 nets) over, the method tking non-monotonic routes into ccount should be proposed. In the method, the monotonic routes obtined by our proposed method is used s n initil solution nd is improved itertively to stisfy the design rule. Fig. 14. The reltionship between the number of nets nd time for monotonic orthogonl netlist generted by relxing the sufficient condition in section IV.A.1. In this experiment, we used the order grph G, where E(G) =E b f El f E h E v E E. We pplied it to problems of 56 size from 5 5to60 60. 100 ptterns were generted in ech size. Two instnces in 44 44, 45 45 could not be completed, since G becme cyclic due to the lterntive edges dded in routing process. The others were completed, nd monotonic routing ptterns were generted. The grph between the number of nets nd verge execution time re shown in Fig. 14, nd n exmple of output is shown in Fig. 15. The lgorithm genertes monotonic routing pttern within 1 second even for 3000 nets problem. The prcticl lgorithm will be obtined if the density is tken into ccount in order selecting. VI. Conclusion We gve the necessry nd sufficient condition for prllel netlist being monotonic, nd proposed routing method for monotonic prllel netlists bsed on this condition. Moreover we gve necessry condition nd sufficient condition for orthogonl netlists being monotonic, nd proposed routing method for monotonic orthogonl netlists bsed on the necessry condition. As our future work, we need to investigte lterntive constrints nd whether there re constrints between four or more nets. Routing methods tht tke routing density into considertion should be proposed. More- References [1] E. S. Kuh, T. Kshiwbr, nd T. Fujisw, On Optimum Single-Row Routing, IEEE Trnsctions on Circuits nd Systems, vol. CAS-26, no. 6, pp. 361 368, 1979. [2] S. Tsukiym nd E. S. Kuh, Double-Row Plnr Routing nd Permuttion Lyout, networks, vol. 12, no. 3, pp. 287 316, 1982. [3] M.-F. Yu nd W. W.-M. Di, Single-Lyer Fnout Routing nd Routbility Anlysis for Bll Grid Arrys, in Proceedings of Interntionl Conference Computer-Aided Design, pp. 581 586, 1995. [4] S. Shibt, K. Uki, N. Togw, M. Sto, nd T. Ohtsuki, A BGA Pckge Routing Algorithm on Sketch Lyout System, The journl of Jpn Institute for Interconnecting nd Pckging Electronic Circuits, vol. 12, no. 4, pp. 241 246, 1997. (In Jpnese). [5] Y. Kubo nd A. Tkhshi, A Globl Routing Method for 2-Lyer Bll Grid Arry Pckges, in Proceedings of ACM Interntionl Symposium on Physicl Design, pp. 36 43, Apri-6 2005. [6] C.-C. Tsi, C.-M. Wng, nd S.-J. Chen, NEWS: A Net-Even-Wiring System for the Routing on Multilyer PGA Pckge, IEEE Trnsctions on Computer-Aided Design of Integrted Circuits nd Systems, vol. 17, no. 2, pp. 182 189, 1998. [7] S.-S. Chen, J.-J. Chen, C.-C. Tsi, nd S.-J. Chen, An Even Wiring Approch to the Bll Grid Arry Pckge Routing, in Proceedings of Interntionl Conference on Computer Design, pp. 303 306, 1999.