Efficient VLSI Layouts for Homogeneous Product Networks

Transcription

1 17 IEEE TRANSACTIONS ON COMPUTERS, VOL. 46, NO. 1, OCTOBER 1997 Effcent VLSI Layouts fo Homogeneous Poduct Netwoks Antono enández and Kemal Efe, Membe, IEEE Abstact In ths pape, we develop genealzed methods to layout homogeneous poduct netwoks wth any numbe of dmensons, and analyze the VLSI complexty by devng uppe and lowe bounds on the aea and maxmum we length. In the lteatue, lowe bounds ae geneally obtaned by computng lowe bounds on the bsecton wdth o the cossng numbe of the netwok beng lad out. In ths pape, we defne a new measue that we call maxmal congeston, that can be used to obtan both the bsecton wdth and the cossng numbe, theeby unfyng the two appoaches. Uppe bounds ae tadtonally obtaned by constuctng layouts based on sepaatos o bfucatos. Both methods have the basc lmtaton that they ae applcable only fo gaphs wth bounded vetex degee. The sepaatos appoach geneally yelds good layouts when good sepaatos can be found, but t s dffcult to fnd a good sepaato fo an abtay gaph. The bfucatos appoach s ease to apply, but t geneally yelds lage aea and we lengths. We show how to obtan stong sepaatos as well as bfucatos fo any homogeneous poduct netwok, as long as the facto gaph has bounded vetex degee. We llustate applcaton of both methods to layout a numbe of nteestng poduct netwoks. uthemoe, we ntoduce a new layout method fo poduct netwoks based on the combnaton of collnea layouts. Ths method s moe poweful than the two methods above because t s applcable even when the facto gaph has unbounded vetex degee. It also yelds smalle aea than the eale methods. In fact, ou method has led to the optmal aea fo all of the homogeneous poduct netwoks we consdeed n ths pape wth one excepton, whch s vey close to optmal. In egads to we lengths, the esults obtaned by ou method tuned out to be the best of the thee methods fo all the examples we consdeed, agan subject to one (and the same) excepton. We gve an extensve vaety of such examples. Index Tems Inteconnecton netwoks, poduct netwoks, VLSI, collnea layout, sepaato, bfucato. 1 INTRODUCTION T 1. We use gaph and netwok ntechangeably. A. enández s wth the Dpatmento de Aqutectua y Tecnooga de Computadoes, Escuela Unvestaa de Infomatca, Cta. Valenca Km. 7, 831 Madd, Span. K. Efe s wth the Cente fo Advanced Compute Studes, Unvesty of Southwesten Lousana, Lafayette, LA E-mal: efe@cacs.usl.edu Manuscpt eceved 31 May 1995; evsed 8 eb o nfomaton on obtanng epnts of ths atcle, please send e-mal to: tc@compute.og, and efeence IEEECS Log Numbe HE Catesan poduct s a well-known opeaton defned on gaphs. 1 When appled to nteconnecton netwoks, the Catesan poduct opeaton combnes a set of facto netwoks nto a poduct netwok. Seveal well-known netwoks ae nstances of poduct netwoks, ncludng the gd, the tous, the hypecube [15], and the genealzed hypecube [4]. Recently, we have obseved a stong nteest n poduct netwoks, snce a vaety of new netwoks based on the Catesan poduct opeaton have been poposed. Howeve, thee s no pape whch studes the VLSI layout complexty of poduct netwoks as a geneal class. Ths poblem s mpotant snce VLSI layout cost ultmately detemnes the feasblty of an nteconnecton netwok. Ths pape makes an attempt to fll ths gap by developng genealzed appoaches to layout homogeneous poduct netwoks. We say that a poduct netwok s homogeneous f all ts facto netwoks ae somophc. Othewse, the poduct netwok s sad to be heteogeneous. All of the above mentoned examples of poduct netwoks ae homogeneous and othes have been ecently ntoduced, lke the poduct of de Bujn netwoks poposed by Rosenbeg [19], the poduct of shuffle-exchange netwoks poposed by Panwa and Patnak [18], and the poducts of complete bnay tees netwoks poposed by Efe and enández [7], [8]. Examples of ecently poposed heteogeneous poduct netwoks ae the hype-de Bujn netwok poposed by Ganesan and Padhan [1], the hype-petesen netwok poposed by Das and Banejee [6], and the banyan-hypecube netwok poposed by Youssef and Naaha [6]. Besdes these woks, the Catesan poduct opeaton has been seen as a potental famewok to unfy the study of nteconnecton netwoks. Ths allows devaton of geneal esults applcable to any poduct netwok. o example, Efe and enández [7] pesented geneal esults on the degee, dstance, damete, outng, embeddng, and pattonng popetes of poduct netwoks. Baumslag and Annexsten [1] developed genealzed off-lne pemutaton outng algothms fo poduct netwoks. El-Ghazaw and Youssef [9] developed fault toleant outng algothms. The nteest n poduct netwoks s due to the elegant mathematcal stuctues, as well as the nceased powe and vesatlty. It s shown n [7], fo example, that homogeneous poduct netwoks constucted fom shuffleexchange, de Bujn, o complete bnay tee gaphs ae computatonally moe poweful than the coespondng facto gaphs. That s, they can emulate the lke-sze facto gaph wth a small constant slowdown, but they can also /97/$ IEEE

2 ERNÁNDEZ AND EE: EICIENT VLSI LAYOUTS OR HOMOGENEOUS PRODUCT NETWORKS 171 emulate seveal othe gaphs wth constant slowdown, whch the lke-sze facto gaphs cannot. These gaphs ae deally suted fo sotng, matx multplcaton, and gaph theoy poblems as they can pefom these computatons wth the same unnng tme complexty as the hypecube, but they cost much less. In addton, heteogeneous poduct netwoks appea to combne the advantages of the dffeent facto gaphs they contan. Ths pape develops genealzed methods to obtan lowe and uppe bounds on the aea and the we length equed by VLSI layouts of homogeneous poduct netwoks. These ae the two most mpotant paametes of a layout, snce a lage aea mples low yeld n the fabcaton pocess and long wes mply lage communcaton delays (see [4] fo the technologcal detals). Ou emphass on homogeneous poduct netwoks s not due to any lmtaton of the methods pesented hee. The obtaned esults may be easly extended to heteogeneous poduct netwoks. We comment on how ths can be done n the conclusons secton of ths pape. The emphass on homogeneous poduct netwoks n ths pape s due to the fact that t smplfes the dscusson. The VLSI layout model assumed n ths pape s the Thompson s model [3]. In ths model, the VLSI layout aea s seen as a two-dmensonal gd nto whch the netwok has to be embedded wth unt congeston. The congeston of embeddng a guest gaph G nto a host gaph H s the maxmum numbe of edges of G whch must be mapped to any edge of H. In the study of VLSI complexty, the host gaph s the two-dmensonal gd and the gaph epesentng the netwok s mapped nto t. The concept of edge congeston s mpotant fo the analyses pesented n ths pape. In patcula, when the guest gaph s the N-node dected complete gaph and the host gaph s an abtay gaph contanng N nodes, then we pay specal attenton to the coespondng congeston of the embeddng. In [7], we coned the tem maxmal congeston fo the fst tme n the lteatue to efe to the congeston of embeddng a dected complete gaph n an abtay gaph. The maxmal congeston of a gaph measues the maxmum equed amount of congeston to map any gaph onto t, because no gaph has moe edges than the dected complete gaph. Maxmal congeston s an ntnsc paamete vald fo any connected gaph, just lke the cossng numbe, the chomatc numbe, etc., ae ntnsc paametes of gaphs. We use uppe bounds on the maxmal congeston to deve lowe bounds on the VLSI layout aea and the maxmum we length fo homogeneous poduct netwoks. We ntally show how to obtan an uppe bound on the maxmal congeston of a homogeneous poduct gaph fom an uppe bound on the maxmal congeston of ts facto gaph. Then we apply Leghton s methods [14], [15] to obtan lowe bounds on the bsecton wdth and the cossng numbe of the poduct netwok. Dect applcaton of esults fom [3] and [14] yelds the desed lowe bounds. The maxmal congeston has been pevously used as an ntemedate value obtaned n ode to compute the bsecton wdth o the cossng numbe of a gaph. Hee, we use t as the basc popety to be known about the facto gaph, because the bsecton wdth o the cossng numbe of a facto gaph do not gve enough nfomaton to compute lowe bounds on the bsecton wdth o the cossng numbe of the poduct gaph. Subsequently, we obtan uppe bounds on the VLSI layout complexty fo poduct netwoks wth bounded numbe of dmensons constucted fom bounded-degee facto gaphs. Ths s done by usng two tadtonal famewoks: sepaatos and bfucatos. We ntally use a specal knd of sepaatos that we call bsectos (because they bsect the gaph) and show that bsectos fo a poduct netwok ae easly obtaned fom bsectos of ts facto gaph. The dect applcaton of esults compled n [4] and [14] allows us to obtan uppe bounds on the aea and the length of the wes. Snce t s not easy to obtan small sepaatos fo an abtay gaph, we pesent a smla esult fo bfucatos. We show how to obtan a bfucato fo a poduct netwok, gven a bfucato fo ts facto netwok. We also consde some specal cases whch mpove the uppe bounds fo some specal gaphs. These esults allow us to deve the desed bounds fom the esults n []. nally, we pesent a unvesally applcable method to obtan effcent layouts fo poduct netwoks based on collnea layouts (layouts wth all the nodes along a lne) of the facto netwok. Befly, we combne collnea layouts fo the facto netwok to obtan a layout fo the poduct. We fst show how to obtan effcent collnea layouts fo the facto netwoks and then we pesent an algothm to combne them to obtan layouts fo poduct netwoks wth any numbe of dmensons. We appled all of these appoaches to seveal wellknown netwoks, as well as othes not pevously ntoduced n the lteatue. The esults show that the method based on collnea layouts yelds optmal aea n most cases (thee was just one nonoptmal case among all the cases that we consdeed, but t was also vey close to optmal). The sepaato appoach has yelded optmal aea layouts n moe cases than the bfucato, but t tuned out to be applcable fo only about half of the cases we consdeed. The bfucato appoach does not geneate optmal aea layouts as often as the othes, but ts layouts ae lage only by a logathmc facto of the sze of the facto gaph. o the maxmum we lengths, the method based on collnea layouts geneates optmal uppe bounds that match the lowe bounds f the numbe of dmensons s consdeed bounded. In all cases but one, the we lengths obtaned by ths method wee shote than the we length obtaned by usng bsectos o bfucatos. DEINITIONS We stat by defnng the class of poduct gaphs consdeed n ths pape. DEINITION 1. The -dmensonal poduct gaph, denoted as PG, of a gaph G s the gaph whose vetces compse all the - tuples x = x x x x 1, such that evey x, 1, s a vetex of G, and whose edges compse all pas of vetces (x, y), such that x and y dffe n exactly one ndex poston and (x, y ) s an edge of G.

3 17 IEEE TRANSACTIONS ON COMPUTERS, VOL. 46, NO. 1, OCTOBER 1997 In ths pape, the numbe of nodes of the facto gaph G s denoted as N. Theefoe, the numbe of nodes of the poduct gaph PG s N. The damete of G s denoted as d and ts maxmum vetex degee s denoted as D. The vetex degee of a node u of G s denoted as D u. We say that an edge (x, y) belongs to dmenson f the nodes ncdent to t dffe only n the th ndex poston. A G- subgaph of PG s sad to be a dmenson- subgaph f any two nodes n the subgaph dffe only n the th ndex poston. Next, we defne seveal popetes of netwoks used n ths pape. We stat by fomally defnng the maxmal congeston of a netwok. An embeddng of a guest gaph G onto a host gaph H s a one-to-one mappng of the vetces of G onto the vetces of H and a mappng of the edges of G nto paths n H. The congeston of an embeddng s the maxmum numbe of paths (mages of the edges of G) that tavese any edge of H. DEINITION. The maxmal congeston of a gaph G s the smallest congeston of any embeddng of the N-node dected complete gaph onto G. In ths pape, we ae manly nteested n an uppe bound on the maxmal congeston of a gaph. Ths can be easly obtaned as long as thee s a outng algothm fo the gaph. Just send packets fom evey node to evey othe node n the gaph and count the maxmum numbe of packets that tace any edge accodng to the outng algothm. o the pupose of uppe bounds, the outng algothm does not even need to be optmal, although bette outng algothms may yeld bette (smalle) uppe bounds. Pevously, uppe bounds on maxmal congeston have been used to obtan lowe bounds on the bsecton wdth and cossng numbe of gaphs. o nstance, n [14], Leghton ntoduced a technque that uses uppe bounds on the maxmal congeston to obtan lowe bounds on the cossng numbe of a gaph. A smla technque has been used n [15] to compute lowe bounds on the bsecton wdth of a few well-known netwoks. Howeve, to ou knowledge, t was neve explctly defned as beng a fundamental popety of a gaph. In ths pape, we use maxmal congeston as the fundamental popety to be known about a facto gaph n ode to deve lowe bounds on the VLSI complexty of ts - dmensonal poduct. DEINITION 3. The bsecton wdth of a gaph G s the mnmum numbe of edges that have to be emoved fom G to obtan two dsjont subgaphs wth the same numbe of nodes (wthn one). DEINITION 4. The cossng numbe of a gaph G s the mnmum numbe of edge cossngs of any dawng of G n the plane. These last two popetes of a gaph have been tadtonally used to obtan lowe bounds on the aea equed by any layout of the gaph. We contnue by defnng the class of sepaatos used n ths pape. DEINITION 5. Let f(x) be a monotoncally nondeceasng functon. A gaph G has an f(x)-bsecto ethe f t has only one node o f, by emovng at most f(n) of ts edges, t can be dvded nto two subgaphs wth the same numbe of nodes (wthn one), both wth f(x)-bsectos. In geneal, sepaatos need not bsect the gaph at each stage. Ou defnton s moe estctve, fo nstance, than the defnton of sepaato used by Leseson [17]. Howeve, Ullman [4] shows how to obtan a bsecto (he calls t stong sepaato) fom sepaatos as defned by Leseson. We wll now defne the concept of bfucato. DEINITION 6. A gaph G has an -bfucato ethe f t has only one node o f, by emovng at most of ts edges, t can be dvded nto two subgaphs, both wth -bfucatos. The defnton of bfucato mples a way to teatvely patton G so that n the th step of ths patton pocess (wth = ntally) no moe than edges ae cut n each patton. Specal cases of gaphs wll be consdeed based on addtonal estctons mposed n ths patton pocess. DEINITION 7. A gaph G has an a-specal bfucato, a 1, a f t has an O max N, N j-bfucato such that no e o moe than O((N/ ) a ) edges ae cut n each patton at the th step of the patton pocess, whee = ntally. Note that, when a = 1/, we have the defnton of N - bfucato, but, fo a π 1/, the patton pocess defned s moe estctve than the one mpled n Defnton 6. We now defne a subclass of gaphs wth a-specal bfucatos. Ths subclass was ognally consdeed n []. DEINITION 8. A gaph G has an a-type B bfucato f t has an a-specal bfucato, whee a > 1/..1 The Thompson s Gd Model In ths pape, we use the VLSI layout model defned by Thompson n [3]. In ths model, the layout aea s dvded nto squae tles of unt aea, placed n a gd fashon. Each tle can hold ethe a secton of we, a node, o a we cossng. The wes of the layout un ethe hozontally o vetcally on ths gd. If two wes ente the same tle they must have dffeent dectons and they cannot change decton n the tle. Obseve that, snce a node s assgned to a tle, the nodes ae not allowed to have moe than fou ncdent wes. When a node has a degee lage than fou, Thompson poposed to model t wth a set of adjacent tles whose pemete s at least the desed degee. Although the smallest aea equed to have a pemete of D u fo a node u has O(D u ) tles, t s much moe ealstc to assume that a node wth vetex degee D u wll eque aea of at least W(D u ) W(D u ). In ths pape, we shall assume that any node u wth vetex degee D u s lad out as a ectangle wth sdes of length at least W(D u ). Unde ths model, the we aea of a layout s the numbe of tles that hold ethe a secton of a we o a we cossng. The length of a we s the numbe of tles tavesed by the we fom ts souce node to ts destnaton node. o technologcal easons [4], the layout aea s defned as the aea of the smallest ectangle that contans all the allocated tles of the layout. Ths value s fully descbed wth the length and the wdth of ths ectangle. In ths pape, the t

4 ERNÁNDEZ AND EE: EICIENT VLSI LAYOUTS OR HOMOGENEOUS PRODUCT NETWORKS 173 wdth of a layout s assumed to be the length of the shote sde of the ectangle and the length of the layout s the length of the longe sde. We also assume that the ectangle s oented n the gd wth the longe sde hozontally placed. g. 1 shows an example of layout fo K 5, the fve-node complete gaph, whee all the nodes ae placed n a hozontal lne. Ths knd of layout s called a collnea layout. Ths layout has wdth 7, length 11, we aea 55, layout aea 77, and maxmum we length 15. g.. Nomal collnea layout fo K 5. We ntally pesent the followng two lemmas, whch allow us to obtan bounds on the bsecton wdth and the cossng numbe of poduct netwoks. Both of these lemmas easly follow fom Theoem 4 n [7], but we nclude the poofs hee fo convenence to the eade, snce they wee not stated n [7] n exactly the same fom that they appea hee. LEMMA 1. If the maxmal congeston of G s k, then the maxmal g. 1. Collnea layout fo K 5.. Collnea Layouts A VLSI layout s called collnea f all the nodes ae placed along a staght lne. We wll use collnea layouts of the facto gaph to geneate layouts fo the poduct gaph. To be able to use them, we mpose seveal estctons to the collnea layouts. We assume that the nodes ae algned hozontally. DEINITION 9. A collnea layout s semnomal f all the nodes n the layout ae placed at the bottom ows of the layout, a node u occupes D ows and D u columns, and all the wes ae lad down above the ow D. DEINITION 1. A collnea layout s nomal f t s semnomal, all the nodes ae adjacent, and all the wes ae lad down as two vetcal sectons connected by a hozontal secton. o these two classes of layouts we can defne two new paametes. DEINITION 11. The wng wdth of a collnea layout s the numbe of ows used to oute the wes n the layout. The value of the wng wdth s always the wdth of the layout mnus the maxmum vetex degee D. DEINITION 1. The bandwdth of a collnea layout s the maxmum dstance, n numbe of nodes, between any two connected nodes. The maxmum we length s closely elated to the bandwdth of a layout as we dscuss late. g. pesents a nomal collnea layout fo K 5 wth wng wdth 6 and bandwdth 4. 3 LOWER BOUNDS In ths secton, we obtan lowe bounds on the layout aea and maxmum we length equed by any layout of PG. congeston of PG s, at most, kn -1. PROO. We show a mappng of the edges of the N -node dected complete gaph nto paths of PG. We fst map the nodes of the dected complete gaph onto the nodes of PG one-to-one. Then, we map the dected edge fom x = x x 1 to node y = y y 1 to the path x Æ y x -1 x 1 Æ Æ y y x 1 Æ y. The th aow epesents the path n the coespondng G-subgaph fom x to y fo = 1,,. By defnton of maxmal congeston, these paths mply at most congeston k n the G-subgaph. Let ( z L z L z1, z L z L z1) be a dmenson- edge of PG. If ths edge s tavesed by a path fom x to y as descbed, then the edge ( z, z ) can be tavesed by, at most, k paths between two nodes of G. If ths dmenson- edge s used fo outng an edge of the dected complete gaph, say fom x to y, then y can dffe fom x n, at most, - 1 symbol postons othe than dmenson. Moeove, each dffeng symbol poston can have N possble values. Theefoe, an edge of G can be tavesed by at most kn -1 paths. LEMMA. If the maxmal congeston of G s k, then the bsecton wdth of PG s at least N kn PROO. The bsecton wdth of the N -node dected complete gaph s N / f N s even and (N - 1)/ f N s odd. Snce, fom Lemma 1, we can embed t onto PG wth congeston at most kn -1, the bsecton wdth of PG has to be at least N + 1 N 1 - = k f N s even, o kn (N - 1)/kN -1 f N s odd, because, othewse, we

5 174 IEEE TRANSACTIONS ON COMPUTERS, VOL. 46, NO. 1, OCTOBER 1997 could bsect the embedded dected complete gaph by emovng fewe edges than ts bsecton wdth, whch s a contadcton. Whethe N s even o odd, the clamed value holds as a lowe bound on the bsecton wdth of PG. We note that t s not possble to obtan a esult smla to Lemma by just knowng the bsecton wdth of G. In othe wods, we cannot say that f G has bsecton wdth B 1, then PG has bsecton wdth at least B, because bsecton of a poduct gaph does not have to use ndvdual bsectons of ts facto netwoks. In ths sense, the maxmal congeston caes moe nfomaton than the bsecton wdth. LEMMA 3. If G has E edges and ts maxmal congeston s k, then the cossng numbe of PG s at least e je je j. N N N EN - - kn PROO. Extendng the esults n [11], [1], [13], t was shown n [] that, f an N-node gaph G has E edges and the N-node complete gaph can be embedded onto G wth no edge havng congeston moe than k, then ts cossng numbe s at least N ( N -1)( N -)( N -3) E -. The k condton on edge congeston specfed hee dectly coesponds to the maxmal congeston of G, thus we conclude that any N-node gaph wth E edges and maxmal congeston of k has cossng numbe at least N( N-1)( N-)( N-3) E -. k We know fom Lemma 1 that the maxmal congeston of PG s at most kn -1. We also know fom [7] that, f G has E edges, then PG has EN -1 edges. Usng these, we obtan the clamed lowe bound on the cossng numbe of PG. In [3], Thompson showed that the squae of the bsecton wdth s a lowe bound (wthn a constant facto) on the we aea equed by any layout of a gaph. Smlaly, Leghton [14] pesented the cossng numbe as a lowe bound on the we aea of any layout of a gaph. Then, we can use any of the last two lemmas to pove the followng theoem, whch s one of the two man esults of ths secton. THEOREM 1. If the maxmal congeston of G s k, then the layout aea of PG s at least W N ( + 1) e j. k The second esult of ths secton gves a lowe bound on the length of the longest we n any layout of a poduct gaph. THEOREM. If the maxmal congeston of G s k and ts damete s d, then the length of the longest we n any layout of PG s at least W N +1 e kd j. PROO. Theoem 5- n [14] shows that any layout of a gaph wth damete D and mnmum layout aea A has some we of length at least A 1/ /3D. The damete of PG s D = d [7] and, fom Theoem 1, ts layout aea s at least W N ( + 1) e j. Theefoe, we can conclude k that any layout of PG has some we of length at least + 1 WeN kj + 1 N = W 3d e kd j. 4 UPPER BOUNDS In ths secton, we fst pesent uppe bounds obtaned by tadtonal famewoks, namely bsectos and bfucatos. We show that, gven a bsecto o a bfucato fo the facto gaph, we can obtan a bsecto o a bfucato fo the poduct gaph. Snce these famewoks ae only applcable to netwoks wth bounded vetex degee, we wll assume that the facto gaph has bounded vetex degee and that the numbe of dmensons of the poduct netwok s also bounded. These assumptons ae not vey estctve because all poduct netwoks obtaned fom fxed degee netwoks can gow wthout nceasng the vetex degee. Sheleka and JáJá [], [1] have nvestgated the use of sepaatos and bfucatos to obtan effcent layouts fo unbounded vetex degee gaphs. Howeve, the knds of sepaatos and bfucatos they use ae so estctve that t does not seem possble to obtan smple geneal esults fo poduct gaphs by usng them. Subsequently, we pesent anothe appoach that s unvesally applcable and does not have any estcton on the vetex degee o on the numbe of dmensons. As we mentoned, ths method of obtanng effcent layouts fo poduct netwoks s based on the exstence of effcent collnea layouts fo the facto netwoks. We show that t s always possble to fnd easonably effcent collnea layouts fo any netwok and pesent seveal technques to do so. 4.1 Uppe Bounds Based on Bsectos The followng theoem pesents the basc esult of ths secton. THEOREM 3. If G has an f(x)-bsecto, then PG has an O(x (-1)/ f(x 1/ ))-bsecto. PROO. We ntally pesent the followng lemma that shall be used n the poof. LEMMA 4. If G has an f(x)-bsecto, then t has at most O(Nf(N)) edges. PROO. Assume, fo smplcty, that N s a powe of two. By the defnton of bsecto, G can be dvded nto two subgaphs by emovng no moe than f(n) edges. Then, we obtan two subgaphs wth N/ nodes each, whch can be bsected by emovng no moe that f(n/) edges fom each. Afte bsectons of ths knd, we obtan subgaphs wth N/ nodes each, whch, n tun, can be bsected by emovng no moe that f(n/ ) edges fom each. Afte applyng the bsecton pocess log N tmes we obtan N solated nodes. The maxmum numbe of edges cut n the whole pocess can be easly computed as, log N-1 log N-1 fn + fn + fn + L + fn af c h e j e j log N-1 = fn = ONfNh. Â =. All logathms ae to the base two. e j c af

6 ERNÁNDEZ AND EE: EICIENT VLSI LAYOUTS OR HOMOGENEOUS PRODUCT NETWORKS 175 The poof of the theoem shows how to dvde PG nto solated nodes by epeatedly applyng bsectons that espect the defnton of O(x (-1)/ f(x 1/ ))-bsecto. Intally, we show how to dvde PG nto dsjont subgaphs and, possbly, some solated nodes. Ths pocess s done n bsecton steps, each of whch cuts O(N -1 f(n)) edges fom ts coespondng gaph. Each of the obtaned subgaphs s the -dmensonal (possbly heteogeneous) poduct of facto gaphs wth ÎN/ nodes and f(x)-bsectos. A patton pocess smla to the one appled to PG can then be appled to each of the subgaphs, to the subgaphs obtaned fom them, and so on, untl all the nodes ae solated. The basc patton pocess consdes two cases, when N s even and when N s odd. Case N even: By defnton of bsecto, each of the G- subgaphs n each dmenson can be bsected by emovng no moe than f(n) edges. We can ntally consde only the G-subgaphs n dmenson one. PG can be dvded nto two subgaphs wth the same numbe of nodes n each by bsectng each of the dmenson-one G-subgaphs. As thee ae N -1 such subgaphs, we have cut no moe than N -1 f(n) edges fom the N -node gaph. Now we can take one of the two subgaphs of PG obtaned and dvde t nto two subgaphs wth same numbe of nodes by bsectng each of ts dmensontwo G-subgaphs. The numbe of edges cut ths tme s no moe than N -1 f(n)/ fom a gaph wth N / nodes. We can contnue ths pocess, bsectng the obtaned subgaphs along each dmenson. When bsectng the subgaphs by dmenson, we ae emovng no moe than N -1 f(n)/ -1 = O(N -1 f(n)) edges fom N / -1 = Q(N )-node gaphs. Afte bsectng the subgaphs by dmenson, we obtan dsjont subgaphs, each beng the - dmensonal poduct of N/-node gaphs wth f(x)- bsectos (because they ae bsectons of gaphs wth f(x)-bsectos). Case N odd: The logc n ths case s smla to the logc n the above case, but we must be caeful because, by smply bsectng each subgaph along a dmenson, we ae not bsectng the whole gaph. What we do n ths case s beakng each G-subgaph n a gven dmenson nto two subgaphs, wth (N - 1)/ nodes each, and one solated node. As the solated nodes ae connected between themselves by the othe dmensons, we also cut these connectons and dstbute the so obtaned solated nodes evenly between the two lage subgaphs. We can ntally take dmenson one. By bsectng each dmenson-one G-subgaph, we cut no moe than N -1 f(n) edges and we obtan two subgaphs wth N -1 (N-1)/ and N -1 (N+1)/ nodes, espectvely. Clealy, PG has not been bsected. Now we can take the subgaph wth the lage numbe of nodes and solate one node along dmenson one fom each of the dmenson-one subgaphs, the same node n each subgaph. Snce we ae assumng that G has bounded vetex degee, we can do so by cuttng a bounded numbe of edges fom each dmenson-one subgaph. Ths leads to a total of O(N -1 ) edges cut. Now we have two subgaphs wth N -1 (N-1)/ nodes each, and an ( - 1)-dmensonal subgaph, somophc to PG -1. om Lemma 4, the facto gaph G that geneates the ( - 1)-dmensonal subgaph has no moe than O(Nf(N)) edges. Theefoe, we can solate the nodes of ths subgaph by emovng, at most, ( - 1)N - O(Nf(N)) = O(N -1 f(n)) edges. As a esult of the above pocess, we have two subgaphs wth the same numbe of nodes and some solated nodes. If we dstbute the solated nodes evenly between the two subgaphs, ou bsecton s done. The total numbe of edges cut has been N -1 f(n)+o(n -1 ) + O(N -1 f(n)) = O(N -1 f(n)) fom the ntal N -node gaph. Ths pocess can be appled to each dmenson, as n the case of N even. In each applcaton, O(N -1 f(n)) edges ae cut fom an Q(N )-node gaph. Afte the gaph has been bsected n ths way along each dmenson, we have dsjont -dmensonal subgaphs, each beng the poduct of (N - 1)/-node gaphs wth f(x)-bsecto, plus seveal solated nodes dstbuted evenly between them. We now have subgaphs of PG, each beng the -dmensonal poduct of facto gaphs, wth ÎN/ nodes and f(x)-bsectos. Note that, n the above descbed pocess, we only use the fact of PG havng the same numbe of nodes along each dmenson and the fact of each facto gaph havng an f(x)-bsecto. Snce the obtaned subgaphs fulfll these equements, the descbed pocess can be appled agan to each of them. Subsequently, the subgaphs obtaned fom them wll also fulfll the equements, and the pocess can be appled to each of them, and so on, untl all the nodes ae solated. Snce, n each bsecton of the whole pocess, the numbe of edges cut does not exceed the lmts mposed by the defnton of f(x)-bsecto fo f(x) = O(x (-1)/ f(x 1/ )), the poof s complete. Once we obtan a bsecto fo the poduct netwok, we ae eady to apply t to obtan bounds on the layout paametes. We can use Theoem 3.5 n [4], whch deves uppe bounds on the layout aea of netwoks wth a gven bsecto, to dectly obtan the followng esult. THEOREM 4. If G has an f(x)-bsecto, then PG can be lad out n a squae of sde O(Nf(N)log N) when =, o sde O(N -1 f(n)) when >. PROO. Theoem 3.5 n [4] states that any m-node gaph wth a g(x)-bsecto and bounded degee can be lad out n an aea of sde H G log 4 m  O max m, g m 4 = e II j K J K J.

7 176 IEEE TRANSACTIONS ON COMPUTERS, VOL. 46, NO. 1, OCTOBER 1997 om Theoem 3, we have obtaned that PG has an O(x (-1)/ f(x 1/ ))-bsecto. PG has N nodes and we can obtan the value of the summaton as log 4 N Â = O f N -1 e j e j O N 4 f N 4 af a f I K J = a-1f NI I H G 4 K J, KJ log 4 N Â = snce f(x) s a monotoncally nondeceasng functon. The value of ths last summaton s O(Nlog N) when =, o O(N -1 ) when > [4]. Theefoe, the value of the fst summaton s O(Nf(N)log N) when =, o O(N -1 f(n)) when >, and the clam follows. The most studed knd of bsectos has been O(x a )- bsectos, fo bounded a. The esults of Leseson [16], [17] and Valant [5] mply that any m-node gaph wth an O(x a )-bsecto can be lad out n aea O(m) when a < 1/, n aea O(m log m) when a = 1/, o n aea O(m a ) when a > 1/. Bhatt and Leseson [3] have also shown that these uppe bounds can be eached wth maxmum we lengths of Od m log m, Od m log m log log m, o O(m a ), espectvely. These esults can be dectly appled to poduct netwoks to obtan the next coollay (note, m s N hee). COROLLARY 1. If G has an O(x a )-bsecto, fo bounded a, then PG can be lad out n an aea of O(N log N) wth maxmum we length O(N log N/log log N) when a = and =, o n an aea of O(N (+a-1) ) wth maxmum we length O(N +a-1 ) othewse. 4. Uppe Bounds Based on Bfucatos The followng theoem and ts coollay pesent the ntal geneal esults of ths secton. Afte these, we pesent addtonal esults applcable to gaphs wth a-specal bfucatos, whch yeld tghte bounds. THEOREM 5. If G has an -bfucato, then PG has an - N 1 6 d + -bfucato. PROO. om Theoem 6 n [], we know that, f G has an - bfucato, then t has an H = 6 d + -bfucato (balanced bfucato) that bsects the gaph at each patton. Then, afte, at most, log N + 1 pattons, G s tansfomed nto N solated nodes. Thoughout the est of the poof, we wll denote 6d+ as H fo bevty. The poof s vey smla to the poof of Theoem 3. We show that, gven PG, we can obtan subgaphs, each beng the -dmensonal (possbly heteogeneous) poduct of facto gaphs wth H -bfucatos and, at most, ÈN/ nodes. We ntally consde dmenson one. To patton PG, we can bsect each G-subgaph n ths dmenson, cuttng no moe than HN -1 edges n total. Each dmenson-one G-subgaph s so dvded nto an ÎN/ node and an ÈN/ -node subgaphs. To follow the wost case, we take the lage of the two obtaned subgaphs. We can patton ths subgaph by dmenson two by cuttng at most HÈN/ N - edges. Ths value s smalle than HN -1. We can contnue n ths way, pattonng the subgaphs by each dmenson. When dvdng the lagest subgaph by dmenson, we cut, at most, HÈN/ -1 N edges, that s, smalle than HN. Afte dvdng by dmenson, each subgaph obtaned has, at most, ÈN/ nodes along each dmenson. If we stat the pocess agan, the next dvson wll cut, at most, H N -1 edges, that s smalle than HN -1. Theefoe, the pocess can be epeated wthout exceedng the numbe of edges allowed by the defnton of bfucato. As n the poof of Theoem 3, we can apply the patton pocess just descbed to each of the subgaphs of PG obtaned, to the subgaphs obtaned fom them, and so on, untl all the nodes ae solated. By epeatng ths pocess, at most, log N + 1 tmes, all the nodes n PG wll be solated, and the theoem follows. We ecall hee that Bhatt and Leghton [] showed that, f G can be lad out n an aea A, t has a A -bfucato. Thus, f G can be lad out n an aea A, then PG has an - N 1 6 d + A -bfucato. om Theoem 5, we can obtan bfucato-based bounds fo the aea and maxmum we length by usng the esults fom []. COROLLARY. If G has an -bfucato, then PG can be lad out n an aea of O(N (-1) log (N/)) wth maxmum we length O N -1 b log N log log N 6+ e c The above theoem and coollay ae unvesally applcable. Howeve, as Bhatt and Leghton [] noted, thee ae gaphs wth specal chaactestcs whch allow to mpove the above bounds. Ths fact s eflected n the followng esults. THEOREM 6. If G has an a-specal bfucato, then PG has an O(N +a-1 )-bfucato. Ths s an ( + a - 1)-type B bfucato f ethe > o = and a >. PROO. Let N be a powe of two fo smplcty. om Theoem 5 n [], we know that G has a patton pocess whee each patton n the th patton step bsects the coespondng gaph wthout cuttng moe than p 6Â O e N s j I a edges, whee log N and p s s= H K g h I KJ j.

8 ERNÁNDEZ AND EE: EICIENT VLSI LAYOUTS OR HOMOGENEOUS PRODUCT NETWORKS 177 the numbe of steps of the ognal patton pocess. Ths summaton s a deceasng geometc sees, that s, essentally on the ode of ts fst tem. Then, G has a patton pocess such that each patton n the th patton step bsects the coespondng gaph wthout cuttng moe than 6O((N/ ) a ) = O((N/ ) a ) edges. The patton pocess s smla to the one pesented n the poof of Theoem 5. We apply a basc pocess log N tmes, pattonng the gaphs tmes n each applcaton. We wll use to count the applcatons of the basc pocess, vayng fom to log N - 1, and j to count the pattons wthn an applcaton of the basc pocess, vayng j fom to - 1. The absolute count of patton steps fo the complete gaph s then k = + j. In the th applcaton of the basc pocess, the sze of the facto subgaphs that we ae consdeng s m = N/ and we cut at most O((N/ ) a ) edges to patton ths subgaph. Then, the kth absolute patton step cuts at most O((N/ ) a (N/ ) -j-1 (N/ +1 ) j ) edges. We can wte + a-1 a -j j N en j en j en j = k- 1 a -af. Snce k - (1 - a) k/ fo, we can conclude that PG has an O(N +a-1 )-bfucato. We stll need to show n whch cases ths s an ( + a - 1)-type B bfucato. Note that 1-a1-af + a-1 N en j = k-a1-af k 1-a1-af. k e j Snce (1 - a)/ (1 - a)/k, then 1 - (1 - a)/k 1 - (1 - a)/ and, theefoe, 1-a1-af en j k 1-a1-af O N k 1 a1 f = k e - - j H G I a K J, e j whee 1 - (1 - a)/ > 1/ (.e., we have a type B bfucato) f ethe > o = and a >. We can now apply the esults of [] combned wth ths theoem to obtan the followng coollay. COROLLARY 3. If G has an a-specal bfucato, then PG can be lad out n an aea of O(N log N), wth maxmum we log N length ON j, f = and a =, o n an aea of e log log N O(N (+a-1) ) wth maxmum we length O(N +a-1 ) othewse. 4.3 Uppe Bounds Based on Collnea Layouts In ths secton, we pesent the collnea appoach to obtan layouts fo poduct netwoks. The collnea appoach has seveal advantages: 1) It gves the optmal aea layouts fo all the cases we consdeed n Secton 5 of ths pape (wth one excepton), and we lengths wee qute close to optmal. ) It depends on obtanng a collnea layout fo the facto gaph, whch s much ease than obtanng good bsectos o bfucatos. The aea of the layout only depends on the wng wdth of the collnea layout. The maxmum we length depends also on the bandwdth of the collnea layout. Below, we pesent seveal ways to obtan nomal collnea layouts wth small wng wdth fo abtay gaphs. 3) It s applcable fo any gaph egadless of the vetex degee, whle the applcablty of bsecto o bfucato based appoaches ae lmted to gaphs wth bounded degee. 4) The aspect ato of layouts ae always O(1), whch s a desable chaactestc fo fabcaton Methods fo Obtanng Nomal Collnea Layouts In ths secton, we ae nteested n obtanng nomal collnea layouts wth small wng wdth and small bandwdth fo the facto gaph G, because these ae the popetes that nfluence the cost/pefomance chaactestcs of the layout of the poduct gaph that we wll obtan. We can devse seveal methods to obtan nomal collnea layouts fo any gaph. st, obseve that any gaph G has a nomal collnea layout of wng wdth at most 1 D Â uœv u, whee V s the set of nodes of G, snce ths s the numbe of wes n the layout and each we eques no moe than one ow. Second, the poblem of fndng an effcent collnea layout s closely elated to the class of poblems known as gaph labelng [5]. Gven a way of labelng the nodes of an N-node gaph wth ntege labels 1,, N (o, equvalently, gven a way of placng the nodes of the gaph on a lne), the maxmum dstance between two connected nodes s the bandwdth of the labelng, whle the maxmum numbe of edges that coss a vetcal lne placed between any two nodes s the cutwdth of the labelng. Thus, f thee s an embeddng of a gaph G onto the N-node lnea aay wth bandwdth b and cutwdth c, t s tval to obtan a nomal collnea layout fo G wth wng wdth c and longest edge of length O(bD + c). o an abtay gaph, we can obtan a labelng whch mnmzes the bandwdth and the cutwdth by usng dynamc pogammng algothms o heustcs. Thd, t s shown n [17] how to constuct nomal collnea layouts fo a gaph G wth an f(x)-sepaato. The layout has wng wdth O(f(N)log N) n geneal, but, f f(x) = W(x a ) fo a >, then the wng wdth s O(f(N)). o gaphs wth -bfucatos, we obtan a smla esult n the followng lemma. LEMMA 5. If G has an -bfucato, then t has a nomal collnea layout wth wng wdth O(). PROO. The constucton of the layout s smla to the constucton shown n [17] fo sepaatos. We use a dvdeand-conque pocess that dvdes the gaph nto two subgaphs, obtans a collnea layout fo each, and econnects the two layouts by addng, at most, as many new ows as the edges whch wee cut n the patton step. om the defnton of bfucato, the dvson pocess s appled at most log tmes befoe we obtan solated nodes, and at step at most edges

9 178 IEEE TRANSACTIONS ON COMPUTERS, VOL. 46, NO. 1, OCTOBER 1997 ae cut to dvde the gaph, fo log. Then, the numbe of ows needed to oute the edges of G log ae at most  = O( ). = nally, we pesent a geneal method to obtan a nomal collnea layout fom an abtay layout. The followng esult shows that thee s always a semnomal collnea layout wth small wng wdth. LEMMA 6. If G has a layout of length l and wth w, G also has a semnomal collnea layout of length O(l + ND) and wng wdth O(w). PROO. We pove the lemma by showng how to tansfom the gven layout nto a semnomal collnea layout wth the clamed dmensons. The tansfomaton s llustated n g. 3, whee we show only the pocess fo one node of the layout. The appeaance of ths node n the ognal layout s shown n g. 3a. We fst tansfom the nodes of the gven layout by addng enough ows so that each node uses at least D u ows whee D u s the vetex degee of the node u whch s beng tansfomed. The wdth of the esultng layout s at most O(w). g. 3b shows the esult of enlagng ou example node to use two ows. We subsequently ceate D new ows at the bottom of the layout. We wll eventually move all the nodes n the layout to these new ows. No othe ows ae added n the est of the tansfomaton pocess, theefoe, the wng wdth of the new layout wll be O(w). At the bottom of g. 3c, the thee new ows ntoduced fo ou example can be obseved. We assumed, then, D = 3. Then, the followng step s appled teatvely untl all the nodes ae n the ceated bottom ows and we have a semnomal collnea layout. The step seaches fom left to ght fo the fst column wth tles assgned to nodes not yet moved. Ths column can have tles fom seveal nodes, f so we take one node abtaly. Let u be the node we have chosen. Ceate D u new columns on the left sde of u. When ceatng these columns, do not stetch the wes ncdent to u acoss the columns just ceated. g. 3c pesents the two new columns ceated n ths step. Then, move u to the bottom ows, afte eszng t to D u D. nally, use the newly ceated columns, as well as the ows ognally allocated to u, to eoute the edges fom the bottom ows. Snce u had at least D u ows and we have D u columns, ths eoutng can be done. g. 3d pesents the fnal esult fo ou example. Ths ends the tansfomaton. Note that the total numbe of added columns s  D uœv u, whee V s the set of nodes of G, and, theefoe, the length of the layout s O(l + ND). The above lemma shows that any layout can be tansfomed nto a semnomal collnea layout wth wng wdth of the same ode as the wdth of the ognal layout. Whle the tansfomaton nceases the length of the collnea layout, we wll see that t s the wdth of the nomal collnea layout whch domnates the layout aea complexty fo g. 3. Tansfomaton of a compact layout nto a collnea layout. the poduct gaph. The collnea layout obtaned can now be compessed to obtan a nomal collnea layout wth, at most, same wng wdth. Ths s shown n the followng lemma. LEMMA 7. If G has a semnomal collnea layout wth wng wdth w and bandwdth b, t also has a nomal collnea layout wth wng wdth, at most, w and bandwdth b. PROO. The ognal layout gves us a possble labelng (.e., the ode n whch the nodes of G can be placed) to obtan the desed wng wdth w. Ths s all we need fo the pupose of obtanng the desed nomal layout. We stat by placng the N nodes touchng each othe along a staght lne. The th node n ths lne coesponds to the th node n the semnomal collnea layout. We then connect these nodes by theesegment wes (two vetcal and one hozontal), as equed by the ognal layout. Snce thee s a semnomal collnea layout of wdth w that uses the same node ode, we can obtan a layout whch has at most w ows used fo wes. The bandwdth of the layout emans the same. Note that, n the above obtaned layout, the length of the longest we s at most w + bd, whee b s the bandwdth of the layout. We fnsh ths secton by pesentng a lowe bound on the wng wdth of any nomal collnea layout fo abtay gaphs. THEOREM 7. If the maxmal congeston of G s k, then the wng wdth fo any nomal collnea layout of G s at least N /k. PROO. Note fst that any embeddng of the N-node dected complete gaph onto the N-node lnea aay eques congeston at least N /. Snce the N-node dected complete gaph can be embedded onto the gaph G wth congeston k, t follows that any embeddng of G onto the N-node lnea aay eques congeston at least N /k, snce, othewse, we can obtan an embeddng of the dected complete gaph wth congeston smalle than N /. Snce the congeston of any embeddng of G onto the lnea aay s a lowe bound on the numbe of ows needed to oute the edges of G, the esult follows The Layout Method fo Poduct Gaphs The followng theoem epesents the man esult of ths secton. The poof gves an algothm to obtan the layout fo a poduct gaph fom a nomal collnea layout of ts facto gaph.

10 ERNÁNDEZ AND EE: EICIENT VLSI LAYOUTS OR HOMOGENEOUS PRODUCT NETWORKS 179 THEOREM 8. If G has a nomal collnea layout wth wng wdth w, then PG has a layout wth squae nodes of sde DÈ/ placed egulaly n N È/ columns of N Î/ nodes each, whee two adjacent columns of nodes ae at dstance w -1  N = -1  N = and two adjacent ows of nodes ae at dstance w. PROO. We show the teatve pocess that can be used to obtan the desed layout. The poof s llustated n g. 4, whch pesents the constucton of a layout fo the thee-dmensonal poduct of two-node lnea aays, whch s somophc to the thee-dmensonal hypecube. g. 4a pesents a nomal collnea layout fo the two-node lnea aay. Intally, we place the N nodes of PG n the layout as squaes of sde DÈ/ n a gd fashon wth N È/ columns of nodes and N Î/ ows of nodes. Each node touches ts neghbo nodes n the layout. The sze of the nodes wll guaantee that thee ae enough connecton ponts n each sde of the node when needed. g. 4b pesents ths ntal stuaton fo ou example gaph. o each ow of nodes we apply the followng teatve pocess. We stat by ceatng w new ows above the ow of nodes. The nodes n the ow ae dvded n N È/ -1 goups of N adjacent nodes each, and the nodes n each goup ae connected usng the ceated ows wth the wes lad down as specfed by the nomal collnea layout of G. Ths completes the connectons fo the fst dmenson of the poduct gaph. We subsequently ceate wn new ows, dvde the nodes n a ow nto N È/ - goups of N adjacent nodes each, and use the wn new ows n goups of w each to connect N nodes of the second dmenson. These nodes ae N nodes apat fom one anothe. In the th teaton, we ceate wn -1 new ows, dvde the nodes n N È/ - goups of N nodes each, and connect sets of N nodes n the th dmenson, each N -1 nodes apat fom one anothe. Ths pocess s appled È/ tmes fo each ow of nodes. The total numbe of wng ows ceated s -1  N w. Ths s the dstance between two ows = of pocessos. Two adjacent pocessos n the same ow ae stll touchng each othe. g. 4c pesents the example layout afte completon of the above pocess. To obtan ths layout, we appled the teatve step twce. The same teatve pocess can be appled to connect the columns. As a esult, we fnd that the columns of pocessos ae at dstance wâ N. Ths -1 = completes the poof. g. 4d shows the fnal layout obtaned fo ou example gaph. g. 4. Layout fo the thee-dmensonal hypecube. om ths theoem, we can obtan bounds on the aea and maxmum we length fo the layout. COROLLARY 4. If G has a nomal collnea layout wth wng wdth w and bandwdth b, then PG can be lad out n an aea of dmensons Q(w N -1 ) Q(w N -1 ) wth maxmum we length Q(bwN - ). PROO. The length of the layout obtaned fom the above theoem along the hozontal dmenson s -1 N D + w  N. = Snce w D/, N Q N I KJ -1 -  = 1 = H I K, and N Î/ -1 È/ fo N and, then ths length s Q(wN -1 ). The length along the vetcal dmenson s I K -1  e -1j. = N D + w N Q wn J = Smlaly, the length of the longest edge s at most -1  -1 - w N + b N D + w N bwn J = Q e j. = -1  = 5 APPLICATION O THE BOUNDS TO VARIOUS NETWORKS In Tables 1 and, we have compled the bounds obtaned by applyng the pesented esults to seveal netwoks. Table 1 pesents the bounds on layout aea, whle Table pesents the bounds on maxmum we length. In ths secton, we wll pesent how these bounds have been obtaned. The second column n both tables pesents the lowe bounds obtaned by dect applcaton of Theoems 1 and. We fst obtan an uppe bound on the maxmal congeston fo each facto netwok. The maxmal congestons of the lnea aay and the complete bnay tee ae easly found to be O(N ). The analyss of lowe bounds on the bsecton wdth n [15] mpled that the maxmal congestons of the shuffle-exchange, de Bujn, buttefly, and cube-connected cycles (CCC) netwoks ae O(N log N). The bounds fo the hypecube ae obtaned fom the fact of beng the poduct of the two-node lnea aay. It can be also seen that the maxmal congeston of K N s O(1) (actually, two). II KJJ K

11 18 IEEE TRANSACTIONS ON COMPUTERS, VOL. 46, NO. 1, OCTOBER 1997 TABLE 1 BOUNDS ON THE LAYOUT AREA OBTAINED BY APPLICATION O THE PRESENTED METHODS UN stands fo unknown, N.A. stands fo not applcable. The uppe bounds maked wth * ae optmal. TABLE BOUNDS ON THE WIRE LENGTH OBTAINED BY APPLICATION O THE PRESENTED METHODS UN stands fo unknown, N.A. stands fo not applcable. The uppe bounds maked wth * ae optmal f s bounded. The thd and fouth columns of the tables pesent uppe bounds obtaned fom bsectos of the facto gaphs. Agan, t s easy to obseve that the lnea aay and the complete bnay tee have O(1)-bsectos. By applyng Coollay 1 dectly, the pesented bounds ae obtaned. Thee ae not, as fa as we know, tght bsectos fo the shuffle-exchange and the de Bujn gaphs. Thus, we pesent the coespondng bounds as unknown. We can easly show that the n log n-node buttefly can be bsected by emovng O(n) edges, esultng n two buttefles wth one less ank and seveal solated nodes. Theefoe, we conclude that the buttefly has an O(x/log x)-bsecto. Smlaly, t can be shown that the cube-connected cycles has an O(x/log x)- bsecto. To obtan the bounds on we length we use x/log x = x a fo some a > and, theefoe, a > n Coollay 1 fo both netwoks. Snce the hypecube can only gow by nceasng the numbe of dmensons, t s consdeed as a netwok wth unbounded numbe of dmensons, and the bsecto appoach cannot be appled to t. Smlaly, ths appoach cannot be appled to the poduct of complete gaphs, snce K N has not bounded vetex degee. The ffth and sxth columns contan the bounds obtaned fom bfucatos of the facto netwoks. The lnea aay and the complete bnay tee have zeo-specal bfucatos. The value of the bfucatos fo the shuffle-exchange and de Bujn netwoks ae obtaned fom known layouts of aea O(N /log N) [14] that mples the exstence of O(N/log N)- bfucatos fo these netwoks []. It s easy to see that the buttefly and the cube-connected cycles have one-specal bfucatos. We then apply Coollaes and 3 to obtan the bounds on layout aea and maxmum we length fo all these netwoks. Agan, the hypecube and the poduct of complete gaphs ae not consdeed. The last column of the tables pesent the uppe bounds obtaned fom collnea layouts fo the facto netwoks. If the nodes of the lnea aay ae lad down n a lne, we obtan a collnea layout wth wng wdth one and bandwdth one. The complete bnay tee has a collnea layout wth wng wdth O(log N) and bandwdth O(N), whch can be obtaned by just labelng the nodes n n-ode. To obtan nomal collnea layouts fo the shuffle-exchange and de Bujn gaphs, we can apply Lemmas 6 and 7 to the optmal O(N/log N) O(N/log N) aea layouts [14] to obtan nomal collnea layouts wth wng wdth O(N/log N), hence, the bounds on the layout aea n Table 1. We do not know the bandwdth of these layouts to obtan a bound on the we length. But, we use the wng wdth and bandwdth of a collnea layout pesented n [3] fo the shuffleexchange gaph, whch has wng wdth O(N/log 1/ N) and bandwdth O(N/log 1/ N). Note, then, that the maxmum we length bounds pesented n Table fo these netwoks may not be achevable wth the optmal aea layout pesented n Table 1. The nomal collnea layout obtaned by placng the anks of the buttefly n ode, one afte the othe, has wng wdth O(N/log N) and bandwdth O(N/log N). A smla appoach can be used fo the cube-connected cycles to obtan the same bounds. The hypecube has, as facto netwok, the two-node lnea aay, whch s lad out wth wng wdth one and bandwdth one (see g. 4a). Also, t s possble to obtan a nomal collnea layout fo K N wth wng wdth O(N ).

12 ERNÁNDEZ AND EE: EICIENT VLSI LAYOUTS OR HOMOGENEOUS PRODUCT NETWORKS CONCLUSIONS AND COMPARISON O RESULTS In ths pape, we have nvestgated bounds on the aea and maxmum we length of layouts fo homogeneous poduct netwoks. We have obtaned lowe bounds, based on the maxmal congeston of the facto netwok, and uppe bounds, based on the exstence of bsectos, bfucatos, o an effcent nomal collnea layout fo the facto gaph. A compason of the aea bounds fo some poduct netwoks s gven n Table 1. The poposed method based on collnea layouts seems to geneate layouts wth optmum aea n most of the cases. Only fo poducts of complete bnay tees, the layout aea s not mnmum, and t s not possble to each an optmal aea layout fo ths netwok usng ths method, snce we would need a nomal collnea layout fo the complete bnay tee, wth O(1) wng wdth. In fact, the layout obtaned fo the poduct of complete bnay tees s also aea optmal fo two dmensons, snce ths netwok has the mesh of tees as a subgaph, whch eques aea W(N log N) fo two dmensons [8]. The layouts obtaned by usng bsectos (when applcable) ae also qute aea effcent, snce they have optmal aea fo moe than two dmensons n the studed cases. The layouts obtaned by usng bfucatos ae not always aea optmal, but ae off by only a polylogathmc facto of N. In Table, we compae the esults of maxmum we length. If the numbe of dmensons s constant, the collnea method obtans bounds that match the lowe bounds. In most cases, t gave bette bounds than the othe two appoaches. The only excepton we have s the poduct of complete bnay tees. When applcable, the use of bsectos seems to gve the same maxmum we lengths as the use of bfucatos. The above analyses suggest the method based on collnea layouts as a vey useful and poweful appoach to the layout poblem fo homogeneous poduct netwoks. Moe eseach may help n fndng nomal collnea layouts wth small wng wdth and small bandwdth fo a vaety of othe facto gaphs. If the poduct netwok s heteogeneous and all the facto gaphs have equal numbe of nodes, t s not dffcult to deve bounds smla to those pesented by usng the man esults of ths pape, wthout efeng to the detaled dscusson n the poofs. One way s to just consde the wost case. o nstance, the lowe bounds pesented n Theoems 1 and ae stll vald f we defne k as the maxmum of the maxmal congestons of the facto gaphs. Smlaly, f f(x) s the lagest asymptotc complexty bsecto of all the facto gaphs, then the poduct gaph has an O(x (-1)/ f(x 1/ ))-bsecto. The esults fo bfucatos and collnea layouts can be genealzed n a smla way. Addtonal detal may ase when dffeent facto gaphs have dffeent numbes of nodes fo dffeent dmensons, o when they have dffeent maxmal congestons. Beng moe caeful and gvng exact bounds (uppe o lowe) n such cases s not moe dffcult, but dscusson gets athe tedous. In fact, we had to deal wth such cases dung the poofs of Theoems 3 and 5. Even though we stated wth a homogeneous poduct netwok, afte the fst bsecton, the poduct gaphs obtaned ae nethe homogeneous no do they contan the same numbe of nodes at each dmenson. ocusng on homogeneous poduct netwoks allowed the tedous pat of dscusson to be lmted wthn the poofs and not n the mansteam dscussons of the pape. The key pont to ealze s that we should consde the smallest bsecton that can be found at each step of bsectng the poduct netwok. ACKNOWLEDGMENT A. enández s contbuton to ths wok was made dung the tme n whch he was at the Unvesty of Southwesten Lousana. REERENCES [1] M. Baumslag and. Annexsten, A Unfed amewok fo Off- Lne Pemutaton Routng n Paallel Netwoks, Math. Systems Theoy, vol. 4, no. 4, pp , [] S.N. Bhatt and.t. Leghton, A amewok fo Solvng VLSI Gaph Layout Poblems, J. Compute and System Scences, vol. 8, pp , [3] S.N. Bhatt and C.E. Leseson, Mnmzng We Delay n VLSI Layouts, MIT VLSI memo 8-86, 198. [4] L. Bhuyan and D.P. Agawal, Genealzed Hypecubes and Hypebus Stuctues fo a Compute Netwok, IEEE Tans. Computes, vol. 33, no. 4, pp , Ap [5].R.K. Chung, Labelngs of Gaphs, Selected Topcs n Gaph Theoy 3, L.W. Beneke and R.J. Wlson, eds., pp Academc Pess, [6] S.K. Das and A.K. Banejee, Hype Petesen Netwoks: Yet Anothe Hypecube-Lke Topology, Poc. outh Symp. ontes of Massvely Paallel Computaton, pp. 7-77, McLean, Va., Oct [7] K. Efe and A. enández, Poducts of Netwoks wth Logathmc Damete and xed Degee, IEEE Tans. Paallel and Dstbuted Systems, vol. 6, no. 9, pp , Sept [8] K. Efe and A. enández, Mesh Connected Tees: A Bdge between Gds and Meshes of Tees, IEEE Tans. Paallel and Dstbuted Systems, vol. 7, no. 1, pp. 1,83-1,93, Dec [9] T. El-Ghazaw and A. Youssef, A Geneal amewok fo Developng Adaptve ault-toleant Routng Algothms, IEEE Tans. Relablty, vol. 4, no., pp. 5-58, June [1] E. Ganesan and D.K. Padhan, The Hype-deBujn Netwoks: Scalable Vesatle Achtectue, IEEE Tans. Paallel and Dstbuted Systems, vol. 4, no. 9, pp , Sept [11] P.C. Kanen, A Lowe Bound fo Cossng Numbes of Gaphs wth Applcatons to K n, K p,q, and Q(d), J. Combnatoal Theoy, vol. 1, pp , 197. [1] D.J. Kletman, The Cossng Numbe of K 5,n, J. Combnatoal Theoy, vol. 9, pp , [13].T. Leghton, New Lowe Bound Technques fo VLSI, Poc. nd Ann. Symp. oundatons of Compute Scence, pp. 1-1, [14].T. Leghton, Complexty Issues n VLSI. Cambdge, Mass.: MIT Pess, [15].T.. Leghton, Intoducton to Paallel Algothms and Achtectues: Aays, Tees, and Hypecubes. San Mateo, Calf.: Mogan Kaufmann, 199. [16] C.E. Leseson, Aea-Effcent Gaph Layout (fo VLSI), Poc. 1st Ann. Symp. oundatons of Compute Scence, pp. 7-81, Oct [17] C.E. Leseson, Aea-Effcent VLSI Computaton, PhD thess, Canege-Mellon Unv., MIT Pess, [18] R.B. Panwa and L.M. Patnak, Soluton of Lnea Equatons on Shuffle-Exchange and Modfed Shuffle Exchange Netwoks, Poc. 6th Alleton Conf., pp. 1,116-1,15, [19] A.L. Rosenbeg, Poduct-Shuffle Netwoks: Towad Reconclng Shuffles and Buttefles, Dscete Appled Mathematcs, vol. 37/38, pp , July 199. [] D.D. Sheleka and J. JáJá, Layouts of Gaphs of Abtay Degee, Poc. 5th Ann. Alleton Conf., Sept

13 18 IEEE TRANSACTIONS ON COMPUTERS, VOL. 46, NO. 1, OCTOBER 1997 [1] D.D. Sheleka and J. JáJá, Balanced Gaph Dssectons and Layouts fo Heachcal VLSI Layout Desgn, Techncal Repot CSE- TR--89, Dept. of Electcal Eng. and Compute Scence, Unv. of Mchgan, Ann Abo, [] O. Sýkoa and I. Vto, On the Cossng Numbe of the Hypecube and the Cube Connected Cycles, Poc. 17th Int l Wokshop Gaph-Theoetc Concepts n Compute Scence, WG 91, G. Schmdt and R. Beghamme, eds., vol. 57, Lectue Notes n Compute Scence, pp schbachau, Gemany: Spnge-Velag, June [3] C.D. Thompson, A Complexty Theoy fo VLSI, PhD thess, Canege-Mellon Unv., Aug [4] J.D. Ullman, Computatonal Aspects of VLSI. Rockvlle, Md.: Compute Scence Pess, [5] L.G. Valant, Unvesalty Consdeatons n VLSI Ccuts, IEEE Tans. Computes, vol. 3, no., pp , eb [6] A.S. Youssef and B. Naaha, The Banyan-Hypecube Netwoks, IEEE Tans. Paallel and Dstbuted Systems, vol. 1, pp , 199. Antono enández eceved the degee of Dplomado en Infomátca n Mach 1988 and the degee of Lcencado en Infomátca n July 1991 fom the Unvesdad Poltécnca de Madd. He eceved the MS degee n compute scence n the fall of 199 and the PhD degee n compute scence n the fall of 1994 fom the Unvesty of Southwesten Lousana, suppoted by a ulbght Scholashp. D. enández s an assocate pofesso n the Depatamento de Aqutectua y Tecnología de Computadoes at the Unvesdad Poltécnca de Madd, whee he has been a membe of the faculty snce He s cuently on leave as a postdoctoal eseache at the MIT Laboatoy fo Compute Scence, suppoted by a gant fom the Spansh Mnsty of Educaton. Hs eseach nteests nclude paallel achtectues and algothms, nteconnecton netwoks, dstbuted systems, and data communcaton. Kemal Efe eceved the BSc degee n electonc engneeng fom Istanbul Techncal Unvesty, the MS degee n compute scence fom the Unvesty of Calfona at Los Angeles, and the PhD degee n compute scence fom the Unvesty of Leeds. D. Efe s cuently an assocate pofesso of compute scence at the Cente fo Advanced Compute Studes, Unvesty of Southwesten Lousana. He was pevously a membe of the faculty of the Compute Scence Depatment at the Unvesty of Mssou-Columba. Hs eseach nteests ae n paallel and dstbuted computng, n whch he has made many sgnfcant contbutons. Hs expetse ncludes paallel algothms and achtectues, nteconnecton netwoks, dstbuted opeatng systems, dstbuted algothms, pefomance evaluaton, and VLSI complexty models. D. Efe has seved on the techncal commttees of many ntenatonal confeences and has gven nvted talks n the U.S., Euope, and Japan. In 1995, D. Efe eceved the Cetfcate of Recognton fom NASA fo hs eseach contbutons. He s a membe of the ACM and the IEEE.