ECE 407 Computer Aided Design for Electronic Systems. Circuit Modeling and Basic Graph Concepts/Algorithms. Instructor: Maria K. Michael.

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1 0 Computr i Dsign or Eltroni Systms Ciruit Moling n si Grph Conptslgorithms Instrutor: Mri K. Mihl MKM - Ovrviw hviorl vs. Struturl mols Extrnl vs. Intrnl rprsnttions Funtionl moling t Logi lvl Struturl moling t Logi lvl Extrnl rprsnttions Intrnl rprsnttions Grph Trvrsl lgorithms nhmrking Ciruit nhmrk ormts Input prsing Rommn t struturs CiruitGrph pth ounting lgorithm MKM - 0: Computr-i Dsign or VLSI

2 hviorl, Funtionl, n Struturl Mols hviorl Contin logi n timing inormtion Funtionl Dsri th iruit logi (untionlity) Struturl Componnt list sriing logi nor timing (usully rrr to s ntlist); om t irnt lvls (x. gtlogi, trnsistor, lyout) hviorl n untionl initions r otn intrtwin MKM - Extrnl Vs Intrnl mols Extrnl Rrs to th mol viw y th usr Exmpls inlu txt-s sriptiv ormts, igrmsgrphil rprsnttions Intrnl Rrs to th mol viw y th omputr suh s t struturs n progrms MKM - 0: Computr-i Dsign or VLSI

3 MKM - 5 Funtionl moling t th gtlogi lvl Truth tls n K-mps Not prtil or iruits with lrg numr o inputs Cnonil è untion hs uniqu rprsnttion Primitiv us Compt rprsnttion o truth tlsk-mps Non-nonil Truth tl F= K-mp Primitiv us F=. x x 0 Funtionl moling t th gtlogi lvl (ont) Sum-o-Prouts (SoP) n Prout-o-Sums (PoS) Cnonil ut not ompt SttFlow tls, ul igrms lso not prtil or lrg iruits Dision Digrms Compt grphil rprsnttions Non-nonil: oolnlogi ntworks, inry xprssion igrms, t. Sp is linr to th iruit siz. Cnonil: inry Dision Digrms; vry ompt or mny rl-li xmpls. MKM - 0: Computr-i Dsign or VLSI

4 Struturl moling t th gtlogi lvl Extrnl rprsnttions: Txt-s ormt Grphil ormt Hr.iruit XOR.inputs.outputs Z C E Z oy NOT D NOT C ND E C ND F D OR Z E F D F MKM - gt typ output signl input signls Grph Prliminris grph G is triplt onsisting o vrtx st V(G), n g st E(G), n rltion tht ssoits vrtis with h g, ll th n-points x G(V,E) z y Unirt Grph V(G) = {x, y, z} E(G) = {,, } MKM - 8 0: Computr-i Dsign or VLSI

5 Grph Prliminris (ont) Dirt Grph (DG): th rltion tht ssoits with h g vrtis is sri y untion ssigning h g n orr pir o vrtis x G(V,E) z y orr pirs V(G) = {x, y, z} E(G) = {,, } = {(x,y), (x,z), (y,y)} MKM - 9 Grph Prliminris (ont) Dirt yli Grph (DG): DG in whih no yls xists Comintionl logi iruits r rprsnt y DGs C Squntil logi iruits r rprsnt y DGs C 5 5 MKM - 0 0: Computr-i Dsign or VLSI 5

6 Struturl moling t th gtlogi lvl (ont) Intrnl rprsnttions Ciruits n rprsnt y grph struturs in linr timsp Two si iruit-to-grph trnsormtions:. G (V,E ): on grph no pr iruit gtprimry input; on g pr iruit lin, othr thn nout stm primry output. è impliit rprsnttion o nout stms. G (V,E ): on grph no pr iruit lin; on g pr pir o gtsnout stms with irt onntion. è xpliit rprsnttion o nout stms MKM - Exmpls o iruit-to-grph trnsormtions C D E F z Z G (V,E ) G (V,E ) D D F F z E E C C Z MKM - 0: Computr-i Dsign or VLSI

7 Exmpls o iruit-to-grph trnsormtions C D E F z Z G (V,E ), V (G ) = n + i E (G ) = l s o n = # gts i = # primry inputs l = # lins s = # stm lins o = # primry outputs G (V,E ), V (G ) = l E (G ) = l MKM - Dt struturs or grphs Two tritionl t struturs us to rprsnt grphs: -D rry with link lists For G(V,E), V =n n E =, with n = # nos n = # gs, rt rry [..n] with vry [i] lmnt pointing to list o ll outgoing gs o i. Sp omplxity is O(n+) -Dimnsionl rry For G(V,E), V =n n E =, with n = # nos n = # gs, rt n n n mtrix M suh tht M[x,y]= i (x,y) E, othrwis M[x,y]=0. Sp omplxity is O(n ), ompt only i n MKM - 0: Computr-i Dsign or VLSI

8 Exmpl o t struturs or grphs z Mp no nms with intgr IDs z 5 MKM jny Mtrix igonl lwys 0 or yli 5 z ours only in topologilly inx DGs 5 5 jny List Exmpl o t struturs or grphs 5 Giv th jny mtrix n list Wht i th grph is unirt? Wht i oth nin n nout ino must xpliitly kpt? MKM - 0: Computr-i Dsign or VLSI 8

9 Grph Trvrsl lgorithms Trvrsl: systmti visiting o grph omponnts (nos n gs) Populr trvrsl mthos rth-irst-srh (FS) Dpth-irst-srh (DFS) Topologil Orr MKM - FS Trvrsl Strting t sour s, visit ll nos with istn k, or visiting ny no with istn k+. istn(s, x ) = min # o gs twn nos s n x Exmpl o vli FS trvrsl s s 5 no nm visiting orr MKM - 8 0: Computr-i Dsign or VLSI 9

10 DFS Trvrsl Strting t sour s, visit nos out o th most rntly visit no v, until ll sussors o v hv n visit; ktrk through no whih v ws visit. Exmpl o vli DFS trvrsl s 5 no nm s visiting orr MKM - 9 Topologil Orr Trvrsl Strting t sour s, visit no v only i ll o th prssors o v hv lry n visit. Exmpl o vli topologil orr trvrsl s 5 no nm s visiting orr MKM - 0 0: Computr-i Dsign or VLSI 0

11 Topologil Orr Trvrsl implmnttion Simpl s: Grph nos lry llinx in topologil orr ( no with inx i only hs prssors with inx < i) In this s, simpl or loop suis Exmpl: or i = 0..n- % n = # nos { visit & pross no i } MKM - Topologil Orr Trvrsl implmnttion (ont) lso, n us quus n rlt oprtors si Quu oprtions: Lt Q th quu strutur n x quu lmnt, - Enquu(Q,x): put x t n o Q - x = Dquu(Q): : stor lmnt t top o Q in x - Init_Quu(Q): initiliz Q s mx siz - Empty_Quu(Q): rturns Tru i Q is mpty - Full_Quu(Q): rturns Tru i Q is ull For topologil trvrsl: - Enquu no whn it hs n visit - Dquu no whn it hs n pross. no is pross i it is no longr n in th trvrsl pross. MKM - 0: Computr-i Dsign or VLSI

12 Topologil Orr Trvrsl lgorithm using Quus Lt grph G(V,E) with PI = st o nos with no prssors n S(v) = st o immit sussors o no v. MKM - Topologil_Trvrsl(G) { Q : quu; v,p : no; } Init_Quu(Q); FOR vry v V { IF ( v PI ) THEN { Enquu(Q,v); mrk v s visit; } ELSE mrk v s not-visit; } WHILE ( Empty_Quu(Q) == Fls ) { p = Dquu(Q); FOR vry no v S(p) { IF ( v not yt visit ND n visit ) THEN { Enquu(Q,v); mrk v s visit; } pross p; } } Exmpl o topologil orr trvrsl using Quus whn no hs n visit whn no hs n pross s Quu s nquu s s Quu quu s s nquu,, MKM - 0: Computr-i Dsign or VLSI

13 Exmpl o topologil orr trvrsl using Quus (ont) s whn no hs n visit whn no hs n pross Quu quu nquu s Quu quu nquu MKM - 5 Exmpl o topologil orr trvrsl using Quus (ont) s Quu whn no hs n visit whn no hs n pross quu s Quu quu nquu MKM - 0: Computr-i Dsign or VLSI

14 Exmpl o topologil orr trvrsl using Quus (ont) s Quu whn no hs n visit whn no hs n pross quu s Quu quu MKM - nhmrking Us to vlut th prormn o th vlop CD tool. llows or ttr shring n ir omprison o rsrh rsults. Enourgs hlthy omptition. Dvlopmnt o goo nhmrks is n importnt rsrh topi. MKM - 8 0: Computr-i Dsign or VLSI

15 Ciruit nhmrks Som o th iruit nhmrks tht hv n us xtnsivly to vlut VLSI tst rlt n synthsis rlt rsrh: ISCS 85: Comintionl iruit proils ISCS 89: Squntil iruits; lso in ull-snn vrsions ITC 99: Lrgr iruits y Torino CD group IWLS 005: 8 signs in VrilogOpnss mny othrs ISCS: Intrntionl Symposium o Ciruits n Systms ITC: Intrntionl Tst Conrn IWLS: Intrntionl Workshop or Logi Synthsis MKM - 9 ISCS 85ISCS 89ITC 99 nh Formt Hr # # 5 inputs # outputs # 0 invrtr # gts ( NNDs ) Ciruit rom th ISCS 85 suit Primry Inputs Primry Outputs Gts INPUT() INPUT() INPUT() INPUT() INPUT() OUTPUT() OUTPUT() 0 = NND(, ) = NND(, ) = NND(, ) 9 = NND(, ) = NND(0, ) = NND(, 9) 0 9 MKM - 0 0: Computr-i Dsign or VLSI 5

16 ISCS 85ISCS 89ITC 99 nh Formt kywor vril # - ommnt lin INPUT(x) - primry input lin with nm x OUTPUT(x) - primry output lin with nm x x = GTYPE(IN, IN,, INk) - gt o typ GTYPE with output x n k inputs, IN INk Vli gt typ GTYPE ND, NND, OR, NOR, XOR, XNOR, NOT, UFF MKM - Prs nh iruit n stor s grph # # 5 inputs # outputs # 0 invrtr # gts ( NNDs ) Ciruit rom th ISCS 85 st INPUT() INPUT() INPUT() INPUT() INPUT() 0 OUTPUT() OUTPUT() 0 = NND(, ) = NND(, ) = NND(, ) 9 = NND(, ) = NND(0, ) = NND(, 9) 9 MKM - 0: Computr-i Dsign or VLSI

17 Rommn t struturs (xmpl or iruit ) 5 no intgr ID no nm no nm no typ 0 no strutur no intgr ID D rry with link lists PI PI PI PI PI 0 NND NND NND 9 NND NND NND MKM - ing RNCH nos to th grph Th nh ormt os not xpliitly in nout stms n rnhs It is impli tht: x x = ND(,, ) y y = NOT() z = OR(, ) z To rnh nos to th grph, in nw no typ RNH, n: RNH ND x x OR RNH z z ND OR MKM - 0: Computr-i Dsign or VLSI

18 ing RNCH nos to th grph (xmpl or iruit ) 5 0 _ 5 _ _ 0 _ PI PI _ RNH _ RNH _ RNH _ RNH _ RNH _ RNH MKM - 5 CiruitGrph Pth Counting Prolm o ounting th # o pths in omintionl iruit rus to moii topologil trvrsl. Lt no v with S(v) = st o immit prssors o v. Th # o pths up to v, in y p(v), is: p(v) = Σ p(i), i S(v) MKM - 0: Computr-i Dsign or VLSI 8

19 Grph Pth Counting lgorithm Input: G(V,E) with PI = st o nos with no prssors, PO = st o nos with no sussors, n S(v) = st o immit sussors o no v. Output: # o pths in G lgorithm Count_Pths(G) Stp : FOR ( vry v V ) IF ( v PI ) p[v] = ; mrk v s visit; ELSE p[v] = 0; mrk v s not-visit; Stp : Run topologil trvrsl lgorithm; moii suh tht whn no v is pross, lult p(v) = Σ p(i), i S(v) Stp : Rturn Σ p(v), v PO MKM - Clult Σ p(v), v PO MKM - 8 0: Computr-i Dsign or VLSI 9

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