Binary Decision Diagram with Minimum Expected Path Length

Transcription

1 Bnary Dcson Dagram wth Mnmum Expctd Path Lngth Y-Yu Lu Kuo-Hua Wang TngTng Hwang C. L. Lu Dpartmnt of Computr Scnc, Natonal Tsng Hua Unvrsty, Hsnchu 300, Tawan Dpt. of Computr Scnc and Informaton Engnrng, Fu Jn Catholc Unvrsty, Tap, Tawan Abstract W prsnt mthods to gnrat a Bnary Dcson Dagram (BDD) wth mnmum xpctd path lngth. A BDD s a gnrc data structur whch s wdly usd n svral flds. On mportant applcaton s th rprsntaton of Boolan functons. A BDD rprsntaton nabls us to valuat a Boolan functon Smply travrs th BDD from th root nod to th trmnal nod and rtrv th valu n th trmnal nod. For a BDD wth mnmum xpctd path lngth wll b also mnmzd th valuaton tm for th corrspondng Boolan functon. Thr ffcnt algorthms for constructng BDDs wth mnmum xpctd path lngth ar proposd. 1. Introducton Bnary Dcson Dagram (BDD) was ntroducd n 1978 [1] as an ffctv way to rprsnt Boolan functons that lads to ffcnt manpulaton [2] and mplmntaton [3]. Gvn a random nput pattrn, w can travrs th BDD to dtrmn th outputng. Thr ar prvous works on mnmzng th numbr of nods n a BDD [4] [5] [6]. Howvr, th numbr of nods s not drctly rlatd to th valuaton tm for a Boolan functon. Rathr, th valuaton tm s drctly rlatd to th total xpctd path lngth of a BDD. In a logc crcut, th probablts of occuranc of th nput varabls ar dffrnt. It follows that th valuaton tm for dffrnt nput combnatons ar also dffrnt. Consquntly, a BDD wth mnmum xpctd path lngth wll also hav mnmum valuaton tm. To obtan a BDD wth mnmum xpctd path lngth, w nd to dvlop (1) mthods for computng th xpctd path lngth of a BDD (2) hurstcs for ordrng th varabls n a BDD to mnmz th xpctd path lngth. Ths papr s organzd as follows. Bsds Scton 1 bng an ntroducton, w wll dscuss basc concpt of BDDs n Scton 2. In Scton 3, w wll propos thr mthods to comput th xpctd path lngth of an ROBDD. Scton 4 dals wth hurstcs to dtrmn varabl ordrng so that th rsultant ROBDD wll hav mnmum xpctd path lngth. Fnally, xprmnt rsults wll b prsntd. 2. Prlmnary 2.1. ROBDD A BDD s a drctd, acyclc graph whch conssts of nods and dgs. Thr ar two typs of nods, trmnal and non-trmnal. A trmnal nod s a Boolan valu or, and a non-trmnal nod s an nput Boolan varabl. Each non-trmnal nod contans two outgong dgs pontng to othr nods, whch s rfrrd to as th parntnod of ths nods. For convnnc, w dstngush th two dgs as thn-dg and ls-dg. A nod whch s connctd from a thn-dg s th thn-chld of ts parnt-nod and from an ls-dg th ls-chld. A Rducd Ordrd BDD (ROBDD) [2] s a BDD wth a chosn varabl ordrng n all paths from th root nod to th trmnal nods and posssss th followng proprts (1) f both th thn-chld and ls-chld of a non-trmnal nod ar dntcal, th non-trmnal nod wll b rmovd from th path, and (2) f thr ar two or mor dntcal subtrs, only on wll b rtand and all parnt-nods whch hav th sam subtr wll shar on subtr as thr chldrn. Th formula rprsntaton of an ROBDD s f parntnod thn thn-chld ls ls-chld, whch s calld an ITE oprator (IF-THEN-ELSE). By usng th ITE oprator, Boolan opratons can b asly prformd on ROBDDs. Fgur 1 gvs an ROBDD for th functon wth th nput varabl ordrng. Not that for th sam functon, dffrnt ordrng of nput varabls wll rsult n dffrnt ROBDDs Expctd Path Lngth of ROBDD Snc th valuaton tm of a ROBDD s drctly rlatd to th path lngth of ROBDD, w want to shortn th path lngths wth hgh probablty,.., to mnmz th xpctd path lngth of th ROBDD. It s ncssary that w b abl to comput a xpctd path lngth of a gvn ROBDD. To bgn wth, w dfn th xpctd lngth of th th path n ROBDD as! ", whr # s th probablty of th th path and $ s th path lngth of th th path. Th total Expctd Path Lngth (EPL) of a ROBDD rootd at nod % s dfnd as &('*), ' 6; ) 6 (1)

2 6 6 6 I N F A B BEPL (a) a C 1- P a P a 0 1 BEPL (a ) 0 a 0 a 1 BEPL (a 1) Fgur 1. An ROBDD xampl. whr < s th numbr of paths of th ROBDD. Followng ths formula, to calculat th xpctd path lngth wll nd drctly numrat all paths. Ths approach s mpractcal bcaus th numbr of paths may grows xponntally. In th followng scton, w wll propos thr ffcnt mthods to calculat xpctd lngth bottom-up mthod, topdown mthod, and mddl-way mthod. 3. Thr Mthods for Computng EPL Snc t s mpractcal to numrat all paths n a ROBDD. W prsnt thr ffcnt mthods n ths scton for computng th EPL of a ROBDD. Thy ar bottom-up mthod, top-down mthod, and mddl-way mthod. Th frst and scond mthods nd to b xcutd only onc ovr all nods. Aftr ntalzng, th thrd mthod wll b actvatd to mprov th prformac of our algorthm Bottom-up Mthod Bottom-up mthod s formulatd by whch th xpctd path lngth can b calculatd from trmnal nods up to root nod. Snc at ach non-trmnal nod, th xpctd path lngths of ts two chld-nods ar alrady computd, w can comput th xpctd path lngth of th non-trmnal nod basd on th xpctd path lngths for ts chld-nods. Bfor w procd to dvlop th formula, w dfn th followng notatons as [ [ [ &"'*),+?>@ ] th xpctd path lngth of all paths from A to trmnal nods 'CB ] th probablty of th th path from nod A to a trmnal nod )DB6 ] th path lngth of th th path from nod A to a trmnal nod A 8 3 ' 6 B I ) B I ' 6 B N ) B N Now, lt b a non-trmnal nod wth ts thn-chld A;E and ls-chld A F as shown n Fgur 2. Th xpctd path lngths of all paths from trmnal nods up to A E and AGF can supposdly b computd as &('*),+?> D1 5*HJI &('*),+?>LK0D1M3 5*HON (2) (3) Fgur 2. Bottom-up mthod for calculatng th xpctd path lngth. whr <QP and <4P ar th total numbr of paths from trmnal nods up to nod A E and A F, rspctvly. Lt th probablty from nod A to A E b R9P. Thn, th probablty from nod A to AGF wll b TSQR P. Th xpctd path lngth from trmnal nods up to nods A s computd as&('*)u+.> V1 3 56W7X8ZY H I B ' 6 B I + ) B6\[^] I [ 3 5*HON 6W7X8 + ],_`Y B ' 6 B N + ) B6a[Q] I 1 Y B +b H I ' 6 B I ) B6\[ I H I ' 6 B I [ + ],_`Y B H N ' 6 B N ) B6 N [ H N ' 6 B N Not that th total path probablty from a non-trmnal nod to trmnal nods s. That s, 3 5*HJI ' 6B I 1 3 5CHcN 6W798 ' 6B N 1 ] From th dntty of quatons (2) (3) (5), w can rduc quaton (4) to &('*)U+.> V1 Y B + &('*)U+.> 8 [^] [ + ],_`Y B + &"'*),+?>LK0 [d] 1 Y B &('*),+?> 8 [ + ],_`Y B &('*)U+.> K [^] From quaton (6), w follow that th xpctd path lngth of a non-trmnal nod at A can b computd by followng stps stp1 multply th probablty of th thn-chld by th xpctd path lngth of th thn-chld whch s computd n th prvous stag and s radly usd. stp2 multply th probablty of th ls-chld by th xpctd path lngth of th ls-chld whch s computd n th prvous stag and s radly usd. stp3 sum rsults of stp1, stp2, and constant 1 (5) (6) (4) 2

3 m I + [ Wth boundary condton that both th path lngth of trmnal nods and ar 0, w can rcursvly calculat th xpctd lngth of th root of an ROBDD by bottom-up mthod. root 3.2. Top-down Mthod Rvrsly, top-down mthod s usd to calculat th xpctd path lngth from root nod down to trmnal nods. Snc at ach non-trmnal nod, th xpctd path lngths of ts parnt nods ar alrady computd, w can comput th xpctd path lngth of an non-trmnal nod basd on th xpctd path lngths for ts parnt nods. Bfor w procd to dvlop th formula, w dfn th followng notatons as [ &"'*),+?fo ] th xpctd path lngth of all paths from root to non-trmnal nod g '*h [ ] th probablty of th j th path from root to nod g ' h [k [ ] th sum of probablts of all paths from root to nod g ) h ] th path lngth of th j th path from root to nod g Obvously, from th abov dfntons, quatons (7) (8) hold. That s, th summaton of probablts of all paths from root to a nod g0l s k ' hnm 1 3 5Co 7X8m (7) whr <Qp s th total numbr of paths from root to nod g0l. Morovr, th xpctd path lngth from root to a nod g0l s &"'*),+?foq@d1 3 5*o 798m ) h m (8) Now, lt g b a non-trmnal nod wth ts r parnt-nods labld gle to gs and th probablts from parnt nods gle to gs to nod g b R p to R p!t, rspctvly, as shown n Fgur 3. W ar to comput th xpctd path lngth of all paths from root to nod g. Frst, th summaton of probablts of all paths from root nod to nod g s ku ' h 1Q3wv6W7X8 Y hnx whr r s th numbr of parnt-nods of nod g. Now, th xpctd path lngth from root to nod g s &('*)U+.fOy1 3 v6.798; GY o x hzx ' h x + ) h x [d] 1 3v6.798; o x + Y hzx ' hzx ) hzx [Y hzx ' hzx (10) By quatons (7) (8) and (9), quaton (10) can b rducd to &('*)U+.fOy1 3v6.798{+ Y hzx 1 3 v6.798 Y h x ku ' hnx &('*),+?f 6 [Y hzx ' hnx &"'*),+?f 6 [ ' hku ku (9) (11) From quaton (11), w follow that th xpctd path lngth of a non-trmnal nod at g can b computd by followng stps stp1 for all parnt nods, multply th probablty of parnt-nod to th nod g by th xpctd path lngth from root to th parnt nod whch s computd n th prvous stag and s radly usd. b 1 b j TEPL p b j b b j TEPL b Fgur 3. Top-down mthod for calculatng th xpctd path lngth. stp2 sum probablts of all paths from root to nod g by quaton (9), whr th sum of probablts of all path from root to a parnt-nod s alrady computd and radly usd. stp3 sum rsults of stp1 and stp2 Basd on th assumptons that th path lngth of root s and th probablts from all paths to root s 1, w can calculat th xpctd lngth of th trmnal nods of an ROBDD by top down mthod. Fnally, th xpctd path lngth of th whol ROBDD s th summaton of th xpctd path lngths of th two trmnal nods Mddl-way Mthod Th thrd mthod s usd to comput th xpctd path lngth of an ROBDD assumng that th xpctd path lngths of all non-trmnal nods at lvl ws} hav bn computd by top-down mthod and th xpctd path lngths of all non-trmnal nods at lvl hav bn computd by bottom-up mthod. Lt b th cut btwn lvls ^S and and thr ar ~ dgs crossng that connct th nods at lvls ^S and as shown n Fgur 4. To comput th xpctd path lngth of th whol ROBDD, w hav to consdr all paths connctd by ~ dgs. Frst, w consdr how to comput th xpctd path lngth connctd by a sngl crossng dg. In ths cas, dg conncts th top nd-nod g l and th bottom nd-nod A l as shown n Fgur 4. Suppos that th probablty of dg b R l and thr ar r paths from root nod to nod g l, and < paths from trmnals nods to Aul as shown n Fgur 4. Thn, th xpctd path lngth of all paths, ƒz!;, whch pass through dg s &('*)U+. ˆy1 3 56W7X8 3 v 7X8 Y q ' 6 B m ) B6 m [ ) hnm [Q] 1 Y q +b v 798 ' 6 B m 3w56W7X8u3 v 7X8 ' 6 B m ' h!m 3w56W7X8u3 v 7X8 ' 6 B m ' h!m b n ' hnm ) hnm ) B6 m [ 3

4 [ 1 Y Y TEPL(b ) b k k p k dg k a k BEPL (a k) C Algorthm fnd-last-cost(œ ) InputŒ Boolan functon; ŽJ u? G # # "(štš ; Œ Bgn (1) fnd an ntal varabl ordrng for ROBDD of ; (2) buld an ntal ROBDD; (3) comput th cost of th ROBDD; (4) whl ( not stop ) do (5) fnd nw varabl ordrng; (6) rstructur th ROBDD accordng to th nw varabl ordrng; (7) comput th cost of th nw ROBDD; (8) updat th cost functon; (9) chck f th stoppng crtron s st; - - Output rturn OŽ c and b X OŽ { ndwhl -{- (10) rturn OŽ O and b 9 OŽ End Ž G 0. G # ((štš ; Fgur 4. Mddl-way mthod for calculatng th xpctd path lngth. 1 Y q +b3v W7X8 3 56W7X8 ' 6 B m 3 v 7X8 3 56W726 ' 6 B m 3 v 7X8 ' h!m ' 6 B m ) h m ) B6 m [ By quatons (2)(5)(8)(9), quaton (12) can b rducd to &('*)U+. ˆy1 Y q + &"'*),+?> q [ &('*)U+.fOq@ ku [ ' hnm k (12) (13) From quaton (13), w follow that th xpctd path lngth of all paths passng through dg connctng nod g0l and A l can b computd by followng stps stp1 sum th xpctd path lngth from root of all paths to nod g l, th xpctd path lngth of all paths from nod A l to trmnals and twc of th total path probablty to nod g l. All of thm ar alrady computd and ar radly usd. stp2 multply th rsult of stp1 and th probablty of dg Fnally, th total xpctd lngth of th whol ROBDD s ƒ 3Š l0 E ƒ` n; 4. Algorthms for Fndng Low Cost ROBDD Our algorthm bgns wth an ntal ordrng. Thn, an tratv loop s ntrd to modfy th ntal ordrng. In ach traton, a nw ordrng s dtrmnd and th old ROBDD s rstructurd basd on th nw varabl ordrng. Nxt, th cost for th nw ROBDD s computd. Th loop contnus tll th stoppng crtron s mt. Th procdur s shown n Fgur 5. Th dtald dscrpton s xpland as follows. In stp 1, an ntal ordrng nds to b dtrmnd to construct an ntal ROBDD. It can b dtrmnd by constructng a mnmum sz ROBDD. In ths cas, algorthms Fgur 5. Th fnd-last-cost algorthm proposd n [5] [6] can b usd. Stp 3 s to comput th cost of th ntal ROBDD. For th computaton of th xpctd path lngth of th ROBDD, bottom-up and top-down mthods proposd n Scton 3 ar usd and all computd data for ach nod s stord at th nod. Stp 5 fnds a nw nput ordrng. Wndow prmutaton algorthm [5] [6] whch fnds a local mnmum n a wndow of sz and sftng algorthm [4] whch looks for a sutabl poston for ach varabl n ach traton can b usd. Wth th nw nput varabl ordrng, stp 6 rstructurs th old ROBDD to a nw ROBDD. Transposton oprator [7] s usd whch rstructurs an ROBDD by svral smpl ROBDD opratons. For xampl, f th ordr of and œ ar swappd n a nw ordrng, th followng formula can b usd. ˆž xzÿ ž@ ˆž w ` b D ;œ xb ž xª ž@ «ž@ Not that swappng s not lmtd to adjacnt varabls. In stp 7, th cost for th nw ROBDD s computd. For th xpctd lngth of th nw ROBDD, smlar to stp 6, t nds not b computd from scratch. Instad, w wll us th nformaton for th old ROBDD to calculat th nw xpctd lngth of ROBDD. Suppos that w hav a nw ordrng that varabls at lvls and j b$ ju ar swappd as shown n Fgur 6. Snc th part of ROBDD abov lvl and th part of ROBDD blow lvl j wll not b changd as sn n Fgur 6, w do not hav to rcomput thm. W only nd to comput th nw xpctd path lngth for th nods n th mddl. On th on sd, th bottom-up mthod calculats th xpctd path lngth from nods at lvl j4 up. On th othr sd, th top-down mthod calculats th xpctd path lngth from nods at lvl S down. Whn varabls on th two sds mt n th mddl, th mddlway mthod can b usd. By usng mddl-way mthod, t prvnts much mor rdundant calculatons than top-down and bottom-up mthods. 5. Exprmntal Rsults Our xprmnt s prformd on a SUN-Ultra Entrprs 150. Softwar platform s basd on th ROBDD packag n 4

5 STP TEPL BEPL nvald lvl lvl j Fgur 6. Computaton for parts lyng btwn th two swappd varabls. SIS [8]. LGSynth93 bnchmark sut s usd n our xprmnt. Exprmnt s conductd to fnd a mnmum xpctd path lngth ROBDD. Frst, th sftng algorthm [4] s usd to fnd a mnmum-szd ROBDD whch can b usd as our ntal ROBDD. Th rason for ths ntal ordrng hurstc s that a small sz ROBDD may lad to a small xpctd lngth ROBDD. In subsqunt stps, wndow prmutaton s usd to gnrat a nw nput ordrng and th xpctd path lngth for th nw ROBDD s computd. Fnally, th probablts of nput varabls ar st to b qual. Tabl 1 shows th rsults. Column <^ªr r, ˆ± ² gvs th rsults of ROBDD wth ntal ordrng whos objctv s to fnd an ROBDD wth mnmum sz. Column <^ªr ² R R AG³c gvs th rsults of ROBDD wth xpctd path lngth as objctv. Columns RDAG³c gv th xpctd path lngth and sz of ROBDDs, rspctvly. In column rato, w comput th rato of th rsults of mnmum nods to th rsults of mnmum xpctd path lngth. It s clar from Tabl 1 that n most cass, th xpctd path lngth can b rducd at th cost of small ncras of th numbr of nods xcpt cordc and f51m. In som cass (con1, vg2, cm150a, tc.), w found that both th sz and th xpctd path lngth of ROBDD ar rducd. On th avrag, th xpctd path lngth s rducd by 25% whl th th numbr of nods s ncrasd by 10%. 6. Conclusons In ths work, w prsnt mthods to gnrat a ROBDD wth mnmzd xpctd path lngth. Wth smallr EPL of a ROBDD, th valuaton tm of a Boolan functon wll b rducd. Thr mthods ar proposd to calculat th xpctd path lngth of a ROBDD ffcntly. Th xprmntal rsults ndcat that th xpctd path lngth of an ROBDD can b rducd at th cost of a small ncras of OBDD sz. Tabl 1. Rsults of xpctd path lngth wth qual nput probablty crcut mn nod mn xp path rato (%) path sz path sz path sz 5xp alu b con cordc x nc sao vg msx cm150a cm151a cm162a cm163a cm85a mux z4ml f51m pcl traffc Avrag Rfrncs [1] S. B. Akrs, Bnary dcson dagrams, IEEE Trans. Comput., Vol. C-27, pp , Jun [2] R. E. Bryant, Graph-basd algorthms for boolan functon manpulaton, IEEE Trans. on Comput., Vol. C-35, No. 8, August [3] K. S. Brac, R. L. Rudll, and R. E. Bryant, Effcnt Implmntaton of a BDD Packag, DAC, [4] R. Rudll, Dynamc varabls ordrng for ordrd bnary dcson dagrams, ICCAD, [5] M. Fujta, Y. Matsunaga, and T. Kakuda, On Varabl Ordrng of Bnary Dcson Dagrams for th Applcaton of Mult-lvl Logc Synthss, EDAC, pp 50-54, Mar [6] N. Ishura, H. Sawada, and S. Yajma, Mnmzaton of Bnary Dcson Dagrams Basd on Exchangs of Varabls ICCAD, pp , Nov [7] K. H. Wang, T. T. Hwang, and C. Chn, Rstructurng Bnary Dcson Dagrams Basd on Functonal Equvalnc, EDAC, pp , Fb [8] E. M. Sntovch, K. J. Sngh, L. Lavagno, C. Moon, R. Murga, A. Saldanha, H. Savoj, P. R. Stphan, R. K. Brayton, A. Sangovann-Vncntll, SIS A Systm for Squntal Crcut Synthss, Elctroncs Rsarch Laboratory Mmorandum No. UCBERL M9241, 4 May