Laplacian spectrum analysis and spanning tree algorithm for circuit partitioning problems

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1 Vol. 6 No. 6 SCIENCE IN CHINA (Seres F) December 003 Laplaca spectrum aalyss ad spag tree algorthm for crcut parttog problems YANG Huazhog ( p) & HU Guazhag ( ). Departmet of Electroc Egeerg, sghua Uversty, Beg 0008, Cha;. Departmet of Mathematcal Scece, sghua Uversty, Beg 0008, Cha Correspodece should be addressed to Yag Huazhog (emal: yaghz@tsghua.edu.c) Receved February 6, 003 Abstract he spectrum of a graph s the set of all egevalues of the Laplaca matrx of the graph. here s a closed relatoshp betwee the Laplaca spectrum of graphs ad some propertes of graphs such as coectvty. I the recet years Laplaca spectrum of graphs has bee wdely appled may felds. he applcato of Laplaca spectrum of graphs to crcut parttog problems s revewed ths paper. A ew crtero of crcut parttog s proposed ad the bouds of the partto rato for weghted graphs are also preseted. Moreover, the defcecy of graph-parttog algorthms by Laplaca egevectors s addressed ad a algorthm by meas of the mmal spag tree of a graph s proposed. By vrtue of takg the graph structure to cosderato ths algorthm ca fulfll geeral requremets of crcut parttog. Keywords: graph parttog, Laplaca spectrum of a graph, partto rato, spag tree of a graph. DOI: 0.360/0yf005 Wth the appearg of system-o-a-chp, may ew problems, such as software-hardware parttog, reusable IP based desg, platform based desg, ad low power desg etc., are becomg more ad more dffcult but crtcal to electroc desg automato. All these problems are cocered wth the parttog of weghted graphs (ether wth edge weghts, vertex weghts, or both). he weghted graphs are more complcated. It s well kow that the crtero of parttog s of great mportace by whch we ca dscrmate the optmal partto from others. I ths paper we wll study the crtero of parttog, ts relatoshp wth the Laplaca spectrum, ad algorthms for crcut parttog. Let G(V, E) be a graph wth vertex set V ad edge set E, V =, A(G) be the adacet matrx of G. D( G) = dag( d, d,, d ) s called the degree matrx of G, whered s the degree of vertex v, =,,f,. he L( G) = D( G) A( G) s called Laplaca matrx of G. he ame Laplaca matrx came from the fact that the coeffcet matrx of dfferece equatos of the drum vbrato problem turs up to be the Laplaca matrx of the correspodg grd graph. he theory o the egevalues ad egevectors of A(G) s called the spectrum theory of graphs whch s oe of the ma topcs of algebrac graph theory []. he theory o the egevalues ad egevectors of L(G) s called the Laplaca spectrum theory of graphs. I recet years the Laplaca spectrum theory has gaed a lot of applcatos ad great progress [,3].

2 60 SCIENCE IN CHINA (Seres F) Vol. 6 he techology ad otatos cocered wth graph theory ca be foud ref. [] or [5]. Improvemet of the crcut parttog crtero A large scale ad complcated system s ofte dvded to small ad smple subsystems whch ca be subsequetly coquered oe by oe. hs methodology ca also be appled to aalyzg large scale ad complcated crcuts. he crcut s frstly mapped to a graph G; the the graph G s parttoed to two or several parts to make the aalyss procedure smpler ad more effcet. hs techque s called the crcut parttog [6]. What s the crtero of parttog? here are some commo requremets. Suppose we wat to partto the vertex set of G to two subsets, A ad B, such that V = A B ad A B =. hs partto s deoted by (A, B), ad subgraphs duced by A ad B are deoted by G[A] adg[b], respectvely. he set of edges betwee A ad B s deoted by E (A, B) whch s called the edge cut of G. Usually we requre the partto to satsfy the followg codtos: (a) Mmum edge cut: the edge cut E(A, B)sassmallaspossble. (b) Uformty: the dfferece betwee A ad B sassmallaspossble. (c) Coectvty: at least oe of G[A]adG[B] s coected. A partto satsfyg all the above three requremets s called the optmal partto. But t s dffcult to obta a partto satsfyg all the three requremets at the same tme. For example, a star graph could ot be parttoed to two coected subgraphs wth the same order. herefore cut rato s proposed as the crtero of parttog [6]. ρ ( AB, ) = E( AB, ) A B. Geerally, the frst Defe parttog rato or cut rato to be ( ) two requremets (a) ad (b) wll be satsfed by meas of reducg the parttog rato. he partto wth the mmal parttog rato may be optmal, but t s ot always true. For example, for a complete graph K the parttog rato of ay partto (A, B) s ρ ( AB, ) = ( ) ( ) ( ) EAB (, ) A B = A B A B =, ad obvously we caot determe whch s optmal by the parttog rato. herefore the above parttog rato s ot a good crtero for graphs wth dese edges. I order to mprove the parttog crtero we defe a ew parttog rato EAB (, ) ρ( AB, ). () A B Now for complete graph K f (A, B) s ay partto wth A = k, the the parttog rato ρ( AB, ) s EAB (, ) ρ( AB, ) =. A B k ( k) I ths case the optmal parttog s obtaed whe k = ad ρ ( AB, ) =. By

3 No. 6 LAPLACIAN SPECRUM ANALYSIS FOR CIRCUI PARIIONING PROBLEMS 6 ρ( AB, ) we ca determe the optmal partto for graphs ot oly wth dese edges but also wth sparse edges. Graphs proposed from practcal problems are usually wth edge-weghts, or vertex-weghts, or both of them. For the graphs wth both edge- ad vertex-weghts we defe the parttog rato to be wab (, ) ρ ( AB, ), ha ( ) hb ( ) where we ( ), e E s the edge-weght fucto, hv ( ), v V wab (, ) = wuv (, ), hv ( ) = hv ( ), V V. ( uv, ) E v V u A, v B () s the vertex-weght fucto, ad It s easy to see that ρ( AB, ) s the partcular case of ρ( AB, ) whe we ( ) =, e E ad hv ( ) =, v V. Relatoshp betwee graph parttog ad Laplaca spectrum of L( G) Let L( G) = D( G) A( G). It s kow that 0 s a egevalue of L( G). All the egevalues are oegatve ad 0 s a sgle egevalue whe G s coected. he egevector that belogs to 0 s = ( ),,,. Let the egevalues of L( G) be 0 < λ λ 3 λ. he secod egevalue λ s amed the algebrac coectvty by Fedler [7]. From ow o we deote the egevalue of A(G) byµ ad the egevalue of L(G) byλ. Fork-regular graphs they have the followg relato λ = k µ. (3) We suppose that G s coected the followg dscusso. he spectrum of G ca be represeted by µ µ µ k Spec( G) =, () m m mk where k s the umber of dfferet egevalues of G, ad µ > µ > > µ k, m s the multplcty of µ. he spectrum of G also ca be represeted by the set of all egevalues of G. he Laplaca spectrum of G s deoted by LSpec(G), a form smlar to eq. (), or by the set of all Laplaca egevalues of G. he spectrum ad Laplaca spectrum of some typcal smple graphs are gve as follows. 0 () he complete graph K : Spec ( k ) =, LSpec ( k ) =. π () he cycle C : Its spectra are µ = cos, = 0,,,, ad ts Laplaca spectra

4 6 SCIENCE IN CHINA (Seres F) Vol. 6 π are λ = s, = 0,,,. () he path P : Its spectra are π are λ = s, = 0,,,. π µ = cos, =,,,, + 3 (v) he Peterse graph O 3 : Spec ( O3 ) =, 5 ad ts Laplaca spectra 5 0 LSpec ( O3 ) =. 5 he algebrac coectvty λ has a mportat applcato the crcut parttog. Hage ad Kahag obtaed the relatoshp betwee ρ( AB, ) ad λ for uweghted graphs [8].We geeralzed t to weghted graphs wth edge weghts ad used t to solve the problems low power crcuts desg [9]. I ths paper we wll further study the relatoshp betwee ρ( AB, ) ad λ for graphs wth both edge- ad vertex-weghts. heorem. Let G(V, E) be a coected graph wth edge weghts: we ( ), e E weghts hv ( ), v V. G d d d A( G) = ( a, ) a, ad wv (, v) = w,, f ( v, v) E, = 0, f ( v, v) E. ad vertex = D ( ) = dag(, ),where d = a,. L( G) = D( G) A( G). λ s the algebrac coectvty of G. he for ay partto (A, B)ofG we have λ λ ρ 3 ( AB, ) h ( ) h max where h = { h v } h = { h v } max Proof. max ( ), m ( ). m m, (5) he matrx L(G) s symmetrc ad oegatve. Because G s coected 0 s a sgle egevalue of L(G). he egevector of 0 s = ( ) we have λ Χ LΧ,,,. herefore Χ ad Χ 0 λ. Χ Χ Let A = γ, B = s. he γ + s =. We costruct a vector X as follows s, v A, x = γ, v B.

5 No. 6 LAPLACIAN SPECRUM ANALYSIS FOR CIRCUI PARIIONING PROBLEMS 63 It s easy to see he we have γ s Χ, Χ Χ =, ad x x, f v A ad v B, = 0, otherwse. = dx a, xx = a, ( x x) = < < Χ LΧ = wv (, v) = wab (, ). v A, v B Because γ s ad γ s = Χ Χ, we obta wab (, ) X LX ρ ( AB, ) = ƒ ha ( ) hb ( ) γ sh O the other sde of eq. (5) we have = max γshmax X LX X X ƒ λ h 3 wab (, ) X LX X LX ρ ( AB, ) = = ha ( ) hb ( ) γ shm γshm X X λ. ( ) hm hus the proof s fshed. Q.E.D. Now let us cosder ρ of uweghted graphs dscussed secto. For the complete graph K the optmal partto s to halve V ad the optmal cut rato s ρ ( AB, ) =. he lower boud of ρ( AB, ) eq. (5) equals exactly. For the cycle C the optmal partto s to halve V ad the optmal cut rato s 3 6 π ρ ( AB, ) =. he lower boud of ρ ( AB, ) eq. (5) s s. Whe thelower 3 6π boud of ρ( AB, ) teds to 5. For Peterse graph O 3 the optmal partto s to halve V ad the optmal cut rato s ρ ( AB, ) =. he lower boud of ρ 3 ( AB, ) eq. (5) s also We ca see from the above examples that the lower boud of ρ( AB, ) the optmal partto rato ad reachable some specal cases. 3 Algorthms for the optmal parttog max. eq. (5) s close to If we use the exhaustg method to search for the optmal partto for a crcut, the complexty of computato s the type of expoet. Usually heurstc algorthms are used to the crcut parttog [0]. hese algorthms ca be classfed to the followg classes: teratve methods [,],

6 6 SCIENCE IN CHINA (Seres F) Vol. 6 geetc methods [3], graph spectrum methods [8], etwork flow methods [], etc. he graph spectrum methods pck up a tal partto by the egevector of the algebrac coectvty. hs kd of method was frstly proposed by Fedler [5] 973 ad a lot of progress [6] has bee made from the o. heorem [6]. Let λ be the algebrac coectvty of L(G) ad X = ( x, x,, x ) the correspodg egevector. For a real umber rƒ0 defe V( r) = { v V xv ƒ r} V( r) = { v V xv r }. he the subgraphs duced by V (r)adv (r) are both coected. he parttog methods by heorem ca get at least oe coected subgraph. For some specal graphs, early halved parttog ca get oly oe coected subgraph. For example, ay bpartto to a star graph K, ca get oly oe coected subgraph whe ƒ. We ca partto G to two parts by heorem ad set r to zero: ) If X does ot have ay zero compoets, weset { v } { v } be ad A= v Vx > 0, B= v Vx < 0. (6) If A ad B are ot empty, subgraphs G[A] adg[b] are both coected. hus we get the partto (A, B) ad edge cut E(A, B). ) If some compoets of X are zero we ca dstrbute the vertces correspodg to these zero compoets to A or B so that G[A] org[b] s coected, ad A ad B are as close as possble. he partto obtaed by ths method may ot be the optmal oe, so other methods should be used to decrease the parttog rato. I order to decrease the parttog rato, esure the coectvty of oe part of a partto, ad keep A B, we propose a algorthm by meas of the mmal spag tree. he mmal spag tree of G s a subgraph m of G such that m satreehavgthemmal weghts ad cotag all vertces of G. he mmal spag tree of G ca be obtaed by the followg algorthm wth polyomal computatoal complexty []. Step. ake a edge e such that we ( ) = m { we ( )}, let = { e } Step. If { e e} e E m =,, s obtaed, choose a edge A) there s o cycle the subgraph Ge [,, e, e + ] B) we ( + ) s as small as possble uder codto A. Step 3. Repeat step utl m =. m. e + from E { e e} \,, duced by { } e,, e, e +, such that Let m be the mmal spag tree of G. he we ca fd the ceter of m by successvely deletg the -degree vertces. By deletg some edges cdet wth the ceter, m ca be parttoed to two subtrees ad wth early equal orders. Let A= V( ) ad B = V( ). We get a partto (A, B). At least oe of the subgraphs G[A] adg[b] s coected, ad A B s also realzed. Furthermore we ca use the teratve method to mprove the partto so that ad

7 No. 6 LAPLACIAN SPECRUM ANALYSIS FOR CIRCUI PARIIONING PROBLEMS 65 ρ (A, B) sassmallaspossble. he weght of the edge cut of the partto by meas of the mmal spag tree s small ad possbly close to the optmal partto. akg the crcut structure to cosderato, more reasoable parttos ca be obtaed at smaller computatoal cost. Furthermore, costrats ca also be embedded to the mmal spag tree based parttog algorthm, ad the optmal partto ca fulfll some specal requremets of crcut desgs. Cocluso he parttog rato ρ ( AB, ) = E( AB, ) ( A B) s ot reasoable specally for the wab (, ) graphs wth dese edges. he ew parttog rato ρ ( AB, ) proposed ths ha ( ) hb ( ) paper s more reasoable for ay weghted ad uweghted graphs. We gve the relatoshp betwee ρ (A, B) ad the Laplaca spectrum so that we ca determe the optmalty of the partto (A, B). After addressg the defects of the algorthms of fdg the optmal partto by meas of the Laplaca egevector X, the mproved crcut parttog methodology whch s a combato of the mmal spag tree ad teratve methods s proposed ths paper. Ackowledgemets hs work was supported part by the Natoal Natural Scece Foudato of Cha (Grat Nos ad ), ad by the Natoal Basc Research Prortes Program (Cotract No. G ). Refereces. Bggs, N. L., Algebrac Graph heory, Cambrdge: Cambrdge Uversty Press, frst edto 97, secod edto, Groe, R., Merrs, R., Suder, V., he Laplaca spectrum of a graph, SIAM J. Matrx Aal. Appl., 990, (): Merrs, R., Laplaca matrces of graphs: A survey, Lear Algebra ad Its Applcatos, 99, 97-98: Body, J. A., Murty, U. S. R., Graph heory wth Applcatos, Lodo: he Macmlla Press LD, Hu, G. Z., Hadbook o Moder Appled Mathematcs, Volume of Dscrete Mathematcs: Combatorcs ad Graph heory ( Chese), Beg: sghua Uversty Press, Alpert, C. J., Kahg, A. B., Recet drectos etlst parttog A survey, Itegrato, 995, 9: Fedler, M., Algebrac coectvty of graphs, Czechoslovak Math. J., 973, 3(98): Hage, L., Kahg, A. B., New spectral methods for rato cut partto ad clusterg, IEEE ras. Computer-Aded desg, 99, : Yag, H. Z., Hu, G. Z., Wag, H., Crtero for reducg power cosumpto by meas of rato-cut crcut parttog, Electroc Letters, 3rd August, 000, 36(6): Legauer,., Combatoral Algorthms for Itegrated Crcut Layout, New York: Wley-euber, Kergham, B., L, S., A effcet heurstc procedure for parttog of electrcal crcuts, Bell Syst. ech. J., 970, 9: Fducca, C. M., Matheyses, R. M., A lear tme heurstc for mprovg etwork parttos, Proc. ACM/IEEE Desg Automato Cof., 98, Bu,. Z., Moo, B. R., Geetc algorthm ad graph parttog, IEEE ras. Comput., 996, 5: Yag, H., Wag, D. F., Effcet etwork flow based o m-cut, balaced parttog, IEEE ras. Computer-Aded Desg, 996, 5(): Fedler, M., A Property of egevectors of oegatve symmetrc matrces ad ts applcato to graph theory, Czechoslovak Math. J., 973, 5(00): Pothe, A. et al., Parttog sparse matrces wth egevectors of graphs, SIAM J. Matrx Aal. Appl., 990, 30 5.