Pau Wao Purdom, Jr., Idiaa Uiersiy G. Nei Hae, Idiaa Uiersiy Bacracig ad Probig Paria suor roided by NSF Gra CCR 9-0394. Absrac We aayze wo agorihms for soig cosrai saisfacio robems. Oe of hese agorihms, Probe Order Bacracig, has a aerage ruig ime much faser ha ay reiousy aayzed agorihm for robems where souios are commo. Probe Order Bacracig uses a robig assigme (a reseeced es assigme o use ariabes) o he guide he search for a souio o a cosrai saisfacio robem. If he robem is o saised whe he use ariabes are emorariy se o he robig assigme, he agorihm seecs oe of he reaios ha he robig assigme fais o saisfy ad seecs a use ariabe from ha reaio. The a each bacracig se i geeraes subrobems by seig he seeced ariabe each ossibe way. I simies each subrobem, ad ries he same echique o hem. For radom robems wih ariabes, causes, ad robabiiy ha a iera aears i a cause, he aerage ime for Probe Order Bacracig is o more ha whe ( ) us ower order erms. The bes reious resu was ( ). Whe he agorihm is combied wih a agorihm of Fraco ha maes seecie use of resouio, he aerage ime for soig radom robems is oyomia for a aues of whe O( 3 ( ) 3 ). The bes reious resu was O( 3 ( ) 6 ). Probe Order Bacracig aso rus i oyomia aerage ime whe, comared wih he bes reious resu of (). Wih Probe Order Bacracig he rage of ha eads o more ha oyomia ime is much smaer ha ha for reiousy aayzed agorihms. Bacracig The cosrai saisfacio robem is o deermie wheher a se of cosrais oer discree ariabes ca be saised. Each cosrai mus hae a form ha is easy o eauae, so ay dicuy i soig such a robem comes from he ieracio bewee he cosrais ad he eed o d a seig for he ariabes ha simuaeousy saises a of he cosrais. Cosrai saisfacio robems are exremey commo. Ideed, he roof ha a robem is NPcomee imies a ecie way o rasform he robem io a cosrai saisfacio robem. Mos NPcomee robems are iiiay saed as cosrai saisfacio robems. A few secia forms of cosrai saisfacio robems hae ow agorihms ha soe robem isaces i oyomia wors-case ime. Howeer, for he geera cosrai saisfacio robem o ow agorihm is fas for he wors case. Whe o oyomia-ime agorihm is ow for a aricuar form of cosrai saisfacio robem, i is commo racice o soe robem isaces wih a search agorihm. The basic idea of searchig is o choose a ariabe ad geerae subrobems by assigig each ossibe aue o he ariabe. I each subrobem he reaios are simied by uggig i he aue of he seeced ariabe. This se of geeraig simied subrobems is caed siig. If ay subrobem has a souio, he he origia robem has a souio. Oherwise, he origia robem has o souio. Subrobems ha are sime eough (such as hose wih o use ariabes) are soed direcy. More comex subrobems are soed by ayig he echique recursiey. If a robem coais he aways fase reaio, he he robem has o souio. Sime Bacracig imroes oer ai search by immediaey reorig o souio for such robems. Bacracig ofe saes a huge amou of ime. Probig This aer cosiders wo agorihms ha are imroemes oer Sime Bacracig. Boh agorihms use he idea of robig if a xed assigme o he use ariabes soes he robem, o addiioa iesigaio is eeded. Our agorihms robe by seig each use ariabe o fase ad esig o see wheher a reaios simify o rue. The wo robig agorihms are sime eough ha i is ossibe o aayze heir aerage ruig ime.
Our rs agorihm, Bacracig wih Probig, uses bacracig, robig, ad o addiioa echiques. I aricuar, durig siig i aways ics he rs use ariabe from a xed orderig o he ariabes. Our secod agorihm, Probe Order Bacracig, is more sohisicaed i is ariabe seecio. I has a xed orderig o he ariabes ad a xed orderig o he reaios. Firs, i checs ha here are o aways fase reaios. If a aways fase reaio is ecouered, he robem is o saisabe ad he agorihm bacracs. Nex, i checs o see if here is a currey seeced reaio. If here is o currey seeced reaio, i seecs he rs reaio ha eauaes o fase uder he robig assigme. (If a causes eauae o rue he he robig assigme soes he robem.) Fiay, he agorihm does siig usig he rs use ariabe of he seeced reaio. 3 Probabiiy Mode The aerage umber of odes geeraed whe soig radomy geeraed robems is oe measure of he quaiy of a search agorihm. We use his measure where our radom robems are formed by he coucio of ideedey geeraed radom causes (he ogica or of ieras, where a iera is a biary ariabe or he egaio of a biary ariabe). A radom cause is geeraed by ideedey seecig each iera wih a xed robabiiy,. We use for he umber of ariabes, ad for he umber of causes. For he asymoic aaysis, boh ad are fucios of. May agorihms hae bee aayzed wih his radom cause egh mode [3, 4, 7, 8,, 6, 8,, ]. Mos of hese aayses ad a few uubished oes are summarized i [7]. A few agorihms hae aso bee aayzed wih he xed egh mode, where radom robems cosis of radom causes of xed egh [,, 4, 0]. This secod robabiiy mode geeraes robems ha are more dicu o saisfy bu erhas more ie he robems ecouered i racice. The secod mode eads o much more dicu aayses. 4 Summary of Resus This secio summarizes he erformace of Probe Order Bacracig ad gies some iuiio as o why Probe Order Bacracig is fas. The simer Bacracig wih Probig Agorihm urs ou o roide o sigica imroeme oer reiousy aayzed agorihms, so we do o discuss i i grea deai. This aer has coour os showig he erformace of robig agorihms. We aso icude coour os for he aroximae erformace aayses of hese agorihms. Each o is for radom robems wih 50 ariabes. The erica axis shows, he robabiiy ha a gie iera aears i a cause, ruig from 0.00 o wih ics a 0.0 ad 0.. A 00 he aerage cause egh for robems is. A 0 he aerage cause egh is 0 ieras. The horizoa axis shows, he umber of causes, ruig from o 50 wih ics a 0 ad 00. Whe is ear 0 or mos robems are riia. Whe is ow mos robems are easy because hey coai a emy cause; emy causes are riiay usaisabe. Whe is high mos robems are easy because ay assigme of aues o ariabes is a souio o mos robems. The regio of hard robems ies i he midde. I mos cases he coours are shaed ie eogaed horseshoes (see Fig. ). The area wihi a horseshoe coour rereses robems ha are more dicu ha he robems ouside he coour. The ouermos coour shows where he aerage umber of odes is 50, he ex ier oe 50, ex 50 3, ad ay 50 4. Ruig ear he ceerie of he horseshoes is a ie ha shows for each ha aue of ha resus i he hardes robems (hose wih he arges umber of odes). I he ess faorabe cases he uer ad ower braches of a coour do o mee (see Fig. ). I hose cases he uermos ad owermos ies show where here is a aerage of 50 odes, he ex ier air 50 odes, ad so o. Agai he ceerie shows he aue ha resus i he arges umber of odes. Occasioay oe of he coours rus aog oe of boudaries (see Fig. 6). The coours do o aways exed o he righ edge of he gure due o dicuies wih oaig oi oerow (see Fig. ). Figure is a coour o of he aerage umber of odes geeraed by Bacracig wih Probig. This o show shows ha Bacracig wih Probig roides o sigica imroeme oer reiousy aayzed agorihms [7]. Figure is a coour o of he aerage umber of odes geeraed by Probe Order Bacracig. I his case he uer ad ower coours oi o form horseshoe shaed cures. Noe ha he regio of hard robems is cosideraby smaer for Probe Order Bacracig ha for Bacracig wih Probig. Exce
50 0? 0? 0?3 Fig.. Bacracig wih Probig. 0 00 50 for he sma regio, hese coours are much beer ha hose of ay oher agorihm for which such coours hae bee ubished. The imroeme is aricuary oiceabe aog he uer coour. Figure 3 shows how he aerage umber of odes for Probe Order Bacracig comares wih he aerage for seera oher saisabiiy agorihms whe, for each aue of ad each agorihm, is se o mae he aerage as arge as ossibe. The horizoa axis is he umber of causes (from o 500 for his grah). The erica axis is he aerage umber of odes from o oer 0 5 wih ic mars for each ower of 0. Of he cures ha were comued o 500, he uermos is Godberg's simied ersio of he ure iera rue [9, 0], ex is Cause Order Bacracig [3], ad owermos is bacracig combied wih Godberg's ersio of he ure iera rue [9]. Of he cures ha so shor of 500 he highes for arge 3
50 0? 0? 0?3 Fig.. Probe Order Bacracig. 0 00 50 is he Fu Pure Liera Rue [8], ex is he Fu Pure Liera Rue modied o igore auoogica causes [8], ex is Probe Order Bacracig [his aer], ad owes is Probe Order Bacracig combied wih Godberg's ersio of he Pure Liera Rue []. These cures show ha here are huge diereces i he aerage umber of odes geeraed by he arious saisabiiy agorihms. The aerage ime for he Probe Order Bacracig-ye agorihms is by far he bes amog he aayzed agorihms whe is o sma. The asymoic aaysis of Probe Order Bacracig shows ha he aerage umber of odes is o more ha, for arge ad >, whe ay of he foowig codiios hod 4
0 5 50 fure fure c b robe urerobe 0 0 Fig. 3. Wors erformace. 0 00 500 ; ()?? (? )? () ()?? (? )? ( )[() () 3?? ] 4 ( ) ()?? () () 5 ; ()
e[(? )? ] ; (3) (? )? f [()? ]e? g (4) The erm i boud () requires ha icreases more raidy ha. Boud (4) requires he assumio ha he imi of is greaer ha ad ie. By seig o miimize he righ side of (4) we see ha for a ad arge, he aerage umber of odes is o more ha whe 550(? )? () (5) Deais This seece shows ha you are readig he echica reor. The exra deais of he echica reor are coaied i secios sarig wih he word `deais' ad edig wih a diamod. Deais I boud (4) he miimum aue of he righ side (whe cosidered as a fucio of ) occurs a. The boud for arge, boud (), is much beer ha ha for ay reiousy aayzed agorihm. The bes reious resu was r? (6) for Iwama's icusio-excusio agorihm []. I he regio bewee bouds () ad (6) Probe Order Bacracig is he fases agorihm wih roe resus o is ruig ime. Agorihms ha reeaedy adus ariabe seigs o saisfy as may causes as ossibe [4] are ee faser o may robems. Those agorihms, howeer, hae dicuy wih robems which hae o souio. They hae bee dicu o correcy aayze, ad i is o cear a his ime wha heir aerage ruig ime is. For he bes reiousy aayzed agorihms here was a arge rage of where he agorihms aarey required more ha oyomia ime. (The word aarey is used because he aayses were a uer boud aayses.) The raio of he arge boudary o he sma boudary was imes ogarihmic facors. For Probe Order Bacracig, oy he ogarihmic facors are ef. I some cases ee he ogarihmic facors are goe ad he raio is cosa (i he imi of arge ). Boud () for sma resus from he fac ha he aerage umber of odes for Probe Order Bacracig is o arger ha he aerage for Sime Bacracig. Whe wih >, he raio of he uer boudary () o he ower boudary () is (? ) us ower order erms. Thus, for arge oy a ery imied rage of eads o robems wih a arge aerage ime. Perhas he regio of greaes ieres is he oe where is roorioa o. Whe is beow 335, boud (3) is beer ha (). Whe he raio of he uer boud () o rs ower boud () is ( )(?? ) us ower order erms. The raio of he uer boud o he secod ower boud (3) is us ower order erms. Preiousy, for sma he bes agorihm was a combiaio of Fraco's imied resouio agorihm [8] for sma ad Iwama's icusio-excusio agorihm [] for arge. Whe is uow, he wo agorihms ca be ru i arae ad soed as soo as a aswer is foud. Each agorihm geeraes o more ha ha odes (regardess of ) whe O( 3 ( ) 6 ). Combiig Fraco's agorihm wih Probe Order Bacracig imroes he boud o O( 3 ( ) 3 ). The echiques of Fraco's agorihm ca be combied wih Probe Order Bacracig, so here is o oger ay eed o hae wo agorihms ruig i arae. This is hefu whe desigig racica agorihms. The basic idea behid robig is od. The idea resembes ha used by Newe ad Simo i GPS [5]. Jus as heir rogram coceraes o diereces bewee is curre sae ad is goa sae, Probe Order Bacracig focuses o a se of roubesome reaios ha are sadig i he way of dig a souio. I aears ha eoe who are good a soig uzzes use reaed ideas a he ime. Fraco obsered ha wo exremey sime agorihms coud quicy soe mos robems ouside of a sma rage of [6]. His agorihm for he regio of high did a sige robe ad gae u if o souio was foud. His agorihm for he regio of ow ooed for a emy cause ad gae u if here was oe. Sice Fraco's agorihms someimes gae u, heir aerage ime was o we deed. A he ime of Fraco's wor i was aready ow ha Sime Bacracig was fas aog he ower boudary (), bu i was o cear how o obai a agorihm wih a fas aerage ime aog he uer 6
boudary (). Sime uses of robig did o seem o ead o a good aerage ime. Probe Order Bacracig was discoered whie cosiderig Fraco's resus [6] ad cosiderig he measuremes of Sosic ad Gu [4] for agorihms ha cocerae o adusig aues ui a souio is foud. Boh of hose agorihms hae dicuy wih robems ha hae o souio. Sime Bacracig imroes oer ai search by oicig whe a robem has o souio due o he resece of a emy cause. Howeer, Sime Bacracig is ufocused i is ariabe seecio. So og as a robem does o hae a emy cause, Sime Bacracig aways roceeds by seecig he ex siig ariabe from a xed orderig. The Cause Order Bacracig Agorihm [3] imroes oer Sime Bacracig by focusig o he ariabes i oe cause of he robem a a ime. This mehod of searchig has he adaage ha i erforms siig o us hose ariabes ha acuay aear i a robem. The Cause Order Bacracig Agorihm roides a framewor for he cosrucio of a robig agorihm ha has good erformace for a wide rage of robems, icudig hose wih o souio. Probe Order Bacracig, ie Cause Order Bacracig, focuses o he ariabes i oe cause a a ime. Howeer, Probe Order Bacracig imroes oer Cause Order Bacracig by oy seecig ariabes from causes which are o saised by he robig assigme. These are he causes sadig i he way of dig a souio. Our simer agorihm, Bacracig wih Probig, acs his feaure; ie he Sime Bacracig Agorihm i seecs ariabes from a xed orderig. Is oy use of robig is o es for a souio before icig a ew ariabe for siig. The aaysis of Bacracig wih Probig shows ha such a aie aicaio of robig does o ead o fas aerage ime for he regio of high or for he regio of ow. For good erformace i aears o be esseia ha a agorihm use robig boh o oice whe here is a souio ad o idicae which causes are ierferig wih soig he robem. The focused aure of Probe Order Bacracig's search ofe eads o a raid souio of a robem. Of course, seig ariabes o saisfy oe reaio someimes causes oher reaios o become usaised. I he wors case, he agorihm may eed o ry amos eery combiaio of aues for he ariabes. Thus, he aerage-case erformace of Probe Order Bacracig is exremey good, bu is wors-case erformace is o a imroeme oer reious agorihms. 5 Pracica Agorihms Probe Order Bacracig was sudied i ar because i is sime eough o aayze. I racice oe was a agorihm ha is fas wheher or o i is ossibe o aayze is ruig ime. There are seera imroemes ha woud ceary imroe Probe Order Bacracig's aerage seed ee hough i is dicu o aayze heir recise eecieess. So he search as soo as oe souio is foud. The aaysis suggess ha his woud greay imroe he seed ear he uer boudary (), bu soig a he rs souio eads o saisica deedecies ha are dicu o aayze.. Carefuy choose he robig sequece isead of us seig a ariabes o a xed aue. Various greedy aroaches where ariabes are se o saisfy as may causes as ossibe shoud be cosidered (see [3, 4]). This is aricuary imora ear he uer boudary (). 3. Probe wih seera sequeces a oe ime. See [5, 5] for a agorihm ha used wo sequeces. This is hefu aog he uer boudary. 4. Carefuy seec which ariabe o se. The aaysis suggess ha his is aricuary imora aog he ower boudary. Variabes i hard o saisfy reaios (shor causes) are more imora ha hose i easy o saisfy reaios. Variabes ha aear i os of reaios are more imora ha hose ha aear i a few reaios. Aarey whe he reaios are causes i is hefu o cosider he umber of causes coaiig a aricuar ariabe osiiey ad he umber coaiig i egaiey [5]. I aears ha ariabe seecio was a maor facor i deermiig he order of aceme of wiig eries i a rece SAT comeiio [5]. 5. Use resouio whe i does o icrease he robem size [8]. 6 Agorihm Saeme The recise form of Probe Order Bacracig ha is aayzed aog wih he rues for chargig ime is gie beow. This ersio of he agorihm is seciaized o wor o saisabiiy robems reseed i coucie orma form. The Bacracig wih Probig Agorihm is a modicaio of his agorihm. 7
A iera is osiie if i is o i he scoe of a o sig. I is egaie if i is i he scoe of a o sig. I he foowig agorihm a ariabe ca hae he aue rue, fase, or use. The osiiey-augmeed curre assigme is he curre assigme of aues o ariabes wih he use aues chaged o rue. The egaiey-augmeed curre assigme is obaied by seig he use aues o fase. The agorihm simies causes by uggig i he aues of he se ariabes, so ha (exce whe simifyig) i is cocered oy wih hose ariabes ha hae he aue use. I his agorihm he se of souios is a goba ariabe ha is iiiay he emy se. Ay souios ha are foud are added o he se. If he robem has ay souios, a eas oe souio wi be added o he se before he agorihm ermiaes. The agorihm may d more ha oe souio, bu i does o i geera d a souios. If he robem has o souio, he he agorihm wi ermiae wih a emy se of souios. Noice ha he agorihm igores auoogica causes. Probe Order Bacracig for CNF robems.. (Emy.) If he CNF robem has a emy cause, he reur (o souio), ad charge oe ime ui.. (Probe.) If here are o a osiie causes (ha is, eery cause has a eas oe egaie iera), he reur wih he egaiey-augmeed curre assigme added o he se of souios ad charge oe ime ui. 3. (Triia.) If eery cause of he CNF robem has oy osiie ieras he reur wih he osiieyaugmeed curre assigme added o he se of souios ad charge oe ui of ime. 4. (Seec.) Choose he rs cause ha is a osiie. Se esures ha here is a eas oe such cause. 5. (Siig.) Le be he umber of ariabes i he seeced cause. (Se esures ha, ad Se 4 esures ha each ariabe occurs i a mos oe iera of he cause.) For sarig a ad icreasig o a mos, geerae he h subrobem by seig he rs? ariabes of he cause so ha heir ieras are fase ad seig he h ariabe so ha is iera is rue. Use he assigme of aues o simify he robem (remoe each fase iera from is cause ad remoe from he robem each cause wih a rue iera). Ay he agorihm recursiey o soe he simied robem. If seig he rs? ieras of he seeced cause o fase resus i some cause beig emy, he so geeraig subrobems. If he oo sos wih h, he charge h ime uis. The cos i ime uis has bee deed o be he same as he umber of odes i he bacrac ree geeraed by he agorihm. The acua ruig ime of he agorihm deeds o how ceery i is imemeed, bu a good imemeaio wi resu i a ime ha is roorioa o he umber of odes muiied by a facor ha is bewee ad, where is he umber of ariabes ad is he umber of causes. The bacrac ree icudes odes for deermiig ha he seeced cause is emy. The comuaio associaed wih hose odes ca be doe quicy, so oe migh wish o hae a uer imi of o he ime uis for Se 5. This woud ead o sma, uimora chages i he aaysis. Bacracig wih Probig reaces Ses 4 ad 5 wih a se ha seecs he rs use ariabe ad geeraes wo subrobems oe where he seeced ariabe is se o fase ad oe where i is se o rue. 7 Exac Aaysis The remaider of his aer cosiss of he aayses of he Bacracig wih Probig Agorihm ad he Probe Order Bacracig Agorihm. Sice he Bacracig wih Probig Agorihm does o oer ay sigica imroeme oer oher reiousy aayzed agorihms, we resric our asymoic aaysis o he Probe Order Bacracig Agorihm. The reader who was more deaied aayses shoud refer o [3]. We ow derie recurrece equaios which gie exac aues of he aerage umber of odes geeraed by each agorihm. 7. Basic Probabiiies For aaysis of robig agorihms i is usefu o diide causes io he foowig caegories emy (o ieras), a osiie ( or more osiie ieras), auoogica (a osiie ad egaie iera for he same ariabe, ossiby wih addiioa ieras), ad oher (ay cause ha does o fa io oe of he recedig 8
caegories). Assigig aues o some ariabes ad he simifyig he cause may chage he caegory of a cause, or i may resu i he cause becomig saised. (Noe ha emy causes remai emy ad a osiie causes eer become oher causes.) The robabiiy ha a radom cause formed from ariabes is oauoogica, coais osiie ieras, ad coais egaie ieras is P (; ; ) ; ;?? The robabiiy ha a radom cause has o ieras is Noe ha ; (? )?? (7) P (; 0; 0) (? ) (8) P (; ; ) (? ) ; (9) because auoogica causes are o coued i he doube sum. Deais? P (; ; ) (? )?? (? ) (? ) (? ) (? ) ( ) (? ) P (; ; ) ; Suose you form a radom cause from ariabes ad he seec oe of he ariabes a radom. The robabiiy ha he cause has a aricuar aue of ad (imyig ha i is o a auoogy) ad ha he seeced ariabe aears i he idicaed way is osiie P (; ; ); egaie?? P (; ; ); eiher P (; ; ) (0) 7.. A-Posiie Causes The robabiiy ha a radom cause is a osiie is Deais P (; ; 0) 0 P (; ; 0) (? ) [? (? ) ] () P (; ; 0)? P (; 0; 0) (? )? (? ) Suose causes are geeraed a radom ui a a osiie cause is roduced. The robabiiy ha he a osiie cause coais ieras is P (; ; 0) A(; ) (? )?? (? ) () P P (; ; 0) If a radom ariabe is assiged he aue rue, he a a osiie cause wi eiher become saised or remai a osiie. The robabiiy ha he cause wi become saised is A(; ) 9? (? ) (3)
Deais A(; ) 0 (? )?? (? )?? (? )?? (? ) 0? (? )??(?) The robabiiy ha he cause has egh ad ha i remais a osiie is? A(; )? (? )?? (? ) P (? ; ; 0) (? )? [? (? ) ] (4) If a radom ariabe is assiged he aue fase, he a a osiie cause wi eiher become emy or remai a osiie. The robabiiy ha he resuig cause wi be emy is (? )? A(; )? (? ) (5) The robabiiy ha he resuig cause wi be a osiie wih egh is Deais A(; )? A(; ) P (? ; ; 0) (? )? [? (? ) ] (6) A(; )? A(; ) (? )???? (? )? (? )?? [? ]? (? )? (? )??? (? ) The aerage egh of a radom a-osiie cause is A(; ) (? )?? (? ) P (? ; ; 0) (? )? [? (? ) ]? (? ) (7) Deais Muiy eq. (3) by. 7.. Oher Causes The robabiiy ha a radom cause is a oher cause is 0 P (; ; ) (? ) [( )? ] (8) Deais P (; ; ) (? )? (? ) (? ) [( )? ] 0
Suose causes are geeraed a radom ui a oher cause is roduced. The robabiiy ha he oher cause coais osiie ieras ad egaie ieras is (; ; ) M(; ; ) P P (; ; ) (? )?? ; ;?? ( ) (9)? P If a radom ariabe is assiged he aue rue, a oher cause may become emy, become a osiie, become saised, or remai a oher cause. The robabiiy ha a oher cause wi become a emy cause is (? )? M(; 0; ) ( )? (0) The robabiiy ha i wi become a a osiie cause wih egh is? M(; ; ) (? )?? ( )? P (? ; ; 0) (? )? [( )? ] () The robabiiy ha i wi be saised is M(; ; ) [( )?? ] ( )? () Deais M(; ; )! (? )??!!(?? )! ( )? (? )! (? )?? (? )!!(?? )! ( )???? (? )??? ( )?? ( )? [( )?? ] ( )? The robabiiy ha i wi become a oher cause wih osiie ieras ad egaie ieras is M(; ; )?? M(; ; ) P (? ; ; ) (? )? [( )? ] (3) Deais M(; ; )?? M(; ; )!!( )!(??? )!?? (? )!!!(??? )! (? )??? ( )?! (? )??!!(?? )! ( )? (? )??? ( ) [? ]? (? )! (? )???!!(??? )! ( )? P (? ; ; ) (? )? [( )? ]
If a radom ariabe is assiged he aue fase, a oher cause may become become saised or remai a oher cause. The robabiiy ha a oher cause wi be saised is ( )? M(; ; ) ( )? (4) Deais M(; ; )! (? )??!!(?? )! ( )? (? )! (? )??!(? )!(?? )! ( )??? (? )??? ( )??? ( )? [( )?? ] ( )? The robabiiy ha i wi become a oher cause wih osiie ieras ad egaie ieras is Deais M(; ; )?? M(; ; ) M(; ; )?? M(; ; )! ( )!!(??? )!?? (? )!!!(??? )! P (? ; ; ) (? )? [( )? ] (5) (? )??? ( )?! (? )??!!(?? )! ( )? (? )??? ( ) [? ]? (? )! (? )??? P (? ; ; )!!(??? )! ( )? (? )? [( )? ] Eqs. (4, 6,, 3, ad 5) show ha i a cases where a oemy cause resus from seig a ariabe associaed wih a radom oemy cause geeraed from ariabes, he resuig cause has he same reaie disribuio as radom causes geeraed from? ariabes. Thus, i is ossibe o base a aaysis o he umber of a osiie causes, he umber of oher causes, ad he umber of ariabes wihou haig o coed wih saisica deedecies. 7..3 Toa Number of Nodes Eq. (8) imies ha a radom redicae wih causes coais a emy cause (ad is herefore soed wih oe ode) wih robabiiy? [? (? ) ] (6) Eqs. (8,, 8) imy ha he robabiiy ha a radom redicae coais zero emy causes, m a osiie causes, oher causes, ad? m? auoogica causes is m; ;? m? (? ) (m) [? (? ) ] m [( )? ] [? (? ) ( ) ]?m? (7)
Deais The robabiiy ha a cause is emy is (? ). The robabiiy ha a cause is a osiie is (? ) [? (? ) ]. The robabiiy ha a cause is a oher cause is (? ) [( )? ]. The robabiiy ha a cause is oe of he aboe, ad hece auoogica, is? (? )? (? ) [? (? ) ]? (? ) [( )? ]? (? ) ( ) If we e T (; m; ) be he aerage ime required o soe a radom robem wih ariabes, m a osiie causes, oher causes, ad o emy causes he by summig a of he cases we see ha he execed umber of odes is? [? (? ) ] m; m; ;? m? (? ) (m) [? (? ) ] m [( )? ] [? (? ) ( ) ]?m? T (; m; ) (8) This formua aies o boh Probe Order Bacracig ad Bacracig wih Probig so og as he corresodig deiio for T is used. 7..4 Heurisic Aaysis Before coiuig wih he exac aaysis of he wo agorihms we wi gie a brief heurisic aaysis for he aerage ime used by Probe Order Bacracig. Igore he fac ha seig ariabes has a eec o causes oher ha he seeced cause. I aricuar, igore he fac ha he oseeced causes ca become emy or saised ad igore he fac ha oce he ariabes of oe cause are se, here coud be fewer ariabes waiig o be se i he remaiig causes. Uder his radica assumio, he umber of subrobems roduced by siig o he ariabes of he seeced cause is he same as he egh of he seeced cause. Eq. (7) imies T (; m; ) is gie by m? (? )?,? (? )?? (9) Deais If each of m causes coais w ariabes, he oa umber of o-roo odes i he imied search ree saises he recurrece N(m) wn(m? ) w `w' is he umber of odes arisig from seig each ariabe i he cause rue ad fase as secied i he Probe Order Bacracig Agorihm. There are w subrobems roduced by siig. Each subrobem has m? causes. The souio o his recurrece is w m?? w? Add for he roo ode ad use eq. (7) for w o obai eq. (9). Puggig io eq. (8) ad summig oer m ad gies a aerage umber of odes of f[? (? ) (? )(? ) ]? [? (? ) ] g (30) Deais Dee w? (? ) 3
The? [? (? ) ] (? ) (m) [? (? ) ] m [( )? ] m; ;? m? m; w [? (? ) ( ) ]?m? m?? w?? [? (? ) ] w? (? ) m [? (? ) ] m [? (? ) ]?m w m? m m w w? [? (? ) ] [(? ) w? w? ] m [? (? ) ]?m m m w w? [? (? ) ] [ (? )(? ) ] w? w?? [? (? ) ] [ (? )(? ) ]? (? )? (? ) The coours for his fucio are gie i Fig. 4. Carefuy comarig wih he rue aswer (Fig. ) we see ha he heurisic aaysis gies eiher a uer boud or a ower boud. For high he aues are oo sma (because chages i cause yes were egeced) ad for ow he aues are oo ow (because he fac ha oseeced causes ca become emy was egeced). This ye of aaysis ca be usefu durig iiia agorihm desig because i is sime o do, ad i ofe gies roughy he righ aswer. Oe mus beware, howeer, ha o some robems a simiar aroach migh gie a radicay wrog aswer. 7..5 Trasiio Probabiiies Suose a redicae is roduced by reeaedy geeraig radom causes from ariabes. Suose he resuig redicae coais m a osiie causes, oher causes, ad o emy causes. Le G(; ) be he robabiiy ha seig a radom ariabe o rue resus i he redicae haig oe or more emy causes. Whe a ariabe is se o rue, oher causes become emy wih he robabiiy gie i eq. (0) whie a osiie causes do o become emy. Therefore, (? )? G(; )?? ( ) (3)? Le F (; m) be he robabiiy ha seig a radom ariabe o fase resus i a redicae wih oe or more emy causes. Eq. (5) imies m (? )? F (; m)?? (3)? (? ) Le D(; i; ; m; ) be he robabiiy ha seig i radom ariabes o fase resus i o causes becomig emy ad oher causes becomig saised. If i 0, ohig haes, so For i, eqs. (5, 4) imy D(; ; ; m; )? D(; 0; ; m; ) 0 (33) m (? )? ( )??? (? ) ( )?? ( )? ( ) (34)? For i >, some oher causes (x) mus be saised whe he rs i? ariabes are se ad he he res (? x) mus be saised whe he as ariabe is se, so D ca be cacuaed from D(; i; ; m; ) x D(; i? ; x; m; )D(? i ; ;? x; m;? x) (35) 4
50 0? 0? 0?3 0 00 50 Fig. 4. Heurisic Aaysis of Probe Order Bacracig. Sice he m idex is cosa i his recurrece ad Y 0<i? (? )??? (? )?i? (? )?? (? ) ; (36) we hae? (? )?i m D(; i; ; m; ) D(; i; ; ); (37)? (? ) 5
where ad D(; ; ; ) D(; i; ; ) x ( )? ( )??? ( )? ; (38) ( )? D(; i? ; x; )D(? i ; ;? x;? x) (39) By examiig sma cases, oe ds D(; i; ; ) [( )? ( )?i ] [( )?i? ]? [( )? ] ; (40) which ca be roed by iducio. Deais For i eq. (40) reads [( )? ( )? ] [( )?? ]? D(; ; ; ) [( )? ] ; which simies o eq. (38). Suose eq. (39) is rue for i ad for i i?. The eq. (39) reads [( ) D(; i ; ; )? ( )?i ] x [( )?i? ]?x x [( )? ] x? x ( )?i?x? ( )?i?? x ( )?i? ( )?i? [( )?i? ]? [( )? ]? [( )? ( )?i ] x [( )?i ]?x ; x x which simies o eq. (40). Le E(; ; ; ; m; ) be he robabiiy ha seig oe radom ariabe o rue resus i o causes becomig emy, oher causes becomig a osiie causes, oher causes becomig saised, ad a osiie causes becomig saised. (Noice ha o oher chages of cause caegory ca occur.) Eqs. (3, 0,, ) imy m E(; ; ; ; m; )? ; ;?? ( )? [( )???? ] ( )?? (4) ( )? ( )? Deais From eq. () he robabiiy ha a oher cause wi become a osiie is? (? )?? [? (? )? ] ( )? ( )? m E(; ; ; ; m; )? ; ;?? m [( )?? ] ( )? (? )??? (? )? (? ) ( )?? [? (? )? ] ( )? ; ;?? [( )?? ]? ( )?? (? ) 6? (? ) m? [? (? )? ] m? [? (? )? ]? (? ) ( )?? [(?? )?? ] ( )? m? [? (? )? ]?? (? ) ( )??? ( )? ( )?
7. Bacracig wih Probig Le T (; m; ) be he aerage umber of odes for a robem soed by he Bacracig wih Probig Agorihm ha has ariabes, m a osiie causes, ad oher causes, ad o emy causes. If m or is zero, he he agorihm sos immediaey, so here is oy oe ode. Thus, T (; 0; ) T (; m; 0) (4) If boh m ad are bigger ha zero, he here are some odes for he subree ha resus whe he seeced ariabe is se o fase, some odes for he subree ha resus whe he seeced ariabe is se o rue, ad oe ode for he roo of he search ree. Whe he ariabe is se o fase, wih robabiiy F (; m) a emy cause is roduced (ad herefore here is oe ode i he subree). Wih robabiiy D(; ; ; m; ), o emy causes are roduced ad of he oher causes become saised, resuig i T (? ; m;? ) as he execed umber of odes i he subree. Whe he ariabe is se o rue, wih robabiiy G(; ) a emy cause is roduced. Wih robabiiy E(; ; ; ; m; ), o emy causes are roduced, oher causes are coered io a osiie causes, oher causes are saised, ad a osiie causes are saised, resuig i T (? ; m? ;?? ) as he execed umber of odes i he subree. For Bacracig wih Probig, addig u he odes from a he cases gies T (; m; ) F (; m) G(; ) D(; ; ; m; )T (? ; m;? ) ;; E(; ; ; ; m; )T (? ; m? ;?? ) (43) 7.3 Probe Order Bacracig Probe Order Bacracig seecs a cause ad he ses he ariabes ha occur i he cause. If he seeced cause has h ariabes, he here is a roo, a ode from seig he rs ariabe o fase, a oeia ode from seig he rs wo ariabes o fase, ad so o. This gies a roo us u o h addiioa odes. I addiio, here is a subree for seig he rs ariabe o rue, oeiay a subree for seig he rs ariabe o fase ad he secod o rue, ad so o. Whe seig he rs few ariabes, some of he oher causes may eauae o fase. Aso, seig he rs few ariabes may resu i he umber of oher causes droig o zero. Eiher of hese eecs may ree a oeia ode from occurrig i he ree. Dee a(; i) as he robabiiy ha he seeced cause coais i or more odes (hus oeiay coribuig a i h ode o he bacrac ree). The, from eq. () we obai a(; i) A(; ) (? )?? (? ) (44) i i Le T (; m; ) be he aerage umber of odes for a robem ha has ariabes, m a osiie causes, oher causes ad o emy causes. For Probe Order Bacracig T (; m; ) D(; i? ; x; m? ; ) G(? i ;? x) i a(; i) x< ;; E(? i ; ; ; ; m? ;? x)t (? i; m?? ;??? x) (45) The iiia is for he roo of he ree. The i idex is for hose odes ha occur as a resu of seig he rs i ariabes from he cause. The facor a(; i) gies he robabiiy ha he seeced cause has a eas i ariabes. The idex x is for he umber of oher causes ha are saised whe seig he rs i? ariabes fase. The sum does o icude x, because o subrobems are geeraed whe he umber 7
of oher causes is reduced o zero. The facor D(; i? ; x; m? ; ) is he robabiiy ha x of he oher causes become saised ad o causes become emy as a resu of seig he rs i? ariabes. The D facor muiies he sum of erms ha reae o he arious ids of odes ha ca resu whe he i h ariabe is se. The foowig he square brace is for he ode ha resus from seig he i h ariabe o fase. The G(? i ;? x) erm gies he robabiiy ha seig he i h ariabe o rue roduces a emy cause. Whe seig he i h ariabe o rue, he idex cous he umber of oher causes ha become a osiie, he idex cous he umber of oher causes ha become saised, ad he idex cous he umber of a osiie causes ha are saised (he seeced cause is o icuded i his cou). The facor E(? i ; ; ; ; m? ;? x) is he robabiiy ha seig he i h ariabe resus i he aues,, ad. The facor T (? i; m?? ;??? x) is he execed umber of odes i he subree ha resus from seig he rs i? ariabes o fase ad he i h ariabe o rue. As wih he reious aaysis, he boudary codiios are. where T (; 0; ) T (; m; 0) (46) Afer a umber of agebraic rasformaios eq. (45) ca be rewrie as T (; m; ) a(; i) Z(; i; m; ) H(; i; ; ; m; )T (? i; ; ) ; i? (? )?i m? Z(; i; m; )? (? ) ( )? ( )?i? ( )? ad ( ) H(; i; ; ; m; )?i? ( )? Deais Dee The eqs. (37, 45) imy T (; m; ) m? ( )? ( )?i? ( )?? ( )?? (? )?i? m ( )? m???? (? )?i m??? [? (? )?i ]? (? )? (? ) ( )? ( )? ( )?i? 0 ( )? m? i a(; i) x< m????? (? )?i?? (? ) Y (; i)? (? )?i? (? ) m??? (? ) [Y (; i? )] m? D(; i? ; x; ) G(? i ; ;? x) ; (47) ; (48)?m E(? i ; ; ; ; m? ;? x)t (? i; m?? ;??? x) 8 (49)
The ey idea i he deriaio is o rs chage idices wih 0 m?? ad 0??? x. Deais Thus, 0? m, m?? x? 0? 0?, ad T (; m; ) i a(; i) x< [Y (; i? )] m? D(; i? ; x; ) G(? i ; ;? x) E(? i ;? m ; m?? x??? ; ; m? ;? x)t (? i; ; ) The oe comues he sum oer x ad ames he coecies o obai eq. (47). Deais Dee Z(; i; m; ) x<[y (; i? )] m? D(; i? ; x; )[ G(? i ; ;? x)] [( )? ( )?i ] x [( )?i? ]?x x Z(; i; m; ) [Y (; i? )] m? ( [( )? ]? x ) x "?x # (? )?i?? ( )?i? ( ) [Y (; i? )]? ( )?i ( )?? (? )?i m?? ( )?? ( )? Dee H(; i; ; ; m; ) x<[y (; i? )] m? D(; i? ; x; ) E(? i ;? m ; m?? x??? ; ; m? ;? x) [( ) [Y (; i? )]? ( )?i ] x [( )?i? ]?x m? (? x [( )? ] x ) x m?? x? m ; m?? x??? ;? m?? [? (? )?i ]? (? )?i ( )?i? m??x???? [( )?i? ] ( )?i??m ( )?i ( )?i?? (? )?i ; 9
H(; i; ; ; m; ) [Y (; i? )] m? x m?? m [? (? )?i ] ( )?i??? (? )?i ( )?i ( )?i? [( )? ( )?i ] x [( )?i? ]?x (? [( )? ] x ) m??? m???? m???? (? )?i?m [( )?i? ] ( )?i? x m??x??? ( )?i? [Y (; i? )] m? ( )? m? m???? m? (? )?i m?? [? (? )?i?m ] ( )?i??? (? )?i ( )?i? ( )?i? m???? ( )? ( )?i x x x ( )?i? [( )?i m??x???? ] (? x) ( )?i? ( )?i? [Y (; i? )] m? ( )? m?? m? m??? m?? [? (? )?i ]? (? )?i ( )?i? ( )? ( )?i m????? ( )?i? m???? ( )? ( )?i? ( )?i? Cace facors o obai eq. (49).? (? )?i?m ( )?i? ( )?i? [( )?i m????? ] ( )?i? 7.4 Aerae Recurrece Time O( 4 ) is eeded o cacuae he aerage umber of odes for Probe Order Bacracig from eqs. (8, 47). The foowig equaios cacuae he same resus i ime O( 4 3 ). T (; m; ; i) [? (? ) i ](? )?i [(? )? (? ) ] m? [(? )? (? ) ] [? (? ) i ](? )?i (? ) m? (? )?? J(? ; m? ;? ; )T (? ; ; ;? ) (? ) m? [(? )? ] T (? ; m;? ; f); (50) 0<f<i 0
T (0; m; ; i) T (; 0; ; i) T (; m; 0; i) T (; m; ; 0) 0 (5) The facor J(; m; ; ) may be cacuaed from he recurrece J(; m; ; ) [(? )? (? ) ]J(; m;? ; ) J(; m;? ;? ) m J(; m; 0; ) (? ) [(? ) ] m? J(; m; ; 0) [(? ) ] m [(? )? (? ) ] (5) The execed umber of odes for a robem wih ariabes ad causes soed by Probe Order Bacracig is (? ) [? (? ) ]?m? T (; m; ; ) (53) m; ;? m? m I order o coec his ersio of he aaysis o he aaysis exressed i eqs. (8, 47) we dee?(; m; ; i) [? (? ) i ](? )?i [(? )? (? ) ] m? [(? )? (? ) ] (54) The we may reae T from eq. (50) o T from eq. (47) ia T (; m; ; )?(; m; ; )[T (; m; )? ] (55) T (; m; ; i)?(; m; ; i) is he execed umber of o-roo odes i he bacrac ree geeraed by a robem wih ariabes, oher causes, m a osiie causes, ad o emy causes such ha he rs a osiie cause may o coai ay of he rs? i ariabes. The rs hree arameers hae he same meaig as he corresodig arameers from eq. (47). The fourh arameer, i, ees rac of he eec seig ieras fase has i shoreig he rs a osiie cause durig he course of he agorihm. The facor?(; m; ; i) is chose so ha o diisios are eeded o eauae eqs. (50{53). Ahough eq. (50) uses four idices, i is a fu-hisory recurrece i oy hree idices m,, ad i. Hece T (; m; ; i) may be cacuaed i sace O( ). Deais Eqs. (50{53) are a aaysis of he same agorihm as eqs. (8, 47). Howeer, i deriig eqs. (50{53) i is ceares o resae he Probe Order Agorihm i a equiae bu more exiciy recursie form. Begi by esabishig a caoica orderig of he ieras ad causes i he redicae. Remoe a auoogica causes. The. (Tes Emy Predicae) If he redicae is emy, reur he souio.. (Tes Emy Cause) If ay cause is emy, reur wih o souio. 3. (Probe Negaiey-Augmeed Souio) If a remaiig causes hae a egaie iera, reur he egaiey-augmeed souio. (Tha is, chec for m 0.) 4. (Paria Probe Posiiey-Augmeed Souio) If o remaiig causes hae a egaie iera, reur he osiiey-augmeed souio. (Tha is, chec for 0.) 5. (Siig Cause-Order Recursio) Cosider he rs osiie iera (accordig o he caoica orderig of he ieras) aearig i he rs a osiie cause a. (Asser he Liera.) Se he iera rue. Simify he redicae. Charge oe ime ui. Recur wih he simied redicae. b. (Negae he Liera.) Se he iera fase. Simify he redicae. Charge oe ime ui. Recur wih he simied redicae. Cocude he agorihm by chargig oe ime ui for he roo. We cou he odes i he bacrac ree by recursiey couig he odes iroduced by each se of he agorihm. We mus ee rac of how may a osiie causes ad how may oher causes remai a each sage of he agorihm. I addiio we mus ee rac of ees ha aec he egh of he rs a osiie cause. Le T (; m; ; i) be he execed umber of odes (excusie of he roo) i he bacrac ree geeraed by a ca o Probe Order Bacracig o a redicae wih ariabes, m a osiie causes, oher
causes, ad o emy causes, i which he rs a osiie cause is draw from he as i ariabes i he caoica isig of ariabes. We sha see ha T (; m; ; i) is reaed o T (; m; ; i) from eq. (50) ia T (; m; ; i)?(; m; ; i)t (; m; ; i). Le Q(; ) be he execed umber of odes i he bacrac ree geeraed by a ca o Probe Order Bacracig o a redicae wih causes ad ariabes. The from eq. (8) we hae m; ;? m? [(? )? (? ) ] m Q(; ) m [(? )? (? ) ] [? (? ) ]?m? T (; m; ; ) I degeerae cases he agorihm does o se ay ariabes. Hece he aura boudary codiios for T (; m; ; i) are T (0; m; ; i) T (; 0; ; i) T (; m; 0; i) T (; m; ; 0) 0 Nodes may be iroduced i oe of wo ways he agorihm Assers he Liera, or he agorihm Negaes he Liera. Le us deoe he execed umber of odes iroduced by hese braches as A(; m; ; i) ad N(; m; ; i), reseciey. Wrie T (; m; ; i) A(; m; ; i) N(; m; ; i) We ow deermie he equaios for A(; m; ; i) ad N(; m; ; i). Deais 7.4. Deriaio of A Suose he rs iera i he rs a osiie cause is assered. Oe ime ui is charged ad he redicae is simied. Probe Order Bacracig is caed recursiey o he simied redicae. Usig eq. (4), he oa umber of odes iroduced by seig he iera rue is gie by A(; m; ; i) E(; ; ; ; m? ; )T (? ; m?? ;?? ;? ) Noice ha A(; m; ; i) is ideede of i. I is usefu o dee A(; m; ) A(; m ; ; i) (oice he shif i he m idex). Deais 7.4. Deriaio of N Suose he rs iera i he rs a osiie cause is se fase. Oe ime ui is charged ad he redicae is simied. Probe Order Bacracig is caed recursiey o he simied redicae. The rs a osiie cause is draw from a ouaio of i ariabes (he as i ariabes i he caoica isig of ariabes). The robabiiy ha a radom cause draw from i ariabes coais o ariabe aearig egaiey is (? ) i. The robabiiy ha oe of he rs i? f? ariabes occurs osiiey i he cause is (? ) i?f?. The robabiiy ha his cause coais he osiie form of he (i? f) h ariabe is. The robabiiy ha a eas oe of he remaiig f ariabes occurs osiiey is? (? ) f. Eq. () gies he robabiiy ha a radom cause is a osiie. Combiig a hese facors, we obai he robabiiy, gie a radom a-osiie cause i i ariabes, ha he rs iera o aear i his cause wi be he osiie form of he (i? f) h ariabe i he caoica isig of he i ariabes, ad ha afer seig he (i? f) h iera fase he cause is o emy (? ) i (? ) i?f? [? (? ) f ] (? )i?f?? (? ) i? (? ) i? (? ) i? (? ) i Saed aoher way, his is he robabiiy ha seig he rs iera fase i he rs a osiie cause wi resu i a subrobem i which he rs a osiie cause is draw from a ouaio of f ariabes. From eq. (34) ad he recedig discussio, seig he iera fase eaes a robem i? ariabes, m a osiie causes,? oher causes, ad o emy causes, i which he rs a osiie cause is draw from he as f ariabes wih robabiiy (? ) i?f?? (? ) i?? (? ) i D(; ; ; m? ; ) Summig oer a cases, he oa umber of odes iroduced by seig he iera fase is gie by N(; m; ; i) 0<f<i (? ) i?f?? (? ) i?? (? ) i D(; ; ; m? ; )T (? ; m;? ; f)
Deais 7.4.3 Fu Se of Equaios for he Aerae Recurrece We ow simify he foowig se of equaios Q(; ) 0<m< 0<?m m; ;? m? [(? )? (? ) ] m [(? )? (? ) ] [? (? ) ]?m? T (; m; ; ); T (; m; ; i) A(; m? ; ) N(; m; ; i); T (0; m; ; i) T (; 0; ; i) T (; m; 0; i) T (; m; ; 0) 0; A(; m; ) N(; m; ; i) 0<f<i E(; ; ; ; m; )T (? ; m? ;?? ;? ); (? ) i?f?? (? ) i? D(; ; ; m? ; )T? (? ) i (? ; m;? ; f) Perform he idex chages 0 m? ad 0?? i A(; m; ). A(; m; ) ;; E(;? m ; m??? ; ; m; )T (? ; ; ;? ) Pug i he deiio for E from eq. (4). m A(; m; )? m ; m??? ; ;;? Cear some fracios o yied A(; m; ) m? [? (? )? ]? (? ) ( )? ( )?? ( )? (? ) (? ) [( )? ] T (? ; ; ;? ) (? )? (? ) (? ) [? (? ) ] m ;;?m [( )?? ] ( )? m m????? (m? ) (? ) m?? [? (? )? ] [( )?? ] m?? T (? ; ; ;? ) I order o guard agais oaig oi oerow due o he facor of ( )? we hae iroduced a facor of (? ) (m) (? ) (m) io he summaio. Muiyig his hrough gies A(; m; ) (? ) m m? (? )? (? )? (m? ) (? )? (? ) m ;; (? ) m?? [(? )?? (? )? ] [(? )?? (? )? ] [(? )? ] [(? )?? (? )? ]??(m?) T (? ; ; ;? ) Coec a he erms ha deed o ad dee m J(; m; ; ) (? ) m? [(? ) ] [(? )? (? ) ]?(m?) ;? (m? ) he, afer rearragig o emhasize seed of comuaio, we may wrie m A(; m; ) (? ) m (? )? (? ) (? )? (? )? [(? )?? (? )? ] [(? )?? (? )? ] J(? ; m;? ; )T (? ; ; ;? ) 3
We aoid exiciy erformig he sum oer i he eauaio of J(; m; ; ) by usig a recurrece for J(; m; ; ). Usig a recurrece for he biomia coecie we hae m? J(; m; ; ) (? ) m? [(? ) ] [(? )? (? ) ]?(m?)? (m? ) m? (? ) m? [(? ) ] [(? )? (? ) ]?(m?) (? )? (m? ) [(? )? (? ) ] m? (? ) m? [(? ) ] [(? )? (? ) ]??(m?)? (m? ) m? (? ) m? [(? ) ] [(? )? (? ) ]??(?)(m?) (? )? (m? ) [(? )? (? ) ]J(; m;? ; ) J(; m;? ;? ) To ge he boudary codiios i eq. (5) oe ha he sum for J(; m; ; ) ca be eauaed direcy whe 0 or 0. m J(; m; 0; ) (? ) [(? ) ] m? J(; m; ; 0) [(? ) ] m [(? )? (? ) ] Now we wor o N(; m; ; i). Pug he deiio for D(; ; ; m?; ) from eq. (34) io N(; m; ; i). N(; m; ; i) 0<f<i Cearig fracios i N(; m; ; i) gies N(; m; ; i) (? ) i?f?? (? ) i?? (? ) i? m? (? )?? (? ) ( )?? ( )? ( )?? ( ) T (? ; m;? ; f)? (? ) (? ) [( )? ]? (? ) f 0<f<i (? ) f (? ) (? ) [? (? ) ] [? (? )? ] m? [( )? ] [( )?? ]? T (? ; m;? ; f) m? (? ) i?? (? ) i Agai, i order o guard agais oaig oi oerow due o he facor of ( )?, we hae iroduced a facor of (? ) (m?) (? ) (m?) io he summaio. Muiyig his hrough ad rearragig gies m? (? ) i? N(; m; ; i) (? ) m? (? )? (? ) (? )? (? )? (? ) i [(? )? ] [(? )?? (? )? ]? 0<f<i? (? ) f [(? )?? (? )? ] m? T (? ) f (? ; m;? ; f) 4
Now ug he deiios for A(; m? ; ) ad N(; m; ; i) io T (; m; ; i). m? T (; m; ; i) (? ) m? (? )? (? ) (? )? (? )? [(? )?? (? )? ] [(? )?? (? )? ] J(? ; m? ;? ; )T (? ; ; ;? ) m? (? ) i? (? ) m? (? )? (? ) (? )? (? )? (? ) i [(? )? ] [(? )?? (? )? ]? 0<f<i? (? ) f [(? )?? (? )? ] m? T (? ) f (? ; m;? ; f) To cear he res of he fracios we redee T (; m; ; i) T (; m; ; i) [? (? ) i ](? )?i [(? )? (? ) ] m? [(? )? (? ) ] T (; m; ; i)?(; m; ; i)t (; m; ; i) Maig his subsiuio yieds equaios (50) ad (53). Deais 7.5 Vericaio of he Recurreces Aside from beig carefu wih he mahemaics, we erformed measuremes o he isure he correcess of he aayses of Bacracig wih Probig ad of Probe Order Bacracig. For each agorihm ad for ad i he rage 6, 6,, we geeraed each of he SAT robems ad coued he umber of odes roduced. A robem wih i ieras has robabiiy i (? )?i. Muiyig he ode cous for each i by he robabiiy gies a oyomia i wih ieger coecies [3]. We used Mae o soe each recurrece (8, 43, 47, 50, 53) agebraicay ad eried ha he oyomias from he recurreces were ideica wih he oyomias geeraed from he corresodig ode cous. We eried ha he wo aayses of he Probe Order Bacracig Agorihm, eqs. (8, 47) ad eqs. (50, 53), rediced he same aues i wo ways. Firs, we used Mae o soe each recurrece agebraicay for 6 ad 6 ad eried ha he formuas were ideica. Secod, we used each recurrece o comue coours for 50 ad 79. The ocaios of he coours were ideica o wihi he recisio o which hey were comued. The wors erformace mached o a accuracy of 9 digis. 8 Bouds Sime uer bouds o he ruig ime for Probe Order Bacracig are ow comued. The aroach is o eimiae idices from he recurrece ui oe has a sime agebraic equaio. To eimiae a idex, we assume ha he uow fucio (T ) has a aricuar deedece o he idex beig eimiaed imes a ew uow fucio of he remaiig idices. By uggig he assumed form io he iiia recurrece (ad erformig oe or wo summaios), we obai a boudig recurrece for he ew fucio. To simify he agebra, we ow dro he erm ha sars wih 0 from he deiio of H [i eq. (49)] ad dro he rs egaie erm from he deiio of Z [i eq. (48)]. These chages ead o a ew T (; m; ), which is a uer boud o he ruig ime of he agorihm. They hae o sigica eec o he comued ruig ime whe is arge. Droig hem ow saes a o of i. I is coeie o rs shif he recurrece by usig T 0 (; m; ) T (; m; )?. From eq. (47) we obai T 0 (; m; ) H(; i; ; ; m; )[T 0 (? i; ; ) ] i a(; i) Z(; i; m; ) 5
i which ca be wrie as T 0 (; m; ) where? (? )?i m? ( ) a(; i) Z(; i; m; )?? ( )?i? (? ) ( )? i H(; i; ; ; m; )T 0 (? i; ; ) a(; i) Z 0 (; i; m; ) The boudary codiios for he shifed recurrece are ad he aerage umber of odes is m; m; ;? m? Deais Summig H oer gies ( )?i? H(; i; ; ; m; ) ( )? ; (56) H(; i; ; ; m; )T 0 (? i; ; ) ; (57)? (? ) Z 0?i m? (; i; m; )? (? ) (58) T 0 (; 0; ) T 0 (; m; 0) 0; (59) (? ) (m) [? (? ) ] m [( )? ] [? (? ) ( ) ]?m? T 0 (; m; ) Summig H oer ad gies ; H(; i; ; ; m; ) [? (? )?i ] ( )? m?? m [? (? )?i ] ( )? m?? m m???? (? )?i?? (? )?m ( )? ( )?i? m?? m [? (? )?i ] ( )? m? ( )?? (? )?i ( )?? (? ) m?? ( )?? (? )?i?? (? )?m ( ) m????? ( )? m????? (? ) m??? (? )?i m???? (? )? (? )?m ( )?? ( )?? (? )? (? )?i? m???? (? ) m??? (? )?i m? ( )?? (? )?i? (? ) ( )? 6 (60)
? (? ) Z 0?i (; i; m; ) ( )?? (? )?i? ( )? ; m?? (? )? (? )?i m? ( )?? ( )?i? (? ) ( )? which simies o eq. (58). 8. Two Idex Recurrece For ay x() dee T (; ) so ha T 0 (; m; ) x() m T (; ) for a m. (We coud icude deedece i x, bu ha does o aear o be usefu.) Dro he 0 erm from he deiio of H (eq. 49) ad rearrage he biomias. Deais To he sum i he direcio, rearrage he biomias i he deiio of H as H(; i; ; ; m; ) ( )?i? ( )? m??? m? (? )? (? )?i? ( )? ( )? ( )?i m????? ( )?? (? ) m?? [? (? )?i ]?m Combie his H wih he deiio of Z 0 (eq. 58). Use he deiio of T (; ) ad sum oer o obai T (; )? (? ) x() max?i m? a(; i) m x()[? (? ) ] i x(? i)[? (? )?i m? ] [? x(? i)] x()[? (? ) ] ( )?i? ( )? ( )? ( )?i? [? x(? i)]? x(? i)(? )?i ( )?? T (? i; ) (6) T (; 0) 0; T (; ) 0 (6) Deais Summig x H oer gies 7
x H(; i; ; ; m; ) ( ) x?i? ( )? m??? m? (? )? (? )?i? ( )? ( )?i? m?? [? (? )?i ]? (? ) ( )? ( )? m?????m ( )?i? ( )? ( )?i? (? x)? x(? )?i ( )? m? x[? (? )?i m? ] (? x)? (? ) ( )?i? ( )? ( )?? (? )?i m???? (? ) x m??? (? )? ( )? ( )?i? (? x)? x(? )?i ( )?? Eq. (57) imies ha we eed x() m T (; ) i a(; i) Z(; i; m; ) x(? i) H(; i; ; ; m; )T (? i; ) ; T (; 0) 0; T (; ) 0; where he bouds mus hod for a m of ieres. Thus, T (; ) max a(; i) Z(; i; m; ) m x() m i x(? i) H(; i; ; ; m; )T (? i; ) ; T (; 0) 0 Usig he sum of x H ad he deiio of Z, we obai eq. (6). So ha his recurrece wi be faorabe, we wish o aoid raisig quaiies ha are aboe o he m ower. Thus, we require [? (? ) ]x()? (? )?i ; (63) ad [? (? ) ]x() [? (? )?i ]x(? i) [? x(? i)] (64) So og as x() is aboe, he ay icreasig fucio of ca be chose for [? (? ) ]x(). If x() obeys he bouds (63, 64), he we may e T (; ) a(; i) x() i ( )?i? ( )? ( )? ( )?i? [? x(? i)]? x(? i)(? )?i ( )? 8? T (? i; ) (65)
Eq. (60) imies he aerage umber of odes is bouded by m; m; ;? m? (? ) (m) x() m [? (? ) ] m [( )? ] [? (? ) ( ) ]?m? T (; ) (? ) [( )? ] f? (? ) ( ) x()(? ) [? (? ) ]g? T (; ) Eq. (66) gies a good boud whe x() is se o he aerage egh of a a osiie cause, eq. (7). Figure 5 shows he bouds ha resu from his aue of x. Deais Figure 5a shows he boud whe a(; ; ) x()? (? ) ad a(; ; ) has he aue comued a he ed of Secio 8.3. Noe he diisio by x() i eq. (65). This is criica o obaiig a aayica udersadig of why Probe Order Bacracig is fas. We are free o se x() arge eough o cace ou he eec of summig oer i (which is where he growh i T (; ) comes from) so og as he facor i eq. (66) which is raised o he? ower is o aboe. This diisio by x() is reaed o he fac ha seecig a a osiie cause resus i a reducio of oe i he umber of a osiie causes (he seig of ariabes ca augme or couerac his reducio). I Bacracig wih Probig we do o hae his edecy o reduce he umber of a osiie causes by, ad hus ha agorihm is ofe much sower. 8. Oe Idex Recurrece For ay x() ad y() dee T () so ha T 0 (; m; ) x() m y() T () for a m ad. The a suiabe T () is ay fucio a eas as arge as he souio o T () x() max a(; i) m; i? (? )?i m? y() x()[? (? ) ] x(? i)[? (? )?i m? ] [? x(? i)] x()[? (? ) ] ( ) [y(? i)? ]( )?i? y(? i)? [? x(? i)]? x(? i)(? )?i y()[( )? ] Deais ; T (? i) Summig x y H oer ad gies x y H(; i; ; ; m; ) x[? (? ) y?i m? ] (? x)? (? ) ( )?i? ( )? ( )? x[? (? )?i m? ] (? x)? (? ) ( ) (y? )( )?i? y? (? x)? x(? )?i ( )? ( )? ( )?i? (? x)? x(? )?i 9? (66) (67)
50 0? 0? 0?3 Fig. 5. Two idex uer imi. 0 00 50 Usig his sum i eq. (6) gies eq. (67). Agai, we wish o aoid raisig quaiies aboe o high owers. Thus, we si hae bouds (63, 64) for x(). I addiio we hae ad y() ; (68) [y()? ][( )? ]? [y(? i)? ][( )?i? ] fx(? i)[? (? )?i ]? g (69) 30
50 0? 0? 0?3 0 00 50 Fig. 5a. Two idex uer imi, imroed x. These bouds for y are saised by [y()? ][( )? ] If x() ad y() obey he bouds, we hae T () x()? i maxf0; fx()[? (? ) ]? gg (70) a(; i)[ T (? i)] (7) 3
Eq. (66) imies he aerage umber of odes is bouded by y() (? ) [( )? ] f? (? ) ( ) x()(? ) [? (? ) ]g? T () f? (? ) ( ) y()(? ) [( )? ] x()(? ) [? (? ) ]g T () (7) If he aue of y() is se by eq. (70), he he umber of odes is bouded by (? ) x()[? (? ) ]?? maxf0; (x()[? (? ) ]? )g T () (73) Eq. (73) gies a good boud whe x() is se o he aerage egh of a a osiie cause, eq. (7). Figure 6 shows he bouds ha resu from his aue of x. Deais Figure 6a shows he boud whe x is gie a imroed aue ha is discussed i Secio 8.3. If oe igores he requireme ha y() saisfy bouds (68, 69) ad us ses x() o he aerage cause size ad y(), oe obais a resu ha is esseiay he same as ha gie by he heurisic aaysis, eq. (30). 8.3 Zero Idices Eq. (7) has oy oe idex, bu i is si raher comex due o he summaio o he righ side. Therefore, we wi agai eimiae a idex from he recurrece. Assume T () is o more ha T for <. (This assumio does o ead o much error whe T is sma; if oe wishes a good aroximaio whe T is arge, oe shoud cosider T () T z ad seec he bes aue for z.) We obai T ( ) T x() i A good choice for x() is oe ha caces he eec of he summaio. For we hae? x() maxfx()[? (? ) ]? ; 0g a(; i) (74)? (? ) ; (75)? (? ) (? )? (? )?? ; (76) (he ess ha or equa comes from he fac ha may be a oieger). Thus, eqs. (73, 76) imy ha he umber of odes is bouded by T ( ) T; (77) which is saised for a if we ae Deais From he deiio of a(; i), eq. (44), (? )? a(; i)? (? ) i i i i? (? )??? (? )? (? ) T () (78) 3 (? )?? (? )