Construction of ROBDDs. area. that such graphs, under some conditions, can be easily manipulated.

Transcription

1 A Study of Composton Schemes for Mxed Apply/Compose Based Constructon of s A Narayan 1 S P Khatr 1 J Jan 2 M Fujta 2 R K Brayton 1 A Sangovann-Vncentell 1 Abstract Reduced Ordered Bnary Decson Dagrams (s) have tradtonally been bult n a bottom-up fashon In ths scheme, the ntermedate peak memory utlzaton s often larger than the nal sze, lmtng the complexty of the crcuts whch can be processed usng S Recently we showed that for a large number of applcatons, the peak memory requrement can be substantally reduced by a sutable combnaton of bottom up (decomposton based) and top down (composton based) approaches of buldng s In ths paper, we focus on the composton process We detal four heurstc algorthms for ndng good composton orders, and compare ther utlty on a set of standard benchmark crcuts Our schemes oer a matrx of tme-memory tradeo ponts 1 Introducton In the last decade s [3] have frequently been used as the representaton of choce to solve varous CAD problems such as synthess, dgtal-system vercaton and testng s are canoncal gven a varable orderng, and hence the nal sze of the representaton s a constant, regardless of the sequence of boolean operatons by whch the was constructed In the tradtonal (apply based) scheme of constructng s, the peak ntermedate memory requrement often far exceeds the nal (canoncal) representaton sze of the gven functon Although we are solely nterested n obtanng the of the output, the ntermedate peak memory requrement often lmts our ablty to construct the output, and also usually dctates the tme requred for constructon It would therefore be very desrable to construct s n a fashon whch reduces the ntermedate peak memory requrement The presence of an ntermedate peak followed by a subsequent reducton n the memory requrements ndcates that Boolean smplcaton has occurred n the crcut We attempt to capture ths Boolean smplcaton by followng the general procedure outlned n [8] We ntroduce sutable decomposton ponts, and perform all computatons on smaller decomposed graphs Thus, we get a decomposed representaton of the output functon, whch captures the Boolean smplcaton occurrng n the crcut Fnally, we compose the decomposton functons nto the of the output, to obtan the canoncal output In ths paper, we focus our attenton on the composton phase We dscuss and demonstrate varous heurstc algorthms to perform composton; theren les our contrbuton In Secton 3, we dscuss the basc dentons and termnology that we use throughout ths paper In Secton 4 we dscuss the decomposton procedure We descrbe our composton technques n Secton 5, and dscuss the results obtaned 1 Department of Electrcal Engneerng and Computer Scences, Unversty of Calforna, Berkeley, CA Fujtsu Laboratores of Amerca, San Jose, CA n Secton 7 We conclude wth Secton 8 where we summarze our contrbutons, and gve drectons for future research n ths area 2 Prevous Work Though BDDs have been researched for about four decades [10, 1], they found wdespread use only after Bryant [3] showed that such graphs, under some condtons, can be easly manpulated These condtons are that the graph s reduced (e no two nodes have dentcal subgraphs), and that a total orderng of the varables s enforced The resultng BDD s called a Bryant ntroduced two symbolc manpulaton procedures - apply and compose, to combne two dentcally ordered s Apply allows s to be combned under some Boolean operaton, and compose allows the substtuton of a varable wth a functon Another mportant development n ths area s the work on Dynamcal Varable Reorderng [14] Local alteratons are performed on the exstng varable order, and a change n the varable order s accepted f t results n a reduced manager sze Ths scheme was shown to requre reduced amounts of memory for most crcuts, at the cost of ncreased executon tme s are typcally constructed usng some varant of Bryant's apply procedure [3] The gates of the crcut are processed n a depth-rst manner untl the s of the desred output gate(s) are constructed However, an exclusvely bottom-up approach s often not the most computatonally ecent technque for constructng ROB- DDs, as dscussed n [8] The prmary focus of [8] was on ndng good decompostons Structural and functonal methods were proposed for ndng these decomposton ponts The functonal method was shown to have more exblty n choosng decomposton ponts and obtaned better results Accordngly,we use the functonal decomposton method n our work [8] dd not dscuss the process of composton; we focus our attenton on ths step We propose varous composton schemes and compare ther eectveness 3 Prelmnares In ths secton we provde the dentons and termnology for rest of the paper For detals on s, please refer to [2, 3, 4] For an ecent mplementaton of the package, [2] s an excellent reference Assume we are gven a crcut representng a boolean functon F F : B n! B o, wth n prmary nputs x 1 ;:::;x n, and o prmary outputs To smplfy the dscusson, we wll focus our attenton on a sngle output G Let G d ( ) represent the of G expressed n terms of a varable set = f 1 ; 2 ;:::; mg Each 2 has a correspondng, bdd, n terms of prmary nput varables as well as (possbly) other j 2, where j 6= Elements of can be ordered such that jbdd depends on only f <j These 2 are called decomposton ponts of G G d ( ) s the decomposed of G n terms of the decomposton ponts

2 The composton [3] of n G d ( ) s denoted by G d ( ):( bdd ) where G d ( ):( 1 bdd ) = bdd :G d ( ) + bdd :G d ( ) G d ( ) represents the restrcton of G d ( ) at =1,and s obtaned by drectng all the ncomng edges to the node wth varable d to ts = 1 branch G d ( ) s also referred to as the -cofactor of G d ( ) Other technques for composton have been proposed [12, 7]; for our purpose we wll consder the approach descrbed above 4 Decomposton Strategy We use the functonal decomposton technque reported n [8] We ntroduce a decomposton varable at ponts where the sze and / or the manager sze ncreases by a large measure s of the remanng crcut are now bult n terms of prmary nputs, prevously ntroduced decomposton varables as well as the newly ntroduced decomposton varable Accordngly, the of the output s bult n a decomposed manner In ths scheme no decomposton s performed when the functon can be processed wthout any memory exploson In our mplementaton, we decompose a functon when the total manager sze ncreases by more than a userspeced threshold, as a result of some Boolean operaton Ths check s only done f the manager sze s larger than a user-speced mnmum Addtonally, a decomposton pont s added when an ndvdual grows beyond another threshold value, also provded by the user Thus, a decomposton pont may be ntroduced when some operaton on the cubes wthn a node causes the threshold to be exceeded In ths case, there s no physcal correspondence between decomposton ponts and crcut nodes 5 Composton Strateges In ths secton we descrbe our algorthms for ndng good composton orders The correspondng to a new decomposton pont( bdd )mayhave n ts support set some j 's, where j<we store ths dependency nformaton n the form of a dependency graph After analyzng ths graph, we choose the best to be composed The choce of the to compose s made on the bass of some cost functon, whch tres to estmate the resultng graph sze after composton Once a varable s composed n G d,we update the dependency graph for subsequent compostons So the algorthm conssts of three man parts: Intalzng the dependency graph Selecton of the to be composed Updatng the dependency graph after composton In the followng secton we descrbe each of these steps n some detal 51 Intalzng the dependency graph The dependency graph s a drected acyclc graph For every decomposton pont (say ), there s a correspondng noden the dependency graph (node ) There s one addtonal node correspondng tog d, called node Gd There s an edge from node to node j f and only f s present n the support of jbdd Smlarly an edge from node to node Gd represents that G d depends on Ths graph s guaranteed to be acyclc snce there s an edge between node and node j only f j> If an edge goes from node to node j then node j s called the parent ofnode and node s called the chld of node j Anodenode s called the mmedate chld of node j f node 3 2 G d a) Before composng varable 4 b) After composng varable 4 Fgure 1: Dependency Graph, showng changes after an update s a chld of node j and no other path exsts between node and node j Fgure 1(a) shows a dependency graph Note that nodes 3, 4 and 5 are mmedate chldren of the G d node, whle 2 s not, even though t s a chld of G d 52 Cost Functon Heurstcs We mplement four separate heurstcs for cost functon evaluaton For every canddate varable that can be composed nto G d,we assgn a cost whch estmates the sze of the resultng composed The varable wth the lowest cost estmate s composed In all our heurstcs, the canddate nodes for composton are the mmedate chldren of G d Ths choce ensures that that a decomposton pont has to be composed only once n the nal graph and we never need more compostons than the cardnalty of the set of decomposton varables Ths results n the mnmum number of compostons to get the monolthc representaton of the target functon Wthout ths restrcton, the composton process wll need O(n 2 ) compostons n the worst case, where n s the number of decomposton ponts Snce composton s the most tme consumng operaton n the entre algorthm, ths reducton n the algorthm complexty s of sgncance Our four heurstcs are detaled below 521 Bound of the Sze of the Composed For any boolean operaton on s f 1 and f 2 t s well known that the resultng s bounded n sze by j f 1 j j f 2 j, where j f j represents the number of nodes n the of f Wth ths result, t s easy to show that the composed G d ( ):( bdd ) s bounded by j G d ( ) j 2 j bdd j Bryant conjectures n[3] that ths bound s actually j G d ( ) j j bdd j The sze of the composed j G d ( ):( bdd ) j s equal to j G d ( ) + G d ( ) j The sze of G d ( ) and G d ( ) are bounded by j G d j j bdd j Ths bound s strong for the product operaton If Bryant's conjecture s true, then the reason why the resultng composton doesn't blow up s that the OR operaton doesn't blow up If ths s the case then the bound for the sze of the composed would be (j G d ( ) j + j G d ( ) j) j bdd j We use ths bound as our rst cost functon to select the composton varable 3 2 G d 1 5

3 Make G d and -array Intalze the dependency graph whle (no of decomp ponts > 0) Select the mnmum cost mmedate chld of node Gd Compose the selected n G d Update the dependency graph repeat Fgure 2: Algorthm: Functonal Decomposton and Composton 522 Support Set Sze of the Composed It s beleved that the sze of a s postvely correlated to the cardnalty of the support set As our second cost functon, we use the ncrease n the support sze of the composed to select the decomposton pont to be composed We choose that decomposton varable whch leads to the smallest ncrease n the sze of the support set of the composed Ths method works well, as we wll see n the results secton 523 Sum of the szes of G d ( ), G d ( ) and In most cases, the product of szes of two functons as an estmate for the sze of ther resultng s very loose Instead of takng the products as the cost functon, we take the sum of the szes of the s as a measure of cost of composng a decomposton pont So we compose a varable whose estmate of j G d ( ) j + j G d ( ) j + j bdd j s mnmum In essence ths scheme favors varables whch have smaller bdd s and result n smaller cofactors G d ( ) and G d ( ) 524 Sze and Number of Occurrences of Our nal cost functon counts the number of nodes n G d that correspond to a varable Ths s a measure of the number of tmes each nstance of the composton wll have to be done on the graph, and s a reasonable estmate of the complexty of composng nto G d Ths count smultpled by the sze of bdd, whch serves as an estmate of the complexty of each nstance of composton It can be shown that f the supports of bdd and G d are dsjont, then bdd can be composed n G d by replacng all occurrences of n G d by bdd The resultng graph sze, n ths case, would be the same as the estmate 53 Updatng the dependency graph Suppose a decomposton pont,, s composed n kbdd Now the dependency graph needs to be updated Ths s done by removng the edge from node to node k and addng new edges from the chldren of node to the node k Also, f node (correspondng to ) does not fan out to any other node after composton then we remove node and all the edges enterng node Fgure 1(b) shows the result of composng 4bdd n G d In Secton 6 we dscuss the algorthm and mplementaton detals, ncludng the data structure used for the dependency graph 6 Algorthm and Implementaton Detals Fgure 2 shows the complete algorthm schematcally G d represents the decomposed verson of the output node n terms of prmary nputs and the decomposton varables The s correspondng to the decomposton ponts are stored n an array denoted by -array At the end of the decomposton procedure, we get the of the output n terms of prmary nputs and ntermedate varables s We also get an array contanng the bdd s The dependency graph s mantaned n the form of an adjacency matrx Ths matrx has n rows and (n+1) columns where n s the number of decomposton ponts ntroduced durng the bottom-up phase of the algorthm Ths s a 0,1 matrx A \1" bt present n th row and j th column, where 1 ; j n ndcates that jbdd depends on A \1" entry n the th row and (n +1) th column ndcates that G d depends on By lookng at the support set of bdd 's and G d we can easly ntalze the dependency graph To check whether a gven s the mmedate chld of G d, we need to check f the (; (n + 1)) th entry s a \1" and all the other entres n the th row are \0"s To update the graph after s composed n G d,wechange the (; n +1)entry to a \0", and for rows havng a non-zero th column entry,wechange the th column entry to \0" and (n +1) th column entry to \1" 7 Results We mplemented the algorthms n the C language, under the SIS [15] programmng envronment Our results were run on a DECsystem 5000/260, wth 440MB memory We ran our experments on the ISCAS85 benchmark crcuts, for outputs whch are consdered hard for tradtonal packages [15] In the tables below, COMP-SIZE corresponds to the method whch estmates cost by the bound on the sze of the composed shown n Secton 521 SUP-SIZE refers to the support set heurstc of Secton 522, whle SUM-SIZE uses the sze estmate of Secton 523 for the composed NUM-PSI uses the number of occurrences of varable n G d tmes the sze of bdd as the cost functon (Secton 524) The Reference entry represents the tradtonal apply based method of buldng s For each scheme, we havetwo sets of experments In the rst set, we use Natural Orderng wth and wthout Dynamc Varable Reorderng [14] These results are summarzed n Tables 1 and 3 The results wth Dynamc Varable Reorderng are shown under the headng 'DR' The second set of experments uses Malk's Orderng [11] These results are presented n Tables 2 and 4 Tables 1 and 2 report the maxmum sze of the manager at any gven pont n the computaton In these tables, the SUP-SIZE heurstc does well n almost all cases on average For some examples, NUM-PSI outperforms SUP-SIZE under Malk's Orderng and DR On average, NUM-PSI and SUP-SIZE perform better than the other methods The reason for ths s that SUP-SIZE requres just two support set computatons, and NUM-PSI requres the number of nodes of the varable presentng d No cofactor operatons are needed for calculatng the cost, as s the case n COMP-SIZE and SUM- SIZE, where the computaton of postve and negatve cofactors of G d wth the varable under consderaton ncreases the total memory usage Tables 3 and 4 report the total runtme for each example Wthout DR, SUP-SIZE and NUM-PSI are about as fast and are much faster than the other two schemes Ths s prmarly due to the derence n the tme taken for cost estmaton [13] In COMP-SIZE and SUM-SIZE, we actually take the cofactors of G d for estmatng the cost Ths s a tme consumng process, requrng the allocaton of new nodes Wth DR, the tme taken by reorderng s the domnatng factor, hence none

4 of the scheme performs consstently better n the presence of DR 8 Conclusons The man conclusons of our work are as follows: Our method performs better than the tradtonal apply based scheme, and s also able to buld the s of functons that the tradtonal scheme fals on We showed that the ntermedate memory requrement s very senstve to the order of composton A bad order of composton may utlze up to twce the memory of a good order We showed that SUP-SIZE and NUM-PSI are reasonably good heurstcs for gudng the composton process The peak memory requrement usng these schemes s generally lower than the tradtonal method as well as other composton methods Under Dynamc Varable Reorderng, none of the schemes performs consstently better than the others n runtme Ths s because Dynamc Varable Reorderng s the domnatng factor n the runtme In the future, we plan to use ths approach to reduce the resources requred n computng transton relatons, reachablty, and also for alternate BDD approaches, ncludng FDDs [9], OKFDDs [5], Typed-Free BDDs [6] and probablstc vercaton as n [7] 9 Acknowledgements Ths work was supported by the Calforna State Mcro Contracts and References [1] Sheldon B Akers Bnary decson dagrams IEEE Transactons on Computers, C-27:509{516, June 1978 [2] K S Brace, R L Rudell, and R E Bryant Ecent Implementaton of a BDD Package In Proc of the Desgn Automaton Conf, pages 40{45, June 1990 [3] R E Bryant Graph based algorthms for Boolean functon representaton IEEE Transactons on Computers, C-35:677{690, August 1986 [4] R E Bryant Symbolc boolean manpulaton wth ordered bnary decson dagrams ACM Computng Surveys, 24:293{318, September 1992 [5] R Drechsler, A Sarab, M Theobald, B Becker, and M A Perkowsk Ecent representaton and manpulaton of swtchng functons based on ordered kronecker functonal decson dagrams 31st Desgn Automaton Conference, pages 415{419, 1994 [6] J Gergov and Ch Menel Ecent analyss and manpulaton of typed free bdds can be extended to read-once-only branchng programs Tech Report 92-10, Unversty of Trer Also to appear n IEEE Transacton on Computers, June 1995, 1992 [7] J Jan, J Btner, D S Fussell, and J A Abraham Probablstc vercaton of Boolean functons Formal Methods n System Desgn, 1, 1992 [8] J Jan, A Narayan, C Coelho, S Khatr, A Sangovann- Vncentell, R Brayton, and M Fujta Combnng Topdown and Bottom-up Approaches for Constructon Techncal Report UCB/ERL M95/30, Electroncs Research Lab, Unv of Calforna, Berkeley, CA 94720, Aprl 1995 [9] U Kebschull, E Schubert, and W Rosenstel Multlevel logc synthess based on functonal decson dagrams European Desgn Automaton Conference, pages 43{47, 1992 [10] C Y Lee Representaton of swtchng crcuts by bnarydecson programs Bell Syst Tech J, 38:985{999, 1959 [11] S Malk, A R Wang, R K Brayton, and A Sangovann- Vncentell Logc Vercaton usng Bnary Decson Dagrams n a Logc Synthess Envronment In Proc of the Intl Conf on Computer-Aded Desgn, pages 6{9, November 1988 [12] K L McMllan Symbolc model checkng: An approach to the state exploson problem PhD Thess, Dept of Computer Scences, Carnege Mellon Unversty, 1992 [13] A Narayan, S P Khatr, J Jan, M Fujta, R K Brayton, and A Sangovann-Vncentell Compostonal Technques for Mxed Bottom-Up / Top-Down Constructon of ROB- DDs Techncal Report UCB/ERL M95/51, Electroncs Research Lab, Unv of Calforna, Berkeley, CA 94720, June 1995 [14] R L Rudell Dynamc Varable Orderng for Ordered Bnary Decson Dagrams In Proc of the Intl Conf on Computer-Aded Desgn, pages 42{47, November 1993 [15] E M Sentovch, K J Sngh, L Lavagno, C Moon, R Murga, A Saldanha, H Savoj, P R Stephan, R K Brayton, and A L Sangovann-Vncentell SIS: A System for Sequental Crcut Synthess Techncal Report UCB/ERL M92/41, Electroncs Research Lab, Unv of Calforna, Berkeley, CA 94720, May 1992

5 # no DR DR no DR wth DR no DR DR no DR DR no DR DR C C C C { { { { { C C C C { { Table 1: Max Manager Sze, Natural Orderng, 1M lmt # no DR DR no DR wth DR no DR DR no DR DR no DR DR C C C C C { { { { { C C C { { { Table 2: Max Manager Sze, Malk Orderng, 1M lmt # no DR DR no DR wth DR no DR DR no DR DR no DR DR C C C C { { { { { C C C C { { Table 3: Total Tme (sec), Natural Orderng, 1M lmt # no DR DR no DR wth DR no DR DR no DR DR no DR DR C C C C C { { { { { C C C { { { Table 4: Total Tme (sec), Malk Orderng, 1M lmt