Augmented Sifting of Multiple-Valued Decision Diagrams

Transcription

1 Augmented Sftng of Multple-Valued Decson Dagrams D. Mchael Mller Rolf Drechsler Department of Computer Scence Insttute of Computer Scence Unverst of Vctora Unverst of Bremen Vctora, BC Bremen CANADA V8W 3P6 GERMANY Abstract Dscrete functons are now commonl represented b bnar (BDD) and multple-valued (MDD) decson dagrams. Sftng s an effectve heurstc technque whch apples adacent varable nterchanges to fnd a good varable orderng to reduce the of a BDD or MDD. Lnear sftng s an etenson of BDD sftng where XOR operatons nvolvng adacent varable pars augment adacent varable nterchange leadng to further reducton n the node count. In ths paper, we consder the etenson of ths approach to MDDs. In partcular, we show that the XOR operaton of lnear sftng can be etended to a varet of operatons. We term the resultng approach augmented sftng. Epermental results are presented showng sftng and augmented sftng can be qute effectve n reducng the of MDDs for certan tpes of functons. 1. Introducton BDDs [1][2][3][8][14][17][18] and MDDs [10][11] are commonl used n a wde varet of applcatons. The varable orderng can sgnfcantl affect the of a decson dagram and there has thus been consderable work on determnng good orderngs. Sftng [15] s a ver effectve technque applcable to BDDs and MDDs. * The of a BDD can be further reduced b a technque called lnear sftng [9]. In ths approach, certan varables are replaced b the XOR of varables so that the realzaton of a sstem of functons F conssts of a lnear preflter made up of XORs that permutes the nput space to a BDD representng a sstem of functons G whch together realze the gven sstem F at lower overall cost. Ths s n fact the lnearzaton scheme dscussed b The work reported n ths paper was supported n part b a Research Grant from the Natural Scences and Engneerng Research Councl of Canada. Karpovsk [7] who gave an analtcal soluton applcable to a sstem wth a small number of nputs. Lnear sftng s a heurstc applcaton of lnearzaton applcable to large problems. The purpose of ths paper s to eamne the etenson of lnear sftng to the MDD case. We consder mod-p sum sftng whch s based on replacng XOR b summaton mod-p and also augmented sftng where a varet of etensons to XOR are consdered. 2. Prelmnares We consder f( 1, 2,..., n),1 m, a sstem of totall-specfed p-valued functons where the are also p-valued. The functons are totall-specfed so each takes on all values 0 p-1. A partcular functon ma take on a subset of the values 0 p-1. In partcular, we shall consder the case of multple-valued nput, bnar output functons. We denote the mod-p sum as and use k = ( + k)mod p,1 k p 1, to denote the k possble cclc negatons. The tpe of functon consdered can be represented b a multple-valued decson dagram (MDD) whch s a drected acclc graph (DAG) wth up to p termnal nodes each labelled b a dstnct value 0,1,,p-1. Ever nontermnal node s labelled b an nput varable and has p outgong edges; one correspondng to each logc value. These are termed the 0-edge, 1-edge, etc. An MDD s ordered f the varables adhere to a sngle orderng on ever path n the graph, and no varable appears more than once on an path from the root to a termnal node. Fndng a varable orderng to mnmze the number of nodes n an MDD s a crtcal ssue. A reduced MDD has no node where all p outgong edges pont to the same node and no somorphc subgraphs. Clearl, no somorphc subgraphs est f, and onl f, no two non-termnal nodes labelled b the same varable, have the same drect descendants. Throughout ths paper we assume all MDDs are reduced and ordered.

2 For a sstem of functon (multple-output problem), we represent the functons b a sngle DAG wth multple top nodes, a structure called a shared MDD. When p = 2, the MDD structure becomes the well-known BDD. We use cclc negaton as an edge attrbute n our MDDs as developed n [10][11] as a generalzaton of edge negatons n BDDs [1][13]. Ever edge n an MDD ponts to a functon. When an edge has an assocated cclc negaton, t means that edge ponts to the cclc negaton of the functon rather than the functon tself. The representaton s normalzed so that there s no cclc negaton on an 0-edge. Our MDDs alwas have a sngle termnal node wth value 0. Note that a cclc negaton ma be requred for realzng a desred output functon. We note that the normalzaton process used n our MDDs dffers from that often used n BDDs [16]. We have chosen the normalzaton rules for our package to best accommodate dfferent values of p and to allow for easer etenson to med-rad MDDs.. Fgure 1 shows an MDD representng the sum (F1) and carr (F2) for the addton of two 3-valued nputs. The three edges from each non-termnal node are drawn sold for the 0-edge, dashed for the 1-edge, and dotted for the 2- edge. A number mmedatel to the rght of an edge ndcates a cclc negaton assocated wth that edge. Fgure 1 MDD representng sum (F1) and carr (F2). 3. Sftng of BDDs and MDDs Sftng s a ver effectve heurstc varable orderng technque developed b Rudell [15] whch s now avalable n commonl used packages such as CUDD [16]. 3.1 Sftng of BDDs The prncpal step n sftng s the nterchange of a par of adacent varables n the current varable orderng. The ke to the effcenc of sftng s that such a varable nterchange can be done as a local operaton affectng onl nodes labelled b the two varables n queston and no others. Use of a unque table [3][16] makes these nodes drectl accessble. In general terms, sftng proceeds as follows: Sftng Procedure ) select a varable a smple heurstc s to choose the varable that labels the most nodes n the BDD, ) sft to the bottom of the BDD b a sequence of adacent varable nterchanges, ) sft to the top of the BDD b a sequence of adacent varable nterchanges, v) durng steps () and () a record s kept of the poston of that elds the smallest node count n the BDD, so now sft back down to that poston, v) repeat steps () to (v) untl each varable has been sfted nto the best poston notng that once a varable s selected for sftng, t s not selected a second tme. There are n! orderngs of n varables. Sftng eamnes 2 on the order of n orderngs, et does etremel well at dentfng good varable orderngs. In the above procedure, each varable s shfted to ts 'best' poston. The whole process can be terated untl there s no further mprovement whch s termed sftng to convergence [16]. All sftng n ths work s sftng to convergence. 3.2 Sftng of MDDs Sftng an MDD requres an effcent means of performng adacent varable nterchange. Such a method was gven b the present authors n [11]. We here brefl outlne ths method as t wll be used n modfed form to mplement mod-p sum and augmented sftng. Full detal can be found n [11]. We consder the nterchange of and where the former mmedatel precedes the latter n the varable orderng and assume for smplct that all non-termnal nodes have p descendants. For each node η labelled, matr T s constructed wth T qr set to (a) the r-th descendant of the q-th descendant of η f the q-th descendant ponts to a node labelled, (b) the q-th descendant of η, otherwse. Gven T formed as descrbed above, the new nodes labelled are constructed usng the columns of T to determne the descendants and then usng the nodes so constructed as the descendants of the new node labelled. In smplest terms, the requred rearrangement s accomplshed b fllng T b rows and then applng t b columns. There are a number of mplementaton ssues to consder whch are gven n detal n [11]. Gven ths method for adacent varable nterchange, sftng of MDDs s readl mplemented usng the same overall approach as for the BDD case.

3 4. Lnear Sftng of BDDs Lnear sftng was ntroduced b Menel, Somenz and Theobold [9] and further dscussed b Günther and Drechsler [4][5][6]. In ths etenson to sftng, the smple nterchange of two adacent varables and n steps () and () of the procedure outlned above s replaced b the followng: (a) Varables and are nterchanged. Let k1 be the number of nodes n the BDD after ths nterchange. (b) Appl the lnear transformaton. Let k2 be the resultng number of nodes n the BDD. (c) If k1 k2 then the transformaton s undone. Undong the transformaton s accomplshed b smpl reapplng t snce t s ts own nverse. Note that the above s descrbed n terms of XOR due to the normalzaton rules we use for decson dagrams. The orgnal descrpton n [9] s n terms of equvalence due to dfferent normalzaton rules. The concept s the same. Fgure 2 llustrates the two basc operatons used n lnear sftng. (a) shows a BDD structure before transformaton. 0-edges are sold and 1-edges are dashed. The nodes labelled f 00 through f 11 are the top nodes of sub-dags representng subfunctons (not shown). (b) shows the effect of nterchangng and whch s to nterchange the subfunctons f01 and f 10. (c) shows the effect of subsequentl applng whch s to nterchange the subfunctons f 10 and f11 n (b). The functon represented b each of the dagrams n Fgure 2 s the same and the representatons of the subfunctons f 00 through f 11 are not affected b the transformatons. Hence both the nterchange of varables and the lnear transformaton are local operatons affectng onl two adacent levels n the BDD. Usng cclc negatons does not change ths localt propert. Smmetr would suggest applcaton of the transformaton should be consdered. In fact, t s snce sftng wll encounter the varables n the two possble orderngs. Trng both and for each orentaton duplcates effort. 5. Mod-p Sum and Augmented Sftng of MDDs We net consder the etenson of lnear sftng to MDDs. We frst address the case of replacng XOR b the mod-p sum whch results n an approach we term mod-p sum sftng. Based on that, we then consder other operatons as etensons to XOR. The full method, whch we term augmented sftng, allows for the consderaton of multple operatons durng a sngle sftng process. (a) ntal structure (b) after nterchange of and (c) after subsequent transformaton Fgure 2 Lnear sftng transformatons. 5.1 Mod-p Sum Sftng of MDDs As the frst step n etendng the dea of lnear sftng to MDDs, we consder the replacement of XOR n wth the mod-p sum. A crtcal dfference to note s that whle XOR s ts own nverse, mod-p summaton s not ts own nverse and to undo a mod-p sum transformaton we must appl mod-p subtracton. Fortunatel, both transformatons can be mplemented usng essentall dentcal matr procedures. In general, the nterchange of the two varables and results n the subfuncton nterchanges f f, s t,0 st, p 1 st ts Smlarl, applcaton of the transform results n the subfuncton substtutons fst fss, t,0 st, p 1 The nterchange of two varables and the transformaton s mplemented usng the matr based procedure descrbed above n Secton 3.2. The method proceeds as llustrated n Fgure 3. In general, consder a node γ labelled. We construct a matr T wth p rows and p columns. For =0,1,, p-1,

4 (a) If the s-edge from γ leads to a node δ labelled, then for t=0,1,,p-1, Ts tt, s set to pont to the node ponted to b the t-edge of δ wth the edge ccles beng the composton of the edge ccles on the s edge from γ and the t edge from δ. (b) If the s-edge from γ leads to a node δ not labelled, then T s tt, s set to the s-edge from γ for t=0,1,,p-1. (a) ntal structure (b) after nterchange of and (c) after transformaton Fgure 3 Mod-p sftng transformatons. Once T s constructed as above, the transformaton s made b settng each s-edge from γ, s=0,1,,q-1 to pont to a node labelled whose t-edge, t=0,1,,p-1, ponts to the node ponted to b T s tt,. Durng ths constructon, the edge ccle operatons are normalsed to ensure there s no ccle operaton on an 0-edge. The complete transformaton s accomplshed b performng the above for all nodes orgnall labelled. In the same fashon as dscussed above for varable nterchange, t s clear that ths s a local transformaton of the MDD affectng onl the and levels. The same procedure s used for the reverse transformaton,.e. to undo a transformaton when t does not mprove the MDD node count. The dfference s that reference s made to Ts tt, where the mod-p dfference s s t = ( s t+ p)mod p. Once the varable currentl beng consdered has been sfted to the bottom and then to the top t must be postoned to eld the smallest decson dagram. As noted n step (v) of the sftng procedure presented n Secton 3.1, for sftng, onl a sequence of downward varable nterchanges s requred. For lnear or mod-p sum sftng, an ordered record must be kept of the transformatons. Puttng the varable under consderaton nto the correct poston, requres the and varable nterchanges be undone n reverse order back to but not undong the nterchanges and transformatons that put the varable nto the best poston durng the sftng process. The bookkeepng requred s straghtforward but the computaton n undong the operatons back to the best poston can be substantal and n general can be equal to the computaton requred n the sftng down and up of the varable. The latter s certanl the case when the orgnal poston s optmal for the varable. 5.2 Other Operatons We confne our attenton to p = 2, 3, 4. The etenson to hgher values of p should be clear. Table 1 shows the sum and dfference operatons modulo-p. The crtcal propertes for the work here are (a) The operatons are reversble so that a transformaton that does not reduce the of a decson dagram can be undone. (b) The (0,0) entr s 0 whch means the subfuncton on the 0-0 path does not move so the transformaton of the decson dagram s a local operaton. Replacng ths subfuncton wth another could requre a normalzaton requrng edge operaton changes hgher n the dagram thereb destrong the localt of the transformaton. Gven (a) and (b), XOR s the onl choce when p = 2. If we requre ust (a) and (b), there are a number of alternatves to mod-p sum when p > 2. For eample, for p = 3, one alternatve par s shown n Table 2. To lmt the number of choces to a reasonable number both n terms of the computaton and the bookkeepng requred, we add a thrd constrant that the generalzatons of must satsf (c) The frst row and the frst column of the table defnng should contan 0, 1,, p-1 n order. Mod-p sum s then the onl generalzaton of for p = 3. For p = 4, mod-4 sum s a proper generalzaton as are the operatons 1, 2, 3 lsted n Table 3.

5 p=2 p=3 p= Table 1 Sun and dfference mo-p for p=2, 3, and Table 2 Alternatve generalzaton for p = Augmented Sftng Our augmented sftng method follows the same computatonal procedure as lnear sftng. The dfference s that whle lnear sftng for p = 2 need onl consder XOR operatons between a par of varables, for p = 3 or 4 our method tres each of the approprate generalzatons of, and when the are dfferent the correspondng generalzatons of. Hence, for ever adacent varable nterchange whle a varable s sfted to the bottom of the MDD and then to the top, the augmented sftng method tres transformatons based on the operatons: p = 2: XOR; p = 3: mod-3 sum, mod-3 dfference; p = 4: mod-4 sum, mod-4 dfference, the 5 dstnct functons n Table 3. At each step, the augmented sftng method chooses the transformaton (f an) from amongst those tred that elds the greatest reducton n the MDD node count. The mplementaton of all these transformatons s as descrbed for mod-p sum n Secton Table 3 Alternatve operatons for p=4. 6. Epermental Results Augmented sftng has been mplemented n the MDD package dscussed b the present authors n [11] and [12]. As noted above, cclc negatons are used. The MDDs are buld usng recursve mplementatons of MIN and MAX and unque and compute tables as dscussed n [12] We here present the results of applng the procedure to a varet of functons usng our MDD package on a Sun Blade 1000 wth one 750 MHz. UltraSPARC III CPU wth 512 Mb RAM. The bnar eamples presented for comparson were also done wth our MDD package wth p = 2 n whch case augmented sftng s lnear sftng. p n out ntal sfted mod-p sfted # transformatons Table 4 Sftng and mod-p sftng of adder functon.

6 Table 4 shows the results for p-valued addton of two n- bt numbers where each eample has 2n p-valued nputs and n+1 p-valued outputs. The column labelled ntal s the number of MDD nodes for the nput orderng an, an 1,..., a1, bn, bn 1,..., b1that s the nputs of the two numbers beng added one followng the other. The column sfted s the number of MDD nodes after sftng s appled. The varable orderng found b sftng a, b, a, b,..., a, b s n n, n 1 n The column mod-p sfted s the node count after mod-p sftng s appled. The node count s substantall reduced b mod-p sftng wth ust a few transformatons. Sftng and mod-p sftng are clearl ver effectve for adders snce the are both smmetrc n correspondng postons for the numbers beng added and also hghl dependent on the operaton. Augmented sftng gves no further mprovement for adders. Multplcaton s a dffcult case for decson dagram representaton, and mod-p and augmented sftng do not help. For eample, multplcaton of two 6-bt bnar numbers, a problem wth 12 nputs and 12 outputs, has 1,158 nodes n the smple one number after the other varable order, and 1,098 nodes after applng sftng. Applng mod-p or augmented sftng elds the same result as sftng wth no transformatons selected. p n ntal mod-p sfted # transformatons Table 5 Mod-p sftng of summaton functons. p n out nta l sfted augmented sfted # transformatons * * * * * * (+ unsfted order converson; * sfted order converson) Table 6 Bnar and quaternar coded adders. Table 5 shows the results for the summaton of n p-valued nputs. The number of outputs n each case s log p ( n p ) and s the p-valued representaton of the arthmetc sum of the nputs. Table 6 s a comparson of the BDD of bnar adders and the of two dstnct MDDs derved from each. Each case has three rows. The frst gves the results for the bnar adders whch are those from Table 4. The second row s for the MDD where each quaternar nput s derved from a par of bnar nputs from left to rght where the bnar nputs are n the order an, an 1,..., a1, bn, bn 1,..., b1,.e. the dgts of the frst number followed b the second whch we call unsfted order. The natural bnar to quaternar converson s used,.e. (00 0;01 1;10 2;11 3) The outputs are left as bnar so the derved functons are quaternar-nput bnar-output and do not represent the quaternar-nput, quaternar-output adder n Table 4. The thrd row of the table s for the MDD constructed n the same fashon but usng the sfted varable order found n the bnar case whch as noted above s a, b, a, b,..., a, b n n, n 1 n A number of observatons can be made. Frst t s clear that basng the converson of bnar nputs to quaternar nputs on the bnar sfted order s better than usng the unsfted order. In partcular, we conecture the MDD constructed from the sfted bnar order has 3n nodes whereas the correspondng lnear sfted BDD has 7n 4nodes where n s the number of bts n each of the bnar numbers beng added.. We also note that augmented sftng s benefcal for the BDDs (n fact lnear sftng) and also for the MDDs

7 derved from the unsfted bnar nputs when n s even. Augmented sftng does not help the MDDs when n s odd because the quaternar encodng combnes the least sgnfcant bt of a wth the most sgnfcant bt of b whch precludes the transformatons found n the even case where ths s not the stuaton. It s also nterestng to note that for the MDDs constructed from the sfted bnar order, augmented sftng of the MDD tself s of no beneft. Ths s the stuaton because the bnar varable parng used to construct the quaternar nputs captures the lneart. Table 7 (at the end of the paper) shows the results for a number of commonl used benchmark problems. Three representatons are presented for each problem: the BDD for the orgnal bnar problem, the 4-valued nput, bnar-output MDD for the gven varable order, and the 4-valued nput, bnar-output MDD for the varable order found b sftng for the BDD. Three scenaros are presented: (A) sftng followed b augmented sftng, (B) sftng followed b mod-p sum sftng (we show onl the case where the result can dffer from scenaro A, and (C) mod-p sum sftng not preceded b regular sftng. We note that n ths paper we are not concerned wth the best wa to transform a bnar problem to a quaternar one. The are here onl used as a source of eamples. However, we do note that n general smaller MDDs arse when the BDD sfted varable orderng s used. The eceptons to ths arse when ths approach pars varables that are not n the support set of the maort of output functons. For eample, ths results n the MDDs derved for eample e64 beng larger than the BDD. The quaternar parng has n fact ntroduced varable dependenc not present n the orgnal problem. That stuaton must be avoded n the case where the obectve s to fnd a good converson of a bnar problem to quaternar. Nevertheless, the BDD sftng order s a good startng pont. 7. Concludng Remarks Ths paper has consdered the augmented and mod-p sum sftng of MDDs. The epermental results presented ndcate these approaches work well for certan classes of functons such as adders and weght functons. We epect the wll work well for man 'arthmetc' tpes of functons wth the notable ecepton of multplers. In general, our results ndcate mod-p sum sftng s as effectve as the more general augmented sftng whle requrng consderabl less computaton. The results also ndcate that as found n [5] for lnear sftng of BDDs, t s best to appl regular sftng followed b mod-p sum or augmented sftng. The regular sftng determnes a good threshold b varable swappng after whch transformatons are appled onl when the further reduce the node count. Optmzaton of our mplementatons of sftng, mod-p sum and augmented sftng s ongong. At present, our mplementatons are rather slow, especall n comparson to a hghl optmsed package such as CUDD [17]. For eample, the problem ape1 (45 nputs and 45 outputs) treated as bnar requres 1.8 CPU sec. for sftng and 4.8 sec. for augmented (lnear) sftng usng our package. When ape1 s converted to a quaternar problem, sftng takes on the order of 1.7 sec. and mod-p sum sftng takes about 2.5 sec. Augmented sftng for ths eamples takes about 23 sec. We have found that mod-p sum sftng takes on the order of 2 to 5 tmes longer than sftng whereas augmented sftng takes on the order of 7 to 10 tmes longer than mod-p sum sftng. In contrast, the tme requred to read and sft ape1 usng CUDD s neglgble on the same machne. A maor part of the dfference s that our package treats a BDD as a specal case of an MDD whch leads to a slower mplementaton due to the fleblt requred, e.g. a varable number of edges from each node, cclc negaton as opposed to smple bnar negaton etc. The MDD package used n ths work s avalable at References [1] K. S. Brace, R L. Rudell and R. E. Brant, Effcent mplementaton of a BDD package, Proc. Desgn Automaton Conference, pp , [2] R. E. Brant, Graph-based algorthms for Boolean functon manpulaton, IEEE Trans. on Computers, V. C- 35, no. 8, pp , [3] R. Drechsler and D. Selng, Bnar decson dagrams n theor and practce, Int. Journal on Software Tools for Technolog Transfer, 3, pp , [4] W. Günther and R. Drechsler, Lnear transformatons and eact mnmzaton of BDDs, IEEE Great Lakes Smposum on VLSI, pp , Lafaette, LA, Feb [5] W. Günther and R. Drechsler, BDD mnmzaton b lnear transformatons, Conf. on Advanced Computer Sstems, pp , Szczecn, Poland, Nov [6] W. Günther and R. Drechsler, Mnmzaton of BDDs usng lnear transformatons based on evolutonar technques, IEEE Internatonal Smposum on Crcuts and Sstems, pp. I: , Orlando, FL, Ma [7] M. G. Karpovsk, Fnte Orthogonal Seres n the Desgn of Dgtal Devces, John Wle and Sons., [8] H. T. Lau and C.-S. Lm, On the OBDD representaton of general Boolean functons, IEEE Trans. on Comp., C-41, No. 6, pp , [9] C. Menel, F. Somenz, and T. Theobold, Lnear sftng of decson dagrams, Proc. Desgn Automaton Conference, pp , [10] D. M. Mller, Multple-valued logc desgn tools, (Invted Address) Proc. 23rd Int. Smp. on Multple- Valued Logc, pp. 2-11, Ma [11] D. M. Mller and R. Drechsler, Implementng a multplevalued decson dagram package, Proc. 28th Int. Smp. on Multple-Valued Logc, pp , Ma 1998.

8 [12] D. M. Mller and R. Drechsler, On the constructon of multple-valued decson dagrams, Proc. 32nd Int. Smp. on Multple-Valued Logc, pp , Ma [13] S. Mnato, N. Ishura and S. Yama, Shared bnar decson dagrams wth attrbuted edges for effcent Boolean functon manpulaton, Proc. ACM/IEEE Desgn Automaton Conference, pp , [14] S. Mnato, Graph-based representatons of dscrete functons, Proc. IFIP WG 10.5 Workshop on the Applcaton of Reed-Muller Epanson n Crcut Desgn, pp. 1-10, [15] R. Rudell, Dnamc varable orderng for ordered bnar decson dagrams, Proc. IEEE/ACM ICCAD, pp , [16] F. Somenz, CUDD: CU Decson Dagram Package, CUDD [17] F. Somenz, Effcent manpulaton of decson dagrams, Int. Journal on Software Tools for Technolog Transfer, 3, pp , [18] A. Srnvasan, T. Kam, S. Malk, and R.E. Braton, Algorthms for dscrete functon manpulaton, Proc. ICCAD, pp , eample p n out ntal sfted augmented sfted Scenaro A Scenaro B Scenaro C # mod-p # drect modp trans. sfted trans. sfted alu * alu * ape * ape * ape * bw * seq * e * duke * mse * mse * mse * sao * sn * (+ unsfted order converson; * sfted order converson) Table 7 Standard benchmark functons. # trans.