Parasitic Aware Routing Methodology based on Higher Order RLCK Moment Metrics

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1 Parastc ware Routng Methodology based on Hgher Order RLCK Moment Metrcs mtava Bhadur and Ranga emur Department of ECECS Unversty of Cncnnat, Cncnnat, OH 4522 {bhadhua, BSTRCT In the mult-ghz frequency doman, nductve and capactve parastcs of nterconnects can cause sgnfcant rngng or overdampng, whch may lead to false swtchng or ncreased delay. Routng technques whch rely on wre length reducton or couplng capactance mnmzaton are unable to obtan the best routng soluton. In ths paper, the nductve and capactve nteractons of the wres are ntroduced through a moment based cost functon. Ths hgher order RLCK moment metrc gudes the routng process of long wres, ensurng that the chosen soluton has the best trade-off between rngng and delay under a monotone sgnal response. nother sgnfcant departure from exstng routng methodologes s to account for the sgnal drecton n the nets whch may ncrease or decrease the effectve nductve and capactve parastcs.. INTRODUCTION ND MOTITION Inductve and capactve parastcs, both self and mutual, whch were gnored n the realm of low-frequency desgns, become mportant at hgh frequences and are a domnant factor n determnng nterconnect delay and crosstalk of the crcut. Tradtonal maze routng always fnds the shortest path, whch may lead to sgnfcant parallel runs between two neghborng wres gvng rse to consderable amount of nductve and capactve couplng. Recent crosstalk-aware routng methods [] [2] have used couplng capactance as a crosstalk measure. But, self and mutual nductve effects lke rngng, can cause sgnal oscllatons (overshoots and undershoots), leadng to a greater settlng tme delay. We provde a motvatng example n ths regard. 2 d C2 m B D C B m Step response at 2 - Case Step response at B - Case2 Step response at C - Case2 Step response at C - Case 5p n Fgure : () Possble -bend routes (B) Sgnal Response at C In ths smple example (Fgure ()), we consder only -bend routes ( L shaped) for smplcty. We consder a stuaton when nets -2 and B-B2 have already been routed. We now try to determne the route for net C-C2. There are two possble canddate routes for ths net. Case: The lower-l route separated by dstance d that couples wth -2 for a length of m and Case2: The (dotted) upper-l route separated by dstance d that couples wth B-B2 for a length of m. (Note: the dstance D s greater than the threshold dstance below whch couplng s consdered, hence the lower-l route does not couple wth B-B2). s the couplng capactance (Cc m ) s equal n both the cases, d t becomes dffcult to pck the best route from these two solutons. In our moment drven routng cost functon, we account for factors such as self and mutual nductance (L, K), drecton of sgnal propagaton, n addton to resstance (R) and self (C) and couplng capactance (Cc) between the neghborng wres, to fnd the best canddate route. In Case, we see that the current flows n opposte drecton (odd-mode of propagaton) and n Case2, the current flows n the same drecton (even-mode of propagaton). From the sgnal response at C n Fgure (B), we observe that Case2 leads to a more overdamped response compared to Case, leadng to a greater net delay. Thus, Case (lower-l) s the better routng choce for net CC2 as t s response s closer to the crtcally-damped condton compared to Case2, leadng to faster response. hgher order moment-based routng methodology s proposed, governed by a cost functon, whch accounts for the R,L,C,K parastcs between neghborng wres and converges to a routng soluton, after a sutable rngng-delay trade off. lne-search routng methodology has been used and customzed to ncorporate our cost functon. Snce our focus s to route long wres connectng dfferent chps dstrbuted over the entre slcon substrate, we do not use maze routng, due to ts expensve memory requrements. 2. RELTED WORK The authors n [3] have used a smple congeston estmaton technque n ther global routng methodology. They have not allowed the number of route edges across a global bn edge exceed the edge capacty. Smlar congeston estmaton strategy have been used by the authors n [4], [5] wth competng cost functon formulaton where the routng cost ncreased ether lnearly or abruptly wth congeston. In [], a routng technque has been proposed that s couplng aware only n terms of capactance. nother work n the area of crosstalk reducton has been done n [2], [6] where the parastc couplng capactance occurs n the cost functon formulaton of the routng methodology and durng layer track assgnment respectvely. The authors n [7], [8], [9] have developed varous perfor-

2 mance drven routng strateges usng a smple Elmore (RC) delay model for the nterconnects. Cong et al. [], have used hgher order RLC model n ther MINOTUR global router but they have not consdered nterwre nductve and capactve couplng. Though all of the above routng methodologes have ther own merts, none of them accounts for all the R,L,C,K parastcs durng routng. 3. RINGING ND WEFORM RESPONSE t hgh frequences, the nductve effect of the nterconnects gves rse to a transmsson-lne phenomenon called rngng that can nadvertently cause the crcut to transt to a wrong state and cause logc falure. Ths phenomenon s shown n Fgure 2(C) and s well explaned n []. Fgure 2(C) shows the sgnal response at the output of the aggressor lne of the crcut models n Fgure 2() and 2(B). Rngng s ether pronounced or subdued dependng on the even or odd mode of current propagaton n the aggressor and vctm wres. The repeated overshoots and undershoots adversely affect the delay of the lne as t ncreases the settlng-tme of the sgnal response. In addton to the rse tme (t r) of the sgnal, the settlng tme (t s) has to be consdered to obtan the actual wre delay (t d ) [], [], resultng n the expresson, t d = t r t s.the Rv Ra Rv Lv La Lv Kav () Kav Ca (B) Cc Cc Ca Ra Cv Cv La m Overshoot Response for even mode Response for odd mode Settlng tme Undershoot 5p n Fgure 2: () Even-mode (B) odd-mode nteracton (C) Rngng phenomenon n output waveform response RC Elmore delay model used n prevous routng papers, s nsuffcent to account for the hgh frequency nductve parastcs of the nterconnects and can lead to naccuracy. In order to account for the self and mutual nductve effects we resort to hgher order moments of the mpulse response to create an accurate delay approxmaton. Usng the Laplace transform of the mpulse response and wth the help of the concept of central moments from probablty theory as descrbed n [2], the second and thrd order moments, measures of dsperson and skewness, are derved as n Eqn. and Eqn. 2. µ 2 = 2m 2 m2 () m and, µ 3 = 6m 3 6 mm2 2 m3 m m 2 (2) There wll always be an nherent amount of dsperson (µ 2) due to lne loss. In contrast, µ 3 can be made to approach the deal value of zero [3], whch ensures mnmum rngng and also less settlng tme (t s) n the expresson t d = t r t s. s we also mnmze µ 2, we are able to control dsperson of the lossy nterconnects, and have a better sgnal transton. In our cost functon we mnmzed both µ 2 and µ 3 as our prmary objectve to obtan the best trade-off between rngng and rse tme delay. More on ths aspect wll be dscussed n Secton PROPOSED METHODOLOGY We propose a routng methodology, whch uses the concepts ntroduced n Secton 3, to fnd the best possble overall routng soluton that has the least parastc nteractons and the best rngng and delay trade-off for all the nets. We descrbe each of the sx steps n and ther objectves below. Rv Ra H(s) = o(s) (s) Lv La Kav (Cc Rv Cc Ra Cv Rv) s ( Cc Lv Cc La 2 Cc Kav Cv Lv Cv Kav) s^2 = (Ca Ra Cc Rv Cc Ra Cv Rv) s (Cc Ca Ra Rv Cv Ca Ra Rv Ca La Cv Cc Ra Rv Cc Lv Cc La 2 Cc Kav Cv Lv ) s^2 (Cc Ca Lv Ra Cc Ca La Rv Cv Ca Lv Ra Cv Ca La Rv Cv Cc Lv Ra Cv Cc La Rv) s^3 Fgure 3: Transfer functon of 2 nteractng wres n even mode 4. Transfer Functon Generaton We demonstrate a smple example n Fgure 3 to generate the moment-metrcs (µ 2 and µ 3) of two nteractng wres, where the top wre s the vctm and the bottom wre s the aggressor. The transfer functonh(s) of ths crcut has been generated usng SP- WIN v3. and s shown n Fgure 3. SPWIN s essentally a symbolc transfer functon generator ( / ) that generates the transfer functon after solvng the Krchoff s node voltage and branch current equatons usng modfed-nodal-analyss (MN) technques. The mportant pont to note n the expresson of H(s) s the ntroducton of couplng terms Cc and K. The symbolc representaton of the second and thrd order moments of ths crcut s gven n Eqn. 3 and Eqn. 4 respectvely. µ 2 = (Ca 2 2CaCc)Ra 2 2CvCcRaRv 2CvKav 2CaLa (3) µ 3 = (2Ca 3 6Ca 2 Cc 6CaCc 2 )Ra 3 (6CvCc 2 Rv 6CaCc 2 Rv)Ra 2 (2CaCcKav 2CaCcLa 6CvCc 2 Rv 2 6Cv 2 CcRv 2 6CvKavCc 6CvCcLv 6Ca 2 La)Ra 6Cv 2 KavRv 6CvCcLaRv 6CvKavCcRv (4) In order to come to a reasonable concluson and an ntutve explanaton from such a complex value of µ 2 and µ 3 we assume the capactance and resstance of the aggressor and the vctm wres to be equal,.e, Ca = Cv and Ra = Rv.However,wehaveusedthe entre expresson of µ 2 and µ 3 and have not mposed the lmtaton of Ca = Cv and Ra = Rv n our methodology. The correspondng reduced equatons are gven below. Cc Ca Cv o

3 Coupled Routes Template I 2 (I) Two wre 3segment nteracton of template and ts Equvalent RLCK Crcut TemplateIII Template II Template I (III) Three wre nteracton s mapped to 2wre nteractons. 2 (II) Templates 2, 3 and 4 and ther varous nteractons gressor Wre ctm Wre Fgure 4: Templates to determne nteractons among wres and an example showng the RLCK elements for Template µ 2 = Ra 2 Ca 2 2Ra 2 CaCc 2Ra 2 CaCc 2CaKav 2CaLa µ 2 = Ra 2 Ca 2 2Ca(La Ka) (5) µ 3 = 2Ra 3 Ca 3 6RaCa 2 La 6Ca 2 KavRa µ 3 = 2Ra 3 Ca 3 6RaCa 2 (La Kav) (6) From Eqn. 5 and Eqn. 6 we can ntutvely nfer that µ 2 and µ 3 are always postve when La, Ka =, whch represents lossy lnes wth overdamped response [4]. lso, µ 2 and µ 3 decreases monotoncally as the self and mutual nductance values ncreases. We also notce that the effectve nductance s the sum of (La Kav) sgnfyng even mode of current propagaton n the aggressor and vctm wres. We can conclude that, when nterconnect self and mutual nductance s not mportant (as s the case at sub-ghz frequences) the analyss corresponds to a smple RC model wth overdamped waveform response. Smlarly, µ 2 and µ 3 sgnfes the nductve effects of the nterconnects, whch results n rngng as descrbed earler n Secton 3. Objectve: The symbolc transfer functon (H(s)) of nterconnects are pre-generated usng SPWIN, whch also accounts for the nearest neghbor mutual nductance and couplng capactance terms. From ths transfer functon we obtan the symbolc expressons of µ 2 and µ 3 as shown n Secton 3. The dfferent ways an nterconnect can nteract wth ts neghborng wre wll be dscussed n secton 4.2. The symbolc generaton of the expressons of µ 2 and µ 3 s a one-tme process and occur outsde the loop of our routng algorthm. Dynamc evaluaton of these symbolc expressons s made wthn our algorthm as the values of K and Cc changes for dfferent canddate routes of a partcular net. 4.2 Templates for Canddate Routes The above analyss shows that nductve effects play a crucal role n determnng the correct delay and output sgnal response. We have extended the above analyss to generate robust transfer functons for the canddate nterconnect structures used n our router. -bend (L-shaped) wre s splt nto two segments whle a 2-bend (Z-shaped) wre s splt nto three segments to model the nets as a smple dstrbuted RLC network. It s mportant to note that durng the routng process, routng of the current net also affects the waveform response of ts neghborng vctm nets. In order to capture accurate parastc nteractons between the aggressor and the vctm wres, we have generated the requred templates (Fgure 4) that cover the varous ways the -bend and 2-bend nterconnects can nteract wth each other. The L-shaped route s a subset of Z-shaped route, hence we generate the templates wth only Z-shaped nets n mnd. By reducng R,L and C of any of the end segments to zero of a Z-shaped route, we can produce the requred template for a L-shaped net. The correspondng transfer functons are pre-generated for these templates and are used on the fly as deemed necessary, based on the way the canddate routes for the current net nteracts wth ts neghborng wres. Fgure 4(III) demonstrates how a 3-wre nteracton for the aggressor net () s broken down nto two, 2-wre nteracton. The transfer functon for each of the correspondng 2-wre nteracton s then used to determne the second and thrd-order central moments and fnally combned to generate the cumulatve µ 2 and µ 3.Forthe3-wrecasewe have assumed that the vctm wres at the top () and the bottom (2) do not nteract wth each other. These four prmtve templates or ther derved reduced forms are used to capture the varous possble nteractons of the -bend and 2-bend wres wth other neghborng wres. Objectve: The templates account for the varous ways the L- shaped and the Z-shaped nets can couple wth other L-shaped and Z-shaped wres. The templates and ther symbolc transfer functons are pre-generated usng SPWIN. Usng these transfer functons the central moments (µ 2 and µ 3) for each of the templates are symbolcally derved usng the symbolc Maple toolbox of Matlab and are stored n a lbrary. Durng the routng of a net our momentdrven routng algorthm chooses the requred template from the lbrary and evaluates ts correspondng moments dynamcally based onthevaluesofr,l,c,ccandk. 4.3 Estmaton of R,L,C,K Quck and accurate estmaton of the electrcal parameters R,L,C, Cc and K s made based on the wre geometry and usng ther correspondng analytcal expressons [2]. We dd not consder the couplng parastcs between orthogonal wres and have used expressons gven n [5] for calculatng mutual nductance of two parallel wres wth unequal length. The accuracy of the self and mutual nductance(l, K) formulas wth feld-solver smulatons s gven n [5]. 4.4 Route Cost Functon The cost functon s formulated wth the knowledge that routng of a partcular net at any nstance of the routng process may affect the sgnal response and delay of ts neghborng wres. Durng the route determnaton process, we consder several possble canddate solutons for a net. Let us consder that there are n nets that have to be routed and at a partcular nstance the th net needs to be routed. The followng steps llustrate the cost functon formulaton.. For each of the k canddate routes of net, symbolc expressons of µ 2 and µ 3 are evaluated usng estmated values of R,L,C,Cc and K. 2. The µ 2 s and µ 3 s of all the affected nets (vctms) that have already been routed are re-evaluated.

4 2 3. For each canddate of net, we compute the overall standard devaton of the already routed and the current net as gven below. Ths gves the cumulatve cost (C cost) for each canddate route and s a measure of the overall dsperson and skewness from the deal value (µ 2d =and µ 3d =). C cost = n= (µn 2 µ 2d) 2 n= (µn 3 µ 3d) 2 4. Store the values of the cumulatve cost functon for each canddate route. 5. Select the canddate route havng the least cumulatve cost and add to the lst of routed nets. 4 canddates C OBSTCLE B C2 B2 Normalzed Cumulatve Cost (Std. Dev.) Cumulatve Cost Plot for Canddate Routes of CC2.5.5 Chosen Canddate Canddate Routes for CC2 Fgure 5: () Possble Canddate Routes of CC2 (B) Normalzed Cumulatve Cost (Std. Dev) for Canddate Routes (CC2) The cost funton thus gves a snapshot of the cumulatve performance n terms of second and thrd-order central moments of all the nets up to the current net. Fgure 5() and 5(B) llustrates the cost functon flow. Out of the 4 canddates, the cost functon chooses canddate as the routng soluton for net CC2, whch yelds the least cumulatve cost [6]. Ths mples that the chosen canddate obtans the best trade-off between rngng and delay among all other canddates and provdes the least dstorton to ts affected neghbors 2 and BB2. Objectve: The objectve of developng such a cost functon can be understood from Secton 4.5 below. 4.5 Rngng and Delay Trade-off s stated n Secton 3, mnmzng the second central moment µ 2, results n reducton of the spread (dsperson) of the waveform response thereby obtanng an output response wth qucker rse tme (t r). But, the faster transton results n more rngng and the sgnal suffers from repeated overshoots and undershoots. Though the rse tme may prove to be benefcal, the ncreased settlng tme adversely affects the sgnal delay (t d )[], []. To mnmze rngng and decrease the settlng tme delay (t s) we also try to mnmze the thrd central moment (µ 3) sacrfcng a bt on the transton rate of the sgnal. We formulate our cost functon as gven below. S mom = n= (µn 2 µ 2d) 2 T mom = n= (µn 3 µ 3d) 2 (8) C cost = S mom αt mom (9) S mom and T mom n Eqn. 7 and Eqn. 8 represents the overall standard devaton of all the affected vctm nets that have already been (7) routed and the current net. Note that we have used normalzed values of S mom and T mom to get C cost. Objectve: The fnal C cost s evaluated by takng a weghted sum of S mom and T mom, whch ensures a relatve trade-off between rngng and transton delay[3]. The C cost of dfferent canddates of net are evaluated and the one that yelds the mnmum cost and closest to the deal value of s chosen as the best route. In our work we have gven equal mportance to rngng and delay mnmzaton of a net, and have chosen α = n Eqn Moment-drven Routng Technque In our routng methodology (Fgure 6) the possble canddate routes of a net are generated, smlar to the level and level lnes of the standard lne search algorthm, whch has been talored to nclude our moment drven cost functon. s the routng of nets wll eventually be guded by our cost functon, we dd not use a maze router because of ts expensve memory requrements and whch may not be sutable for routng mult-chp modules over a large chp area. ny orderng of nets can be handled by the router Fgure 6: Template-based moment drven routng methodology Inputs: P [..2n] : Lst of Pns and Pn Types N[..n] : Lst of Ordered Nets O[..k] : Intal Obstacle Lst sym µ 2[..4] : Pre-generated symbolc 2nd order moments sym µ 3[..4] : Pre-generated symbolc 3rd order moments Output: R[..n] : Routng Soluton Begn RoutedNets = φ for [..n] C[..j] Generate Canddate(N[],O[..k ]) for k [..j] T d Choose Template(C[k], RoutedNets) Parastcs Calc Parastcs(C[k], RoutedNets) (µ2[k],µ3[k]) Eval Moments(P arastcs, sym µ 2[T d],sym µ 3[T d]) end for Std dev[..j] CostF uncton(µ2[..j],µ3[..j]) R[] Select C[x] wth least Std Dev RoutedNets RoutedNets R[] Update Obstacle lst O wth R[] end for End and should be gven as an nput to the program. The drecton of sgnal s also known at ths pont as we know the termnals to be ether an nput, output or a bdrectonal pn. For a undrectonal current flow between an nput-output pn par of a net, we follow the prncple descrbed n Secton 4.4 before, whle for a par of bdrectonal pns, the canddate route s chosen by modfyng the cost functon slghtly. In that case, the second and thrd central moments are evaluated consderng the current flow n ether drecton for each of the canddate routes. The route, whch yelds the mnmum cost n both the cases s selected the wnner after dong a cumulatve measure of the two moments. We have consdered only 2-pn nets that can be connected usng ether a -bend or a 2-bend route. Majorty of the nets n a MCM crcut have only two pns. Ths can be observed from the

5 data gven n [], of two ndustral mult-chp module benchmarks. We see that for mcc, 75.8% of the total number of nets are 2- pn nets, whle for mcc2 t s about 94.%. We focus on routng these two pn nets usng our methodology. Mult-termnal nets can be broken down nto a number of two-termnal nets as descrbed by Borah et. al n [7], n ther edge-based stener-tree routng methodology. The resultng two termnal nets can then be routed usng our methodology. The nspraton of usng L-shaped and Z- shaped technque fnd ts orgn n other related work by Kastner et.al n [], [3] and by Smey et. al n [2]. pre-specfed step-sze s used to generate k possble 2-bend canddate solutons for the current net that needs to be routed. Our mplementaton s flexble to accommodate other methods to generate canddate solutons as well. For -bend canddates, there can be at most two routes (ether a upper-l or a Lower-L route) to choose from. s we nspect k2 possble canddates for each of the n nets n our sequental algorthm, the routng complexty s Θ(kn), ncontrasttoθ(k n ) complexty n the method descrbed n [2]. We would also lke to pont out that the symbolc transfer functon generaton usng SP- WIN s a one-tme process, occurs outsde the algorthm loop and takes a few seconds for each of the templates. 5. RESULTS ND DISCUSSIONS We have randomly dstrbuted pars of pns n our experments and have routed the termnal pars usng our moment-drven routng algorthm. Our cost functon s formulated based on the second and thrd order central moments (µ 2 and µ 3). Durng moment evaluaton we have accounted for the nearest neghbor capactve and nductve couplng parastcs as well as the self resstance, capactance and nductance of each net. We compare our routng methodology wth the followng. Competng pproach: In ths routng approach we have formulated a cost functon, whch takes nto account the couplng capactve parastcs between neghborng wres for each of the probable canddate routes durng the route determnaton phase of a net. We decouple the from our moment-drven approach n the followng ways.. The cost functon does not nclude moment computatons. 2. Resstve and Inductve (self and mutual) parastcs are gnored n the cost functon. 3. Couplng capactve parastcs are accounted n the cost functon. s the objectve of the cost functon of the s to mnmze the couplng capactve parastcs of the canddate route wth ts neghborng wres, t tres to fnd a route wth maxmal separaton from ts neghborng wres. Fgure 7(a), (b), (c) and (d) shows the step response for 2, 4, 7 and 9 nets usng the two approaches. s the worst net determnes the maxmum speed of the crcut, we chose to do a comparatve analyss on t. The worst net n s the one wth the hghest cost based on the moment metrcs, whle the worst net n the s the one wth the maxmum capactve couplng parastcs. For the Hspce smulaton, we extract all the parastcs (R, L, C, Cc and K) for the worst nets n both the approaches, even consderng the mutual nteractons wth ther neghborng nets. The rse tme of the step response s set at 34ps, whch means a sgnfcant frequency of.34/34ps =GHz [8]. The sgnfcant frequency should not be confused wth the operatng frequency, as t s dependent only on the rse tme and not on the clock perod. In other words, t sgnfes the edge rate of transton whch helps to capture the effect of reactve components (L and C) n the crcut. More n-depth analyss of ths measure s gven n [9]. The graphs clearly show that our method results n better performance for the worst net. Whle the competng method resulted n overdamped (more delay) waveforms for the 2, 4, 7 and 9 routng cases, we were able to obtan a better qualty response for all the routng cases usng. The nferor response for the can also be analyzed ntutvely. s mnmzaton of couplng parastcs tres to maxmze the separaton between each canddate route and ts neghbors, the technque mght not yeld good results as the routng densty ncreases. For future nets, there mght not be enough space left to maxmze the separaton objectve and may result n undesrable ncrease n couplng parastcs and an overdamped (decayed) response. Due to the area constrant on the routng regon, we observe that as the densty of nets ncreases (from 2 to 9), the output response progressvely deterorates. The proposed method obtans a routng soluton based on a cost functon that fnds the best trade-off between rngng (µ 3) and delay (µ 2). Ths ensures a sgnal response wth better sgnal qualty n terms of rse tme, settlng tme, sgnal overshoots and undershoots. For the responses n Fgure 7, we observe that the sgnal settles above.7 (9% of.3) after at most one undershoot. In the 4 and 9 routng nstances t settles even faster, snce the undershoot level does not go below the.7 threshold mark. s the competng method s not drven by ths cost functon, the routng soluton s not guaranteed to be of good qualty and can suffer from ether a sgnal underdamp or an overdamp. Smlar promsng results are also obtaned for the 3, 5, 6, 8 and net cases. In order to make the benchmarks more strngent, we kept the area of the routng regon for all the benchmarks constant. In realty, for MCM crcuts as the number of nets ncreases, the chp area and the area of the routng regon ncreases too [],[2]. mcc wth 68 2-pn nets and mcc2 wth pn nets are dstrbuted on a 45x45 and a 52.4x52.4 mm 2 substrate respectvely. In order to make the routng problem more ntensve and strngent, we kept the routng area constant across our benchmarks. The horzontal and vertcal segments of the routes are n dfferent metal layers and the mnmum wdth and spacng rules requred to avod DRC volatons are mplemented n the routng methodology. The routng algorthm has been developed n C, and we have used SPWIN to generate the one-tme symbolc transfer functons for the templates used. The real tme taken to complete the routng of all the nets n each nstance s comparable for both our momentdrven and the. For the net routng case t took 2.5 mnutes usng whle t took 8.7 mnutes usng the on a Sun Ultra system wth 256MB memory. The dfference n tme s due to the extra computaton tme requred to calculate the moments n. Thus wthout a sgnfcant ncrease n tme, we obtan a soluton wth better sgnal qualty and response. Drect comparson of our results wth [] s not possble as [] dealt wth only a few percentage of nets wth stener tree connectons and dd not report on the worst case consderng the routng of all the nets. 5. Dscusson on Worst Case nalyss We have assumed the sgnals n all the wres swtchng at the same tme to model a worst case scenaro. Recent trends hghlght the mportance of stochastc modelng of the swtchng n wres, argung that n realty not all wres swtch smultaneously and consderng smultaneous swtchng mght lead to the underestmaton of the rse tme of the sgnals. Though much attenton s beng gven to the stochastc swtchng dstrbuton, we consdered the worst case scenaro wth the reasonng that the routng obtaned under ths condton wll also be effectve under stochastc swtchng.

6 m 6m 4m.8.6 less delay, sgnal reaches.3 faster.4.2 more delay, bad sgnal qualty 8m 6m 4m 2m -2m.8 faster transton, good qualty decayed response, bad sgnal qualty 8m 6m 4m faster response, good qualty decayed response, nferor qualty m 6m 4m 2m faster response, good qualty decayed response, nferor qualty 2m -4m 2m -6m -2m n 2n 3n n 2n 3n n 2n 3n n 2n 3n Fgure 7: Hspce smulaton result for the worst net n our and for (a) 2 (b) 4 (c) 7 and (d) 9 nets 6. CONCLUSION Performance-drven routng s ncreasngly becomng mportant as crcut and mult-chp-module (MCM) desgns progress towards the deep-submcron (DSM) regme. Interconnect parastc optmzaton cannot reman restrcted to a smple RC or a RLC model. t mult-ghz frequences, the crosstalk among nteractng wres become mportant. In ths paper, we have proposed a routng methodology, whch takes nto account the nductve and capactve couplng parastcs between neghborng wres durng the cost functon evaluaton of the moment-drven routng process. The proposed approach ensures a routng soluton for a number of randomly dstrbuted pn-pars, wth good sgnal qualty and mnmum rse tme delay. The trade-off between rngng and delay s effectvely captured usng a moment-drven cost functon. Routng obtaned usng only capactve couplng n the cost functon precludes the trade-off and yelds an nferor soluton n terms of sgnal qualty and delay. cknowledgement: We thank Natonal Scence Foundaton for partly supportng ths work under the award number CCF REFERENCES [] R. Kastner, E. Bozorgzadeh and M. Sarrafzadeh. Couplng ware Routng. In Internatonal SIC/SOC Conference, pages , 2. [2] R. M. Smey, B. Swartz and P H. Madden. Crosstalk Reducton n rea Routng. In Desgn utomaton and Test n Europe Conference, pages , 23. [3] R. Kastner, E. Bozorgzadeh and M. Sarrafzadeh. Predctable Routng. In Internatonal Conference on Computer ded Desgn, pages 3, 2. [4] R. T. Hadsell and P. M. Madden. Improved Global Routng through Congeston Estmaton. In Desgn utomaton Conference, pages 28 3, 23. [5] J. Cong and P. Madden. Performance Drven Mult-Layer General rea Routng for PCB/MCM Desgns. In Desgn utomaton Conference, pages , 998. [6] T. Ho et. al. Fast Crosstalk- and Performance-Drven Multlevel Routng System. In Internatonal Conference on Computer ded Desgn, pages , 23. [7] J. Llls et.al. New Performance Drven Routng Technques Wth Explct rea/delay Tradeoff and Smultaneous Wre Szng. In Desgn utomaton Conference, pages 395 4, 996. [8] S. Hur,. Jagannathan and J. Llls. Tmng-Drven Maze Routng. In Tran. on Computer-ded Desgn of Integrated Crcuts and Systems, pages , 2. [9] J. Hu and S. Sapatnekar. Tmng-Constraned Smultaneous Global Routng lgorthm. In Tran. on Computer-ded Desgn of Integrated Crcuts and Systems, pages 25 36, 22. [] J. Cong, C. Koh and P. Maddden. Interconnect Layout Optmzaton Under Hgher Order RLC Model for MCM Desgns. In Tran. on Computer-ded Desgn of Integrated Crcuts and Systems, pages , 2. [] H. B. Bakoglu. Crcuts, Interconnectons, and Packagng for LSI. ddson-wesley, 99. [2]. Bhadur and R. emur. Moment-Drven Couplng-ware Routng Methodology. In Great Lakes Symposum on LSI, pages , 25. [3] R. Gupta, B. Krauter, L. T. Plegg. Transmsson Lne Synthess va Constraned Multvarable Optmzaton. In Tran. on Computer-ded Desgn of Integrated Crcuts and Systems, pages 6 9, 997. [4] Y. Lu et. al. Mn/Max On-Chp Inductance Models and Delay Metrcs. In Desgn utomaton Conference, pages , 2. [5] M. Xu, L. He. n Effcent Model for Frequency-Dependent On-Chp Inductance. In Great Lakes Symposum on LSI, 2. [6]. Bhadur and R. emur. Inductve and Capactve Couplng ware Routng Methodology Drven by a Hgher Order RLCK Moment Metrc. In Desgn utomaton and Test n Europe, pages , 25. [7] M. Borah et. al. n edge-based heurstc for Stener routng. In Tran. on Computer-ded Desgn of Integrated Crcuts and Systems, pages , 994. [8] C. K. Cheng, J. Llls, S. Ln and N. Chang. Interconnect nalyss and Synthess. John Wley & Sons, Inc, 2. [9] H. W. Johnson and M. Graham. Hgh-Speed Dgtal Desgn - Handbook of Black Magc. Prentce Hall, 993. [2] K. Khoo and J. Cong. n Effcent Multlayer MCM Router Based on Four-a Routng. In Tran. on Computer-ded Desgn, pages , 995.

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