A New and Efficient Congestion Evaluation Model in Floorplanning: Wire Density Control with Twin Binary Trees

Transcription

1 New and ffcent ongeston aluaton Model n loorplannng: Wre ensty ontrol wt Twn nary Trees Stee T. W. La epartment of S Te nese Un. of H.K. Satn, N.T., Hong Kong twla@cse.cuk.edu.k angelne. Y. Young epartment of S Te nese Un. of H.K. Satn, N.T., Hong Kong fyyoung@cse.cuk.edu.k rs. N. u epartment of Iowa State Unersty mes, I cncu@astate.edu bstract s tecnology moes nto te deep-submcron era, te complexty of VLSI crcut desgn grows rapdly, especally n te nterconnectons between modules. Terefore, nterconnect optmzaton as become an mportant concern n floorplannng today. Most routablty-dren floorplanners [2][6][8] use grd-based approac tat ddes a floorplan nto grds as n global routng. ongeston s estmated as te expected number of nets passng troug eac grd. ltoug ts approac s drect and accurate, t s not effcent wen dealng wt complex crcut contanng tousands of nets. In ts paper, an effcent and nnoate routablty-dren floorplanner usng twn bnary trees (TT)[9][10] representaton s proposed. Te congeston model we used s te wre densty on te alf-permeter boundary of dfferent regons n a floorplan. Tese regons are defned naturally by te TT representaton. In order to ncrease te effcency of our floorplanner, a fast algortm for te least common ancestor (L) problem n [1] s used to compute te wre densty. rom te expermental results, te number of unroutable wres can be reduced n a sort tme. 1. Introducton 1.1. Motaton In te deep-submcron era, te complextes of VLSI crcuts are growng rapdly. Te nterconnectons between modules wll become longer and denser n te future. Terefore, nterconnect optmzaton n floorplan desgn as become eer more mportant tan before. s floorplannng s at te begnnng pase of te VLSI desgn cycle, an nterconnect-optmzed floorplan wll faor te applcablty and performance of te later desgnng stages lke placement, global routng, detaled routng, etc, and, most mportantly, allow tmng closure to be aceed earler. Recently, some routablty-dren floorplanners [2][6][8] are proposed. Most of tem use te grd-based approac to measure te congeston of a floorplan. In ts approac, a floorplan s dded nto grds as n global routng. t eac grd, te expected number of nets passng troug s recorded as a wegt to measure congeston. ltoug ts approac s drect and smple, suc knd of routng-orented estmaton s tme consumng f t s performed n eac teraton of te smulated annealng process n a floorplanner. It s mpractcal for complex crcut desgns. Terefore, a new and fast congeston ealuaton model usng a sutable floorplan representaton wll be ery useful reous work Recently, seeral floorplanners are proposed to consder routablty n te floorplannng pase. In paper [2], a floorplan s dded nto grds and congeston s estmated at eac grd by assumng tat eac wre s routed n eter L-sape or Z-sape. Tey use smple-geometry routng to plan te wres n reasonable tme. In paper [8], a realstc global router s used to ealuate te congeston of eac placement soluton. In paper [6], a probablstc metod s proposed to estmate congeston and routablty. floorplan s dded nto a 2-dmensonal grd structure and congeston s estmated at eac grd. Smlar approaces are also proposed n [4] and [5]. ltoug te aboe congeston ealuaton models ae been sown to be effecte n reducng nterconnect cost, ter computatonal costs are ery g Our contrbuton In order to prode a smple and effcent congeston ealuaton metod oter tan te complcated grd-based approac, an ndrect congeston ealuaton model, wre densty, s proposed. Instead of estmatng te congeston at eac grd usng global routng, we measure congeston as te wre densty passng troug te boundary of dfferent regons n a floorplan. It s because a floorplan, tat as a g wre densty on aerage, as a greater cance of ang congeston problem. n example s sown n fgure 1. We use twn bnary trees (TT) as te floorplan representaton because te regons to be ealuated can be naturally defned by te TT representaton. or a floorplan wt n modules, n 1 regons

2 are defned by eac tree. In order to prode more regons for ealuaton, we ae constructed an addtonal par of trees, wc s te mrror of te orgnal par of trees. In order to ncrease te effcency of our floorplanner, we ae made use of a fast algortm [1] for te least common ancestor (L) problem to compute te wre densty. xpermental results ae sown tat an nterconnect-optmzed floorplan of a complex crcut can be obtaned n less tan fe mnutes. Ts paper s dded nto seen sectons. In secton 2, a bref reew of te TT floorplan representaton wll be gen. Secton 3 wll ge an oerew of our floorplanner. In secton 4 and 5, te deas and mplementaton detals of te wre densty congeston ealuaton model wll be descrbed and explaned. nally, expermental results wll be sown n secton 6. loorplan oundary of g wre densty loorplan gure 1. Hg wre densty n floorplan. 2. Twn bnary trees In our floorplanner, we use twn bnary trees (TT) as our floorplan representaton. Te TT floorplan representaton s frst proposed n paper [9]. It sows an one-to-one mappng between TT and mosac floorplan. Recall tat te defnton of twn bnary trees s as follows: efnton 1 Te set of twn bnary trees T T n Tree n Tree n s te set: T T n = {(t 1,t 2 ) t 1,t 2 Tree n and θ(t 1 ) = θ c (t 2 )} were Tree n s te set of bnary trees wt n nodes, and θ(t) s te labellng of te bnary tree t. Te labellng of a bnary tree t can be obtaned by performng an n-order traersal on t. Wen te traersed node as no left cld, a bt 0 s added to te sequence. Smlarly, f t as no rgt cld, a bt 1 s added to te sequence. Te frst 0 and te last 1 n te labellng are omtted. If a par of trees (t 1,t 2 ) are twn bnary to eac oter, ter labellngs wll be te complement of eac oter,.e., θ(t 1 ) = θ c (t 2 ). Gen a mosac floorplan, we can construct a par of trees (t 1, t 2 ) by traellng along te slcelnes of. Te root of t 1 s te upper rgt corner of te packng. y connectng te upper rgt corners of all te modules, te orzontal slcelnes represent te tree edges connectng from a parent to ts left cld, wle te ertcal slcelnes represent te tree edges connectng from a parent to ts rgt cld. Te constructon of t 2 can be done smlarly by connectng te lower left corners of all te modules. It as been sown tat te par of trees constructed n ts way must be twn bnary to eac oter. lso, t s obsered tat te n-order traersal of te par of trees are te same[10]. n example s sown n fgure 2, θ(t 1 ) = and θ(t 2 ) = 01101, so θ(t 1 ) = θ c (t 2 ). lso, ter n-order traersals are bot t 1 t 2 gure 2. onstructon of TT. 3. Oerew of our floorplanner In ts secton, we wll ge a bref ntroducton to our routablty-dren floorplanner wt te new wre densty congeston ealuaton model. Our floorplanner s based on te TT floorplan representaton and smulated annealng s used. Gen a canddate floorplan soluton, te total wre lengt of te nets s estmated by te alf-permeter boundng box approac. Te congeston cost s estmated by te wre densty wc s computed as te number of nets passng per unt lengt of te boundary of dfferent regons. Tese regons are defned by te TT representaton naturally and erarccally. Te estmaton of wre densty wll start from te leaf nodes and follow te post-order traersal of te tree. ac tree can prode n 1 samples,.e., n 1 regons, for wre densty estmaton. In order to obtan more samples, two addtonal trees are constructed from te orgnal par of TT to prode a total of 4(n 1) wre densty alues. 4. Wre densty model In order to mproe te routablty of a floorplan soluton n an effcent way, an ndrect but effecte congeston ealuaton model s used. Ts model ams at measurng congeston as te wre densty (number of nets per unt lengt) on te boundary of dfferent regons n a floorplan. efnton 2 Gen a TT (t 1, t 2 ), te regon R() coered by module n t {t 1,t 2 } s te rooms occuped by module and te modules n te subtree rooted at n t. s sown n fgure 3, te regon R() coered by module n t 1 ncludes all te rooms occuped by module,, and. We can obtan n 1 wre densty alues for a tree wt n nodes. It s because R(root) s te wole packng and tere wll be no nets passng troug te boundary of te packng

3 Te followng ges te equaton to calculate te wre densty of R(): = N (1) were s te wre densty of R(), N s te total number of nets passng troug te boundary of R() and s te normalzed alf-permeter of R(). Te detals of te computaton of N and wll be gen n te comng sectons. We coose TT as te floorplan representaton n our floorplanner because t can defne te regons for ealuaton naturally. lso, a lot of fast and smple tree algortms can be used n our congeston ealuaton. We start te estmaton of wre densty from te leaf nodes and follow te post-order traersal of te tree to compute te terms N and at eac node. y dynamc programmng, te nformaton computed at te cldren can be used to compute te wre densty at te parent omputaton of N gure 3. ormaton of R(). Te term N, wc s te total number of nets passng troug te boundary of R(), can be computed as follows: N = N + N l() + N r() M (2) were l() s te left cld of, r() s te rgt cld of, N s te number of nets connected to module, M s an offset for adjustments due to net mergng and net completon, and t s computed as follows: M = 3 j=2 ( j 1)m j + 3 j c j (3) j=2 were m j and c j are te number of nets merged and completed at. Te alue j s te number of subnets of a sngle net tat meet at. It can be eter two or tree. Te adjustment for net mergng m j s needed because te repeated countng of an dentcal net n N, N l() and N r() wll oer-estmate te term N. or j = 2, two subnets comng from R(l()), R(r()) or module of a sngle net are merged. or j = 3, tree subnets comng from R(l()), R(r()) and module of a sngle net are merged. Te term m j s multpled by j 1 because we need to keep one countng n N rater tan j countng. n example s sown n fgure 4. In fgure 4, we consder te stuaton wen we reac module durng te post-order traersal. We use tck sold lnes to represent merged nets. Tere s one net merged between module and R(), one between module and R(), and one between R() and R(), so m 2 = 3. Tere s also one net merged between module, R() and R(), so m 3 = 1. Smlarly, te adjustment for net completon c j s needed because te repeated countng of an dentcal net n N, N l() and N r() wll oer-estmate te term N. Te alue j n c j as te same meanng as tat n m j. Te term c j s multpled by j because te net as completed and all te countng sould be elmnated. In fgure 4, we use tck dotted lnes to represent completed nets. Tere s one net completed between module and R(), two nets completed between module and R(), tree nets completed between R() and R(), so c 2 = = 6. Tere s also one net completed between module, R() and R(), so c 3 = 1. nally, M = m 2 + 2m 3 + 2c 2 + 3c 3 = 3 + 2(1) + 2(6) + 3(1) = 20 In fgure 4, N s computed as N + N + N - M were N = 10, N = 13, N = 11 and M = 20. s a result, N = = 14. Tere are 14 nets passng troug te boundary of R(). Te alue of N can be obtaned easly as te net specfcaton s gen n te floorplannng pase. Howeer, te term M wll ary for dfferent packngs, a nae metod to compute M wll mpose an O(mn) tme complexty were n s te total number of nets and m s te total number of modules. It s mpractcal for complex crcuts. Terefore, we ae made use of an effcent algortm for te least common ancestor (L) problem to compute M. etals of te mplementaton wll be gen n secton 5. R() R() Net Mergng Net ompleton gure 4. n example of computng N omputaton of Te term, wc s te normalzed alf-permeter of R(), can also be computed easly by followng te post-order traersal of te tree. s te tree edges of a TT represent te wdt and egt of te rooms occuped by te modules, we wll separate te alf-permeter of regon R() nto te orzontal ( ) and ertcal ( ) portons to make te operaton smple. Te pseudo-code s gen as follows: Halfermeter(tree t) 1. or j = 1 to n were (π(1),π(2),...,π(n)) s te post-order traersal of t 2. = π( j) 3. If s a leaf node 4. = w 5. =

4 6. If as left cld l() only 7. = w + l() 8. = max(, l() ) 9. If as rgt cld r() only 10. = max(w, r() ) 11. = + r() 12. If as bot left and rgt cld, l() and r() 13. = max((w + l() ), r() ) 14. = max(( + r() ), l() ) 15. = cp wdt + cp egt In te pseudo-code, w and are te wdt and egt of te room occuped by module. Te computaton of () s dded nto four cases. Lne 3-5 s te case were module s a leaf node as n fgure 5(a). gure 5(b) sows te case of lne 6-8 were module as a left cld l() only. gure 5(c) sows te case of lne 9-11 were module as a rgt cld r() only. Lne s te last case were module as bot left cld l() and rgt cld r() as n fgure 5(d). nally, on lne 15, and are normalzed by te cp wdt and cp egt respectely to mantan a unform order of magntude. s dynamc programmng s appled n te computaton, te tme complexty of Halfermeter(t) to compute te normalzed alf-permeters of all te (n 1) regons s only O(n). w w (a) (c) r() r ( ) r ( ) l ( ) l ( ) l ( ) l ( ) w l() w l() (b) (d) r() r ( ) r ( ) gure 5. ases n computaton Usage of mrror TT fter dscussng te computaton of N and, we can ealuate te wre densty for t 1 and t 2. y te caracterstc of te TT representaton, te computed from t 1 represent te wre denstes of te boundares facng te upper rgt drecton, wle tose computed from t 2 represent te wre denstes of te boundares facng te lower left drecton. ac tree can ge n 1 statstcal samples for te wre densty ealuaton were n s te number of modules. In order to ncrease te effecteness of our congeston model, a par of mrror TT, wc are based on te orgnal par of TT, are constructed. Te mrror TT can be magned as te TT constructed from a packng wc s rotated 90 counterclockwse. Togeter wt te mrror TT, our congeston model can ge 4(n 1) wre densty alues wc consder n four routng drectons (upper rgt for t 1, lower left for t 2, upper left for t 3 and lower rgt for t 4 ). s suffcent statstcal samples are consdered, te routablty of a packng can be estmated correctly. 5. Implementaton 5.1. ffcent calculaton of N In ts secton, a detaled explanaton of usng te L algortm to compute N wll be gen. Recall from secton 4.1 tat te major dffculty of computng N s te g computatonal cost of computng te term M. Instead of computng M for eac module one by one, we are gong to compute all M ncrementally by stng eac net one by one. Let s look at te example n fgure 6. In ts example, we need to fnd te nodes, and were adjustments are needed due to net mergng and completon of net p. Net p wll merge at node, and, and fnally complete at. Te nodes were adjustments are needed are L(u, ), were (u, ) are some module pars n a net. or a net wt k modules, k 1 Ls sould be found for adjustments. It s obsered tat we cannot get te correct Ls were adjustments are needed by just pckng te module pars arbtrarly. or example, te Ls obtaned by smply selectng te tree adjacent module pars from te orgnal net specfcaton of p n fgure 6 are L(,) =, L(,) = and L(,) = wc are not te correct set of Ls {,,} were adjustments are needed. Terefore, te followng lemma s used to fnd te correct set of Ls were adjustments are needed for a net p. Lemma 1 Gen a tree t wt n nodes (representng n modules) and a net p connectng k modules (m 1,m 2,...,m k ). Te set of nodes L p n t were two or more subnets of p meet (adjustment s needed) s k 1 L p = {L(m π(),m π(+1) )} =1 were (m π(1),m π(2),...,m π(k) ) s a permutaton of te k modules obtaned by followng te pre-order traersal of t. (In fgure 6, te permutaton of te modules connected by p followng te pre-order traersal s () and L p = {,,}.) roof: Te proof s done by nducton on te dept of te tree t. Te pre-order traersal of t of dept n + 1 can be expressed as n n were s te root, n and n represent te pre-order traersal of te left subtree of rooted at and te rgt subtree of rooted at wt dept smaller tan or equal to n respectely, and n s te

5 larger alue of te depts of te left and rgt subtree of. ecause of te lackng of space, we wll sow te proof for te case were ae bot left and rgt subtrees only. Te cases were as left subtree or rgt subtree only can be proed smlarly. Wen n = 1, te pre-order traersal of t s ( 1 1 ) as sown n fgure 7(a). or te tree t were ae bot left and rgt subtrees, 1 and 1 represent and respectely and te pre-order traersal s. onsder te case for a net p = {,,}, te subnets of p wll meet (twce) at node. Te permutated p s {,,}, and te Ls found accordng to te lemma are correct snce L(, ) = and L(,) =. Te cases were p = {,}, p = {,} and p = {, } can be proed smlarly. Hence, te proposton s true wen n = 1. ssume tat te proposton s true wen n = k, and te pre-order traersal of t s k k as sown n fgure 7(b). Wen n = k + 1, te pre-order traersal of t wll be k+1 k+1. We can re-wrte t as ( k k )( k G k ) as n fgure 7(c). Let f and l be te frst and last node of te permutated subnet of p n k k respectely, and f be te frst node of te permutated subnet of p n k G k. or te case were net p resdes n te left and rgt subtrees of and node, te Ls found from te left and rgt subtrees of are correct accordng to te nducte ypotess. Tere s one more node tat te subnets of p wll meet (twce), wc s, and t wll be found correctly accordng to te lemma snce L(, f ) = and L( l, f ) =. Te cases were net p resdes n te left or rgt subtree of completely, p resdes n te left subtree of and node, p resdes n te rgt subtree of and node, and p resdes n te left and rgt subtrees of but not node can be argued smlarly. Hence, te proposton s true for n = k + 1. Q... L(, ) Net completon Net mergng L(, ) L(, ) Net p={,,, } - reorder Tree Traersal: - Net-lst: p={,,, } - ermutated Net-lst n preorder: gure 6. Usng L to compute N. (a) (b) gure 7. roof of our M computaton. (c) G fter obtanng te set L p of a net p, we can update te alue of te correspondng M. s sown n fgure 6, M, M and M wll be ncremented by 1 because net p wll be merged wen tey are sted. nally, M of te sallowest module n te set L p wll be furter ncremented by 1 because te net s gong to be completed tere. In fgure 6, ts sallowest module s. Te same operaton wll be performed for eac net to compute all M. nally, we can apply equaton (2) to compute all N for wre densty computaton Solng te L problem effcently In paper [1], an effcent and smple L algortm s proposed. It reduces te L problem to te Range Mnmum Query (RMQ) problem. y applyng te Sparse Table (ST) algortm for te RMQ problem, te L problem can be soled n constant tme wt a preprocessng tme of O(nlogn) usng dynamc programmng ost uncton Te cost functon for te smulated annealng process of our floorplanner s sown as follows: cost = + αw + β (4) were s te cp area of te floorplan, W s te total wre lengt estmated, s te summaton of all te wre densty alues of te floorplan, and α and β are te wegts to descrbe te mportance of tese tree terms. In our floorplanner, α and β are set suc tat te rato of te mportance of tese tree terms are : W : = 2 : 2 : omplexty ffcency s one of te major adantages of our wre densty congeston model. Recall from equaton (1) tat te computaton of te wre densty W s dded nto two parts, N and. Te operatons needed to compute N for all are te constructon of te L sparse table (O(nlogn)), te computaton of M for all (O(k)) were k s te total number of pns n all nets, and te computaton of equaton (2) (O(n)), so te tme complexty of computng N wll be O(nlogn)+O(k)+O(n) = O(nlogn)+O(k). Usually, te magntude of k s muc greater tan tat of n, so we treat te tme complexty of computng N as O(k). Secondly, te tme complexty of computng for all s O(n). s a result, te tme complexty of our congeston estmaton metod s O(k) only. 6. xpermental results We ae mplemented two floorplanners for testng. One s a tradtonal floorplanner wtout consderng congeston, te oter one s a routablty-dren floorplanner usng our wre densty model. ot floorplanners are based on te TT floorplan representaton and te smulated annealng approac. Tree MN bencmarks {am33, am49, playout}

6 are used. In addton, tree data sets {n2000, n2500, n3000} are created to demonstrate te performance of our floorplanner for complex crcuts. Te detaled specfcatons of te data sets are sown n table 1. Te experments are performed usng a wt a entum III 1GHz processor and 1G memory. We use a smple global router to ealuate te performance of te floorplanners. xpermental results are sown n table 2. Te term unroutable wre s te wre tat cannot be routed n te sortest Manattan dstance due to te lmtaton of te wre capacty at eac grd n te global router. Te term congeston s te aerage number of nets per µm 2 of te top 10% most congested grds reported by te router. We use te data n paper [7] to compute te parameters n te router. We use te feature alues of te 0.18µm tecnology for am33 and am49. or te oter data sets, we use te feature alues of te 0.13µm tecnology. Te results sow a sgnfcant reducton n te number of unroutable wres. Te results n congeston are smlar for te two floorplanners. It sows tat our floorplanner s actually more effcent n dstrbutng te wres unformly snce more wres are beng routed (te number of unroutable wres as decreased) wtout ncreasng te congeston measures. esdes, te wre capacty of eac grd s actually bounded to aod oer-congeston n te router and te results sow tat te wrng capacty of eac grd s more fully utlzed by our floorplanner. noter mportant result from te experment s tat te runtme of our floorplanner for a complex crcut wt tree tousand nets (n3000) s less tan fe mnutes. It demonstrates bot te effecteness and effcency of our wre densty model n reducng te nterconnect cost. ata Number of Number of Number Modules IO ns of Nets am am playout n n n Table 1. Specfcatons of te data sets. 7. oncluson In ts paper, we present a new congeston model usng wre densty as a measurement. We use TT as te floorplan representaton because te regons for ealuaton can be defned by te TT representaton naturally, te fast and smple tree algortms, for example, te L algortm, can facltate te effcency of our congeston model. y usng te regons defned by te TT and te mrror TT, suffcent samples can be taken for congeston ealuaton. Te tme complexty of te wole congeston estmaton metod s lnear wt respect to te total number of pns n all nets. x- ata ead Wre Number of ongeston Runtme Space Lengt Unroutable (# of nets (s) (%) (10 3 µm) Wres per µm 2 ) loorplanner wt wre densty control am am playout n n n loorplanner wtout consderng congeston am am playout n n n Table 2. xpermental results of our floorplanner. perments ae sown tat ts congeston ealuaton model s effcent and effecte wen dealng wt complex crcuts. Te number of unroutable wres can be greatly reduced n a sort tme. References [1] M.. ender and M. arac-olton. Te L problem rested. In Latn mercan Teoretcal INformatcs, pages 88 94, [2] H. M. en, H. Zou,. Y. Young,.. Wong, H. H. Yang, and N.. Serwan. Integrated floorplannng and nterconnect plannng. In roc. Int. onf. On, pages , [3] X. Hong, G. Huang, Y. a, J. Gu, S. ong,. K. eng, and J. Gu. orner block lst: n effecte and effcent topologcal representaton of non-slcng floorplan. In roc. Int. onf. On, pages 8 12, [4]. Hung and M. J. lynn. Stocastc congeston model for VLSI systems. Tecncal Report SL-TR , Stanford Unersty, [5] J. Lou, S. Krsnamoorty, and H. S. Seng. stmatng routng congeston usng probablstc analyss. In Int. Symp. yscal esgn, pages , [6]. W. Sam and.. Y. Young. Routablty dren floorplanner wt buffer block plannng. In Int. Symp. yscal esgn, pages 50 55, [7]. Sylester and K. Keutzer. Gettng to te bottom of deep submcron. In roc. Int. onf. On, pages , [8] M. Wang and M. Sarrafzade. Modelng and mnmzaton of routng congeston. In roc. of san-acfc, pages , [9]. Yao, H. en,. K. eng, and R. Graam. Restng floorplan representatons. In Int. Symp. yscal esgn, pages , [10].. Y. Young,.. N. u, and Z.. Sen. Twn bnary sequences: non-redundant representaton for general nonslcng floorplan. In Int. Symp. yscal esgn, pages , 2002.