A Multilevel Analytical Placement for 3D ICs

Transcription

1 A Multlevel Analytcal Placement for 3D ICs Jason Cong, and Guoje Luo Computer Scence Department Unversty of Calforna, Los Angeles Calforna NanoSystems Insttute Los Angeles, CA 90095, USA Tel : (30) Fax : (30) e-mal : {cong, gluo}@cs.ucla.edu Abstract - In ths paper we propose a multlevel non-lnear programmng based 3D placement approach that mnmzes a weghted sum of total wrelength and TS va number subject to area densty constrants. Ths approach relaxes the dscrete layer assgnments so that they are contnuous n the z-drecton and the problem can be solved by an analytcal global placer. A ey dea s to do the overlap removal and devce layer assgnment smultaneously by addng a densty penalty functon for both area & TS va densty constrants. Expermental results show that ths analytcal placer n a multlevel framewor s effectve to acheve trade-offs between wrelength and TS va number. Compared to the recently publshed transformaton-based 3D placement method [], we are able to acheve on average % shorter wrelength and 9% fewer TS va compared to ther cases wth best wrelength; we are also able to acheve on average 0% shorter wrelength and 50% fewer TS va number compared to ther cases wth best TS va numbers. I. Introducton Three-dmensonal (3D) IC technologes can offer the potental to sgnfcantly reduce nterconnect delays and mprove system performance. Furthermore, the shortened wrelength, especally that of the cloc net, also lessens the power consumpton of the crcut. 3D IC technologes also provde a flexble way to carry out the heterogeneous system-on-chp (SoC) desgn by ntegratng dsparate technologes, such as memory and logc crcuts, rado frequency (RF) and mxed sgnal components, optoelectronc devces, etc., onto dfferent layers of a 3D IC. Devce layers n a 3D IC are connected usng through-slcon vas (TS va). However, TS vas are usually etched or drlled through devce layers by specal technques and are costly to fabrcate. A large number of the TS vas wll ncrease the area overhead and the cost of the fnal chp. Also, under the current technologes, TS va ptches are very large compared to the szes of regular metal wres, usually around 5-0μm. In 3D IC structures, TS vas are usually placed at the whtespace between the macro blocs or cells, so the number of TS vas wll not only affect the routng resource but also affect the overall chp or pacage areas. Therefore, the number of TS vas n the crcut s constraned and needs to be controlled durng physcal desgn. In recent years, 3D IC physcal desgn attracts more and more attenton. Along wth the technology updates, there are several publshed wors targeted on the 3D placement problem. A thermal-drven force-drected 3D placement method [] was proposed, where the temperature profle s nterpreted as thermal forces to gude the cell placement. In ther wor, a 3D force-drected placement engne s used to place each cell n a true 3D space, wth z poston beng a real number. Roundng s needed for layer assgnment, whch may ntroduce roundng errors. A foldng/stacng based 3D placement [] s proposed to reuse the D placement results and perform devce layer assgnment and other optmzatons. A partton-based approach [3] was also appled to the 3D context, where the temperature and TS va counts and thermal effect are modeled n the mn-cut objectve together wth the total wrelength. Although t s a convenent way to consder these effects wth ths partton-based method, recent comparatve analyss suggests that partton-based methods are not as compettve as analytcal methods for modern D placement problems [4]. A quadratc programmng approach [5] for 3D placement was also proposed, whch fnds an overlap-free placement by modelng the cell dstrbuton wth a dscrete cosne transformaton based cost functon. It shares the same problems as [], whch only dstrbute the cells n a cubod space but not nto layers. All these technques try to explore the trade-off among wrelength, TS va number and temperature. But the qualty of the soluton may be sub-optmal n cases. The goal of our wor s to frst develop a hgh-qualty solver for the smple 3D placement problem wth objectve of wrelength and TS va number, so that t can be used as a basc engne to consder other constrants and/objectves n 3D placement. In partcular, we develop a 3D placement approach usng a nonlnear optmzaton method to handle the 3D global placement. The man dea s to do the overlap removal and devce layer assgnment smultaneously by addng a densty penalty functon for both area & TS va densty constrants. The mnmzaton of ths densty penalty functon have the tendency to acheve overlap-free condton n (x,y)-drecton and also legal devce layer assgnment n z-drecton. The TS va number s also consdered by addng TS va number penalty functon to the objectve. The contrbutons of our wor are lsted as followng: A novel densty penalty functon s proposed to do the cell dstrbuton n a 3D placement regon. The mnmzaton of ths functon would result n a close-to-legal 3D global placement. A formal proof s also gven. Our method releves the roundng problem of prevous analytcal 3D placement methods, and thus reduces the effort for the detaled placement phase. We observe that multlevel scheme s effectve to control

2 TS va number, whch provdes extra TS va number reducton to the weghtng factor. Experments are performed to compare wth [] on the trade-off curves of wrelength and TS va number. The expermental results show that our method outperforms thers by achevng on average % shorter wrelength and 9% fewer TS vas compared to ther cases wth best wrelength; we are also able to acheve on average 0% shorter wrelength and 50% fewer TS va number compared to ther cases wth best TS va numbers. The remander of ths paper s organzed as follows. Secton II formulates the 3D placement problem and descrbes the placement flow. Secton III ntroduces our 3D analytcal placement engne and gves formal proofs for the equvalence between the mnmzer of our densty penalty functon and a legal soluton. Secton IV descrbes the multlevel framewor wth the analytcal placement engne. Secton V presents the expermental results to demonstrate the effect of our method on the trade-offs between wrelength and TS va number. Fnally, secton VI concludes the paper and dscusses about the future wor. II. Problem Formulaton and 3D Placement Flow A. Problem Formulaton Gven a crcut represented as a hypergraph H = ( V, E), the placement regon R (scaled to [0,] [0,] ), and the number devce layers K, the tas of 3D placement problem s to assgn every cell v V a trple ( x, y, z ), whch ndcates that ths cell s placed on the devce layer z {,,, K} wth ts center at ( x, y) R. The objectve s to mnmze the weghted sum of total wrelength and TS va number, under non-overlap constrants. mnmze ( le ( ) + α ve ( )) e E () subject to ( non-overlap constrants) We use the tradtonal half-permeter model for wrelength calculaton, where the wrelength le ( ) of a net e s calculated as follows: le () = maxx x + maxy y () j j v, vj e v, vj e Because the routng nformaton s unnown durng the placement process, TS va number s also estmated through a smlar model. The TS va number of a net s calculated as the heght of the boundng cube of the cells belong to that net. The TS va number ve ( ) of a net e s calculated as follows: ve () = maxz z (3) v, vj e The weghtng factor α s used to acheve the trade-offs between the wrelength le ( ) and the TS va number ve ( ). In modern analytcal placers (e.g. those descrbed n [6]), the non-overlap constrants are usually transformed to densty constrants durng global placement, and are legalzed durng detaled placement. The formal descrptons wll be gven n Secton III.D. B. 3D Placement Flow j The overall placement flow s shown n Fg.. The global placement starts from scratch, or taes n the gven ntal placement. The global placement ncorporates the analytcal placement engne (Secton III) nto the multlevel framewor that s used n [7]. The global placement s then processed layer-by-layer wth the D detaled placer [8] to obtan the fnal placement. Intal Netlst Coarsenng Fnest Netlst Coarsest Netlst Interpolaton Fnal Placement Fg. Our 3D Placement Flow III. Analytcal Placement Engne The analytcal placement engne solves problem () by transformng the non-overlap constrants to densty penaltes. mnmze ( le ( ) + α ve ( )) e E (4) subject to Penaltyxyz (,, ) = 0 The wrelength le ( ) (Secton III.B), the TS va number ve () (Secton III.C), and the densty penalty functon Penalty( x, y, z) (Secton III.D) wll be descrbed n the followng subsectons n detal. In order to solve ths constraned problem, penalty methods [9] are usually appled: OBJxyz (,, ) = ( le ( ) + α ve ( )) + μ Penaltyxyz (,, ) e E (5) Ths penalzed objectve functon s mnmzed at each teraton, wth a gradually ncreasng penalty factor μ to reduce the densty volatons. It can be shown that the mnmzer of equaton (5) s equvalent to problem (4) when μ f the penalty functon s non-negatve. A. Relaxaton of Dscrete Varables As mentoned n Secton II.A, the placement varables are represented by trples ( x, y, z ), where z s a dscrete varable n {,,..., K }. The range of z s relaxed from the set {,,, K} to a contnuous nterval [, K ]. After relaxaton, a nonlnear analytcal solver can be used n our placement engne. The relaxed soluton s mapped bac to the dscrete values before the detaled placement phase. B. Log-sum-exp Wrelength Relaxaton N Intalze/Update Penalty Factor Mnmze the Penalzed Objectve Converge? Y Fnest Level Done? Y Layer-by-layer Detaled Placement N

3 The half-permeter wrelength le ( ) defned n equaton () s replaced by a dfferentable approxmaton wth log-sum-exp functon [0]: le ( ) η(log exp( x / η) + log exp( x / η) v e v e (6) + log exp( y / η) + log exp( y / η)) v e v e For numercal stablty, the placement regon R s scaled nto [0,] [0,], thus varables of ( x, y ) are n the range between 0 and, and the parameter η s set to 0.0 n mplementaton as []. C. TS Va Number The TS va number ve ( ) estmaton defned n equaton (3) s also replaced by the log-sum-exp approxmaton: ve ( ) η(log exp( z / η) + log exp( z/ η)) D. Densty Penalty Functon v e (7) v e whch extends the densty functon (8) from ntegral layer assgnments to the defnton (9) for relaxed layer assgnments. It s obvous that (9) s consstent wth (8) when the layer assgnments { } z are ntegers. An example of how ths extenson wors for a 4-layer 3D placement s gven n Fg. 3. The x-axs s the relaxed layer assgnment n z-drecton, whle the y-axs ndcates the amount of area to be projected n the actual devce layers. The four curves colored n red, green, blue and purple represents devce layer,, 3 and 4 respectvely. As n ths example, a cell s temporarly placed at z =.36 (yellow trangle) between layer and layer 3. The bell-shaped densty projecton functons project 80% of ts area to layer (green) and 0% of ts area to layer 3 (blue). In ths way, we establsh a mappng from a relaxed 3D placement to the area dstrbutons n dscrete layers. The densty penalty functon s for overlap removal n both the (x,y)-drecton and the z-drecton. The mnmzaton of the densty penalty functon should lead to a non-overlap placement n theory. Assume that every cell v has a legal devce layer assgnment (.e. z {,,, K} ), then we can defne K densty functons for these K devce layers. Intutvely, the densty functon D ( uv, ) ndcates the number of cells whch cover the pont ( uv, ) on the -th devce layer. It s defned as: D( uv, ) = d(, ) z : uv (8) = whch s the sum of the densty contrbuton d ( u, v ) of cell v assgned to ths devce layer at pont ( uv, ). The densty contrbuton d ( u, v ) s one nsde the area occuped by v, and s zero outsde ths area. An example s gven n Fg. showng the densty functon wth two overlappng cells. v u D(u,v) Fg. an Example of the Densty Functon Durng global placement, t s possble that cell v stays between two devce layers, so that the varable z [, K ] s not algned to any of two devce layers. We borrow the dea from the bell-shaped functon [] to defne the densty functon for ths case: D( uv, ) = η( z, ) d( uv, ), for K (9) where ( z ) z η(, z) = ( z ) < z (0) 0 otherwse We call (0) the bell-shaped densty projecton functon, = 0 = = Fg. 3 an Example of the Bell-shaped Densty Projectons Inspred by the quadratc penalty terms n D placement methods [-3], we defne ths densty penalty functon to measure the amount of overlaps: Pxyz (,, ) = D( uv, ) dudv ( ) K = () Lemma. Assume the total area of cells equals the placement area (.e. area ( v ) = K, no empty space), every legal placement ( x, y, z ), whch satsfes D ( u, v ) = for every and ( uv, ) wthout any non-nteger z, s a mnmzer of Pxyz (,, ). The proof of Lemma s trval and thus s omtted. Therefore, mnmzng Pxyz (,, ) provdes a necessary condton for a legal placement. However, there exst mnmzers that cannot form a legal placement. An example s shown n Fg. 4, where placement (b) also mnmzes the densty penalty functon but t s not legal. Fg. 4 Two Placements wth the Same Densty Penaltes To avod reachng such mnmzers, we ntroduce the nterlayer densty functon: E( u, v) = η( + 0.5, z) d( u, v), for K () and also the nterlayer densty penalty functon:

4 K Qxyz (,, ) = ( E(, ) ) uv dudv = (3) Smlar to the densty penalty functon Pxyz (,, ), the followng Lemma s also true. Lemma. Assume the total area of cells equals the placement area, every legal placement s a mnmzer of Qxyz (,, ). Combnng the densty penalty functons Pxyz (,, ) and Qxyz (,, ), we defne the followng densty penalty functon: Penalty( x, y, z) = P( x, y, z) + Q( x, y, z) (4) Theorem. Assume the total area of cells equals the placement area, every legal placement ( x, y, z ) s a mnmzer of Penalty( x, y, z), and vce versa. Proof. It s obvous that every legal placement s a mnmzer of Penalty( x, y, z) by combnng Lemma and Lemma. We shall prove that every mnmzer ( x, y, z ) of Penalty( x, y, z) s a legal placement. From the proof of Lemma and Lemma, we now the mnmum value of Penalty( x, y, z) s acheved f and only f D ( u, v ) = and E ( u, v ) = for every and ( uv, ). Frst, f all the components of z are ntegers, t s easy to see the placement s legal, because all the cells are assgned to a certan devce layer, and for any pont ( uv, ) on any devce layer there s only one cell coverng ths pont (no overlaps). Next, we show that there does not exst a z wth a non-nteger value (proof by contradcton). If a cell v has a non-nteger z, we now that there are K cells coverng K ( x, y ) because D (, ) x y = K. Accordng to the = pgeonhole prncpal, among these K cells there are at least two cells v, v wth the z-drecton dstance z z <, snce all the varables { z } are n the range of [, K ]. Wthout loss of generalty we may assume z z, therefore there exsts an nteger {,, K} such that ether z (, + 0.5] and z (,.5) +, or z ( 0.5, ] and z ( 0.5, + ). It s easy to verfy that n the former case z ( + 0.5) + z ( + 0.5) < and E ( x, y ) η( + 0.5, z ) + η( + 0.5, z ) > ; n the later case z + z < and D( x, y ) η(, z ) + η(, z) >. Both cases lead to ether E( x, y ) > or D( x, y ) >, whch conflct wth the assumpton that ( x, y, z ) s a mnmzer of Penalty( x, y, z). Therefore there does not exst a non-nteger z, and every mnmzer of Penalty( x, y, z) s a legal placement n the z-dmenson. In the analytcal placement engne, the denstes D ( uv, ) and E ( uv, ) are replaced by smoothed denstes D ( uv, ) and E ( uv, ) for dfferentablty. As n [], the denstes are smoothed by solvng Helmholtz equatons: D( uv, ) = ( + ε ) D(, ) uv u v (5) E( uv, ) = ( + ε ) E(, ) uv u v And the smoothed densty penalty functon K Penalty( x, y, z) = ( D (, ) ) u v dudv = K (6) + ( E (, ) ) u v dudv = s used n our mplementaton, whose gradent s computed effcently wth the method n [4]. IV. Multlevel Framewor The optmzaton problem below summarzes our analytcal placement engne: ( le () + αve ()) e E K mnmze + μ ( ( D (, ) ) u v dudv = (7) K + ( D (, ) ) u v dudv = ) ncrease μ untl the densty penalty s small enough Ths analytcal engne s ncorporated nto the multlevel framewor n [7], whch conssts of coarsenng, relaxaton, and nterpolaton. The purpose of coarsenng s to buld a herarchy for the multlevel dagram, where we use the best-choce hypergraph clusterng [5]. After the herarchy s set up, multple placement problems are solved from the coarsest level to the fnest level. In a coarser level, clusters are modeled as cells and the connectons between clusters are modeled as nets, so that there s one placement problem for each level. The placement problem at each level s solved (relaxed) by the analytcal engne (7). These placement problems are solved n the order from the coarsest level to the fnest level, where the soluton at a coarser level s nterpolated to obtan an ntal soluton of the next fner level. The cell wth hghest degree n a cluster s placed n the center of ths cluster (C-ponts), whle the other cells are placed at the weghted average locatons of ther neghborng C-ponts, where the weghts are proportonal to the connectvty to those clusters. V. Expermental Results Our experment s performed on the IBM-PLACE benchmar [6], whch s a standard cell crcut wthout I/O ports, as was done n []. In ths experment, we also assume a 4-layer mplementaton of 3D IC. The floorplan sze s scaled from [6] by dvdng the orgnal area by 4, and then enlargng t to obtan 0% whte space. Fller cells wthout connecton to any other cells are added to ensure the exstence of feasble solutons for the equalty constrants n problem (4).

5 TABLE I Benchmar Characterstcs and 3D Placement Results from [] Crcut #cell #net D WL LST (r=0%) LST (8x8 wn) Foldng- WL WL WL bm bm bm bm bm bm bm bm bm bm geo-mean normalzed TABLE II Expermental Results for our Multlevel Analytcal Placement Method wth α=0 -Level Placement (flat) -Level Placement 3-Level Placement Crcut Runtme (mn) Runtme (mn) Runtme (mn) bm bm bm bm bm bm bm bm bm bm geo-mean normalzed In TABLE I, we lst the statstcs of the 0 out of 8 crcuts n the benchmar, whch are the only crcuts wth avalable expermental results n []. We also lst partal results from the prevous transformaton-based 3D placement method [] for later comparson. Here only the results of the three typcal transformaton schemes are lsted, ncludng the local-stacng transformaton LST (r=0%), wndow-based transformaton LST (8x8 wn) and a foldng transformaton Foldng-. The geometrc averages are computed to measure the overall results. These results lsted are for trade-offs between wrelength and TS va number wthout awareness of temperature, thus the followng comparsons are vald. TABLE II presents the results for our multlevel analytcal placer wth the TS va weght α = 0. Due to the comparablty and the page lmt, we only show our results on those 0 crcuts n the benchmar. Three sets of results are collected, for -level placement, -level placement and 3-level placement, respectvely. The -level placement s to run the analytcal placement engne drectly wthout any clusterng, whle -level or 3-level placements construct a -level or 3-level herarchy by clusterng. The wrelength after global placement (), the wrelength after detaled placement (), the number of TS vas (), and the runtme are all collected. In TABLE II, we see that wth the same weght for TS va number, -level placement acheve the shortest wrelength, whle the 3-level placement acheve the fewest TS va number. We compare our multlevel analytcal placement method and the prevous transformaton-based placement method [] by comparng our -level placement wth the LST (r=0%) (the best wrelength case), and comparng out -level placement wth the LST (8x8 wn), and compare the 3-level placement wth the Foldng- method (the best TS va case). From the data shown n TABLE I and TABLE II, t s clear that our -level placement acheves on average % shorter wrelength and 9% fewer TS va than LST (r=0%) ; our 3-level placement also acheves on average 30% shorter wrelength wth slghtly fewer TS va number than Foldng-. In order to obtan a more complete comparson, dfferent trade-off curves are generated for the crcut bm3 n [6]. In the curves -Level, -Level and 3-Level, the TS va number control s acheved by ncreasng the weghtng factor α, from 0 to In the meantme, the results of bm3 wth LST (r=0%), LST (r=0%), LST (8x8 wn), Foldng-4 and Foldng- n [] are also connected by a curve to vsualze the trade-offs. In ths example, t s very clear that the analytcal placement engne outperforms the transformaton-based method both n wrelength and TS va number. Moreover, the multlevel framewor enables a sharp reducton of TS va number, wthout much degradaton n the wrelength. Be aware that the results for other crcuts have smlar behavors. We also notce that n the comparson between Foldng- and our 3-level placement, the latter does not acheve as few TS va number as Foldng- for two large crcuts. It s because the analytcal engne mnmzes the objectve (( le) + 0 ve ()) and the fxed weghtng factor α = 0 e E maes the results scale dfferently from the Foldng-

6 method. By measurng the results wth ths objectve functon, the 3-level placement stll outperforms the Folndg- method. Fg. 5 Trade-off Curve for the Crcut bm3 Moreover, to demonstrate the ablty of controllng the TS va number wth our multlevel analytcal 3D placer, we run the experments by a 3-level placement wth α = 000. The expermental results are shown n TABLE III. Compared to Foldng-, ths set of results acheves on average 0% shorter wrelength and 50% fewer TS va number. In addton, the TS va number obtaned by 4-way mncut usng hmets [7] s also lsted n ths table for reference, whch focuses on TS va mnmzaton only by mnmzng the total cutsze. Our TS va numbers are very close to the mncut results whch can be vewed almost as a lower bound. TABLE III Expermental Results for 3-Level Placement wth α=000 3-Level Placement 4-way Mncut Crcut cutsze bm bm bm bm bm bm bm bm bm bm geo-mean normalzed VI. Conclusons and Future Wor In ths paper a multlevel analytcal 3D placement algorthm s proposed, whch mnmzes the weghted sum of wrelength and TS va number wth careful handlng of overlap removal and devce layer assgnment n an analytcal solver. The overlap removal and devce layer assgnment are handled by a densty penalty functon, whose mnmzer s guaranteed to be a legal placement n the z-dmenson. Expermental results demonstrate that our multlevel framewor combned wth the analytcal engne s very effectve n controllng the TS va numbers. We note that the multlevel placement does not obtan wrelength as short as the flat placement. A possble explanaton s that clusterng lmts soluton space by requrng all cells n a cluster to be n the same layer. Further study to overcome ths lmtaton s underway. Acnowledgements Ths research s partally supported by IBM under a DARPA subcontract, and supported by Natonal Scence Foundaton under grants CCF and CCF References [] J. Cong, G. Luo, J. We, and Y. Zhang, Thermal-Aware 3D IC Placement va Transformaton, Proceedngs of the th Asa and South Pacfc Desgn Automaton Conference, pp , 007. [] B. Goplen and S. Sapatnear, Effcent Thermal Placement of Standard Cells n 3D ICs usng a Force Drected Approach, Proceedngs of the 003 IEEE/ACM Internatonal Conference on Computer-Aded Desgn, pp. 86, 003. [3] B. Goplen and S. Spatnear, Placement of 3D ICs wth Thermal and Interlayer Va Consderatons, Proceedngs of the 44th Annual Conference on Desgn Automaton, pp , 007. [4] G.-J. Nam, ISPD 006 Placement Contest: Benchmar Sute and Results, Proceedngs of the 006 Internatonal Symposum on Physcal Desgn, pp , 006. [5] H. Yan, Q. Zhou, and X. Hong, Effcent Thermal Aware Placement Approach Integrated wth 3D DCT Placement Algorthm, Proceedngs of the 9th Internatonal Symposum on Qualty Electronc Desgn, pp. 89-9, 008. [6] G.-J. Nam and J. Cong, Modern Crcut Placement : Best Practces and Results, Sprnger, New Yor, 007. [7] J. Cong and J. Shnnerl, Multlevel Optmzaton n VLSICAD, Kluwer Academc Publshers, Boston, 003. [8] J. Cong and M. Xe, A Robust Mxed-Sze Legalzaton and Detaled Placement Algorthm, IEEE Transactons on Computer-Aded Desgn of Integrated Crcuts and Systems, vol. 7, no. 8, pp , August 008. [9] J. Nocedal and S.J. Wrght, Numercal Optmzaton nd ed., Sprnger, 006. [0] W.C. Naylor, R. Donelly, and L. Sha, Non-lnear Optmzaton System and Method for Wre Length and Delay Optmzaton for an Automatc Electrc Crcut Placer, US Patent , October, 00. [] T. Chan, J. Cong, and K. Sze, Multlevel Generalzed Force-drected Method for Crcut Placement, Proceedngs of the 005 Internatonal Symposum on Physcal Desgn, pp. 85-9, 005. [] A.B. Kahng, S. Reda, and Q. Wang, Archtecture and Detals of a Hgh Qualty, Large-scale Analytcal Placer, Proceedngs of the 005 IEEE/ACM Internatonal Conference on Computer-Aded Desgn, pp , 005. [3] T.-C. Chen, Z.-W. Jang, T.-C. Hsu, H.-C. Chen, and Y.-W. Chang, A Hgh-qualty Mxed-sze Analytcal Placer Consderng Preplaced Blocs and Densty Constrants, Proceedngs of the 006 IEEE/ACM Internatonal Conference on Computer-Aded Desgn, pp. 87-9, 006. [4] J. Cong and G. Luo, Hghly Effcent Gradent Computaton for Densty-constraned Analytcal Placement Methods, Proceedngs of the 008 Internatonal Symposum on Physcal Desgn, pp , 008. [5] C. Alpert, A. Kahng, G.-J. Nam, S. Reda, and P. Vllarruba, A Sem-persstent Clusterng Technque for VLSI Crcut Placement, Proceedngs of the 005 Internatonal Symposum on Physcal Desgn, pp , 005. [6] [7] G. Karyps and V. Kumar, Multlevel -way Hypergraph Parttonng, Proceedngs of the 36th ACM/IEEE Conference on Desgn Automaton, pp , 999.