A Fast Algorithm for Computing the Deceptive Degree of an Objective Function

Transcription

1 IJCSNS Iteratoal Joural of Computer See ad Networ Seurty, VOL6 No3B, Marh 6 A Fast Algorthm for Computg the Deeptve Degree of a Objetve Futo LI Yu-qag Eletro Tehque Isttute, Zhegzhou Iformato Egeerg Uversty, 454 Zhegzhou, Cha Abstrat I ths paper we preset a fast algorthm for omputg the deeptve degree of a objetve futo We dsuss theoretal foudatos of the fast algorthm ad how to get the polyomal represetato of a futo quly uder the odto of that the futo value of every put s ow We prove a fast deso theorem of whether a moomal has deepto about a varable t, whh maes omputg the deeptve degree of a futo easer I the fal, we desrbes the fast algorthm ad aalyses ts omplety Itroduto Se Goldberg trodued the oto of deepto geet algorthms GAs), deepto has ome to be wdely regarded as a etral feature the desg of problems that are dffult for GAs Though a umber of dfferet deftos of deepto as well as types of deepto have bee proposed the GA lteratureeg, see [,3,4,5,6]), the relatoshp betwee a deeptve problem ad a dffult problem s stll ot easy to be eplaed Some outereamples [7,8,9,] show that deepto s ether eessary or suffet to mae a problem dffult for a GA For there s o geerally aepted defto of deepto, how to defe deepto s stll a top that deservers sruty I [] we preseted a ovel quattatve measure metr for the "degree of deepto" of a problem We preseted a ew defto for the deeptve degree of a futo We vestgated the relatoshp betwee the best soluto ad the moomal oeffets of a futo, ad we gave theorems ad epermets that showed the usefuless of the ew defto For t s a omple wor to omputg the deeptve degree of a futo aordg to the ew defto [], we have to osder whether there s a fast algorthm for omputg the deeptve degree of a futo I ths paper we wll preset a fast algorthm for omputg the deeptve degree of a futo whh was defed [] The remader of the paper s orgazed as follows: Seto revews the ew defto ad the ma results [] The et two setos show theoretal foudatos of the fast algorthm Seto 3 dsusses how to get the polyomal represetato of a futo quly uder the odto of that the futo value of every put s ow Seto 4 proves a deso theorem of whether a moomal has deepto about a varable t, whh maes omputg the deeptve degree of a futo easer Seto 5 desrbes the fast algorthm ad aalyses the omplety of the algorthm Seto 6 summarzes the paper The ew defto ad the ma results [] I ths seto we wll revew the ew defto for the deeptve degree of a futo [] Seto trodues the otato to smplfy the epresso ths paper Seto gves the defto ad propertes of a rtal value whh plays a mportat role the ew defto of deepto Seto 3 desrbes the ew defto whh was preseted [] Some results [] whh may be useful for the fast algorthm s gve lemma form Notato Wthout loss of geeralty, we osder Ω = {,} as the searh spae for GAs ad a pseudo-boolea futo f : {,} R as a ftess futo The goal s to fd a best strg or soluto) whh s orrelatve to the mamum of the futo f Beause the ew defto of deepto was preseted aordg to the relato betwee the best strg ad moomal oeffets of a gve futo, we must osder the futo polyomal form Every pseudo-

2 IJCSNS Iteratoal Joural of Computer See ad Networ Seurty, VOL6 No3B, Marh 6 boolea futo a be epressed polyomal form By oveto, f wll deote a pseudo-boolea futo polyomal form: f ) = a + a + L+ a + La L + L+ a L L L where a L s alled the oeffet of the moomal L, s alled the degree of the moomal L, ad the mamum of the degrees of all moomals s alled the polyomal degree of f f, ) f a L wll deote f wthout the moomal L = L K wll always represet a subset of = θ for all K ad θ {, } For eample, {,, L, }, ad f [, θ ); K] wll deote f where let f ) = , the f,,3) = + + 3, f [,),,)] = = + 3 A shema has bee wdely studed the feld of GAs, ad s also a mportat oept the ew defto of deepto A shema orrespods to a subset of the searh spae Ω = {,}, or more presely a hyper-plae of Ω A addtoal symbol "*" represetg a wld ard "" or "") s used to represet a shema For eample, f =4,the strgs ad are the two elemets of the shema S=* No-* postos a shema are alled defg postos Defg postos ad ther orrespodg values a be used to get aother represetato of a shema For eample, S[,),3,),4,)] s also used to represet the shema S=* The latter represetato of a shema s ofte used ths paper Defto ad propertes of a rtal value A rtal value plays a mportat role the ew defto of deepto Now we gves the oept ad propertes of a rtal value of a moomal wth respet to a varable Defto [] For a gve futo f, λ s alled a rtal value of the moomal wth respet to the varable f for arbtrary ε >, all the best strgs of f, ) + λ + ε ) are luded the shema S [,)] ad oe of the best strgs of f, ) + λ ε ) s luded the shema S [,)], whh meas that all the mamum of f, ) + λ + ε ) satsfy = ad oe of the mamum of f, ) + λ ε ) satsfes = Remar If there ests a rtal value of the moomal wth respet to the varable f, the the rtal value s uque If multple rtal values of the same moomal wth respet to all varables est, f est, they are equal If oe of the best strgs of f s luded the shema S [,)], the there ests a rtal value of eah moomal wth respet to the varable I other words, If there does t est a rtal value of the moomal wth respet to the varable, the all the best strgs of f must be luded the shema S[,)] Lemma [] If a rtal value of the moomal wth respet to the varable of f s, the f L f L ma{,, )[,)]} = ma{,, )[, )]} If there does t est a rtal value of the moomal wth respet to the varable f, we say the rtal value s where deotes a egatve fte umber The followg result a be proved Lemma [] If the oeffet of the moomal s greater tha rtal values of the moomal wth respet to all varables, the the best strg must be luded S[,),,), L,,)] If the oeffet of the moomal s less tha oe of rtal values of the moomal wth respet to all varables, the the best strg must t be luded S,),,), L,,)] [

3 IJCSNS Iteratoal Joural of Computer See ad Networ Seurty, VOL6 No3B, Marh 6 3 The ew defto of deepto I the followg, we dsussed whether a moomal has deepto about a varable aordg to the relato betwee a moomal oeffet ad a rtal value Defto [] If a varable a moomal satsfes oe of the followg three odtos, we say that the moomal has deepto about the varable ) If a rtal value of the moomal wth respet to the varable s ; ) If a rtal value of the moomal wth respet to the varable s postve ad less tha the moomal oeffet; 3) If a rtal value of the moomal wth respet to the varable s egatve ad greater tha the moomal oeffet Obvously these three odtos otradt eah other ad at most oe a be satsfed Lemma 3 [] Let f have oly oe best strg, the the moomal does t have deepto about every varable t f ad oly f the best strg of f, ) s the same as the best strg of f Remar: Lemma ad 3 are help to uderstad the ew defto of deepto For a moomal, dfferet bts betwee the best strg of f ad the best strg of f, ) a be used to dept the GA dffulty If t s easy for a GA to fd the best strg of f, t may be ot easy for the GA to fd the best strg of f, ) oly beause of these dfferet bts, ad ve vsa I fat, for a moomal, f the best strg of f s dfferet from the best strg of f, ), oe of the two best strgs must be luded the shema S[,),,), L,,)] Whe we assume that the best strg of f s luded the shema S[,),,), L,,)], the the umber of varables for whh the moomal has deepto a reflet the dffulty of a GA evolvg to the shema S[,),,), L,,)] Furthermore, f the best strg of f s, we a reflet the evaluato of the GA dffulty of f by the umber of varables about whh the moomal has deepto For smplty, we oly dsussed the deeptve degree of a futo that has oly oe best strg Defto 3 [] Let the best strg of a futo be, for a moomal whose oeffet s t equal to, the umber of varables about whh the moomal has deepto s alled the deeptve degree of the moomal The mamum of deeptve degrees of moomals of the futo s alled the deeptve degree of the futo Let deote the bary XOR operato o bts ad the btwse XOR operato o strgswhe the best strg of a futo s t L, the deeptve degree of the futo a be omputed as follows Defto 4 [] Let the best strg of f be, ad g,,, ) = L f,,, ) L, the we defe the deeptve degree of g to be equal to the deeptve degree of f From Defto 3 ad 4, the deeptve degrees of every futo whh has oly oe best strg a be omputed, the deeptve degrees of futos rage from to Lemma 4 [] For a arbtrary ostat C degree of f ) {,}, the deeptve degree of C) f s equal to the deeptve Lemma 5 [] The deeptve degree of a futo s t greater tha the polyomal degree of the futo

4 IJCSNS Iteratoal Joural of Computer See ad Networ Seurty, VOL6 No3B, Marh How to get the polyomal represetato of a futo quly I order to get the fast algorthm for omputg the deeptve degree of a futo, we must resolve two problems whh are theoretal foudatos of the fast algorthm, oe s how to get the polyomal represetato of a pseudoboolea futo quly, aother s how to omputg the deeptve degree of a moomal quly I the followg, we wll dsuss how to resolve the frst problem 3 A theory for gettg the polyomal represetato If we ow the polyomal represetato of a pseudo-boolea futo, we a get the value of the futo for every strg put) easly But f we ow the value of a pseudo-boolea futo for every strg put), how to get the polyomal represetato of the futo quly s worth to osder Theoretally we a get the polyomal represetato of a futo as follows Frst, a pseudo-boolea futo f : {,} R, should be wrtte the small tem presetato: = f ) = f ) L ) where =, ) {,}, =, ) {,},,,,, ) =,,, ) f L L L =, f,, ),, ) The, let =, =, =,, For every =, ), We substtute for ad for ) Fal, through the ormal polyomal operatos over real fled, we a get the polyomal represetato of f ths form: f ) = a + a + L+ a + La L + L+ a L L L Now we dsuss how to get the polyomal represetato of a futo by a omputer program Frst, we eed ode all oeffets the polyomal represetato of f The ode rule s as follows: The ode umber of every oeffet s a -legth bary umber f =, ) s luded the lower ordate set of a oeffet, the th bt of the orrespodg ode umber s The th bt s aother ase For stae, the orrespodg ode umber of a s,, L,) =, the orrespodg ode umber of a s,, L,) =, the orrespodg ode umber of a s,,,) L = +, the orrespodg ode umber of a L s,, L,) = Obvously,every oeffet has ad oly has a ode umber, oeffets are orrespodg to ode umbers I the followg theorem, we wll desrbe how a small tem f ) L of f has fluee o polyomal oeffets of f Theorem Let array A[] represet polyomal oeffets of f,for every {, }, K = { =,, L, ) =,, ) {,}, {,}, =,, } If K, the small tem f ) L of f has o fluee o A[] If K, the small tem f ) L w ) w ) of f maes A[] rease ) f ) Where w ), w ) are the umber of,,,,, L, ), L ) respetvely Proof Obvously, f f ) =, the small tem f ) L of f has o fluee o A[] for every {, Now we osder oly the ase of ) } f For =, =, =,, ),the f =,all moomals gotte from f ) L must lude ; f,oly half of moomals gotte from

5 4 IJCSNS Iteratoal Joural of Computer See ad Networ Seurty, VOL6 No3B, Marh 6 ) L lude ) L has fluee o A[] where = ) L has fluee o A[] where K f f f So f =, f ) L has fluee o A[] where = ; f =, or = For all =,, L ),we a get that ad f ) L has o fluee o A[] where K f ) L has fluee o A[] where K ad mae A[] hage The orrespodg moomal of A[], whh s gotte form f ) L, s the produt of a f ), w) oes, w ) w ) mus oes ad w) varables So f w ) w ) ) L maes A[] rease ) f ) 3 A proedure for gettg the polyomal represetato Now we a gve a proedure for gettg the polyomal represetato of a futo uder the odto of that the futo value of every put s ow The orete steps are as follows: Frst, let array A[] represet polyomal oeffets of f, the rage of s from to -, set A[]= for every The,for every {, }, f f ), we w ) w ) mae A[] rease ) f ) for every K I the fal, the value of A[] s the value of the orrespodg moomal oeffet Proedure desrbes the orete steps a smlar omputer program laguage Proedure beg for to - do A[] ; for to - do f f ) ) beg for to do for to do for to do w ) w ) A[] A[] + ) f ) ed ed 4 How to ompute the deeptve degree of a moomal quly If we wat to ompute the deeptve degree of a futo, we must ompute the deeptve degree of every moomal of the futo aordg to Defto 3 ad 4 the we must ompute the orrespodg rtal values aordg Defto ad, whh s a omple wor Now we wll preset a ew method to ompute the deeptve degree of a moomal 4 A ew method to ompute the deeptve degree of a moomal I the followg, we wll preset a deso theorem of whether a moomal has deepto about a varable t Furthermore, we a get the deeptve degree of a moomal by some smple omparsos L, ad for every =,, ),let partbest[ ] = ma{ f ) S[,)]} If a < L, the the moomal does t have deepto a > L ad partbest[ ] < ma{ f )} a L, the the moomal Theorem Let f have oly oe best strg whh s about every varable t If

6 IJCSNS Iteratoal Joural of Computer See ad Networ Seurty, VOL6 No3B, Marh 6 5 does t have deepto about the varable If a > L moomal has deepto about the varable ad partbest[ ] ma{ f )} a L, the the Proof If a < L, suppose the moomal has deepto about the varable If a rtal value of the moomal wth respet to the varable s, aordg to Defto, the best strg of f s luded the shema S [,)],whh otradts the odto of that the best strg of f s If a rtal value of the moomal wth respet to the varable s egatve ad greater tha the moomal a L, aordg to Defto, the best strg of f s also luded the shema S [,)],whh also otradts the odto of that the best strg of f s So If a < L, the moomal does t have deepto about every varable t If a > L ad partbest[ ] < ma{ f )} a L,suppose the moomal has deepto about the varable Aordg to Defto, a rtal value of the moomal wth respet to the varable s or postve ad less tha the oeffet a L If a rtal value of the moomal wth respet to the varable s, Lemma shows that: We a get ma{ f, )[,)]} ma{ f, )[, )]} ma{ f )} a L ma{ma{ f, = ma{ f, )[,)]} = partbest[ ] whh otradts the odto of that =, )[,)]},ma{ f, )[,)]}} partbest[ ] < ma{ f )} If a rtal value of the moomal wth respet to the varable s postve ad less tha the oeffet a get a L a L ma{ f, )[,)]} ma{ f, )[,)]}, aordg to Defto, we The ma{ f )} a ma{ma{ f ma{ f, )[,)]} L, )[,)]},ma{ f, )[,)]}} = partbest[ ] whh otradts the odto of that partbest[ ] < ma{ f )}, the the moomal a L If ad > a L partbest[ ] < ma{ f )} So If a L a L a L > does t have deepto about the varable partbest[ ] = ma{ f )}, we wll prove that a rtal value of the moomal wth respet to the varable s For arbtraryε, < ε < a L, we get ma{ f, ) + ε L = ma{ f, ) + a ) L L + ε a L } L } ad

7 6 IJCSNS Iteratoal Joural of Computer See ad Networ Seurty, VOL6 No3B, Marh 6 ma{ f )} ma{ a L ε) L ma{ )} ε) = f a L = partbest ] + ε [ = ma{ f, L, )[,)]}+ ε The the best strgs of f, ) + ε the best strg of s L,we get ma{ f, )[,)]} = partbest[ ] = ma{ f )} a L } are luded the shema S [,)] O the other had, se = ma{ f, )[,)]} ma{ f, )[, )]} ε L The oe of the best strgs of f, ) ε L s luded the shema S [ )], Aordg to Defto ad, we get that a rtal value of the moomal wth respet to the varable s ad the moomal has deepto about the varable If ad partbest[ ] > ma{ f )}, we wll prove that a rtal value of the moomal a L > a L wth respet to the varable s postve ad less tha the oeffet a L Frst, se the best strg of s L,we get ma{ f, )[,)]} > ma{ f, )[, )]} ma{ f )[,)]} > ma{ f,, )[,)]} L Let λ = ma{ f, )[,)]} ma{ f, )[,)]}) / λ = ma{ f )[,)]} ma{ f, )[,)]}) / Obvously, ma{ f, )[,)] + a ) } L λ L > ma{ f, )[,)]} > ma{ f, )[,)] + λ L } Furthermore, the futo ma{ f, )[,)]} s a ostat ad the other futo ma{ f, )[,)] + λ L } s a reasg futo about the varable λ So there must est a λ < λ λ a λ a ),, for arbtrary ε >, all the best strgs of f, ) L < L + λ + ε ) are luded the shema S [,)] ad oe of the best strgs of f, ) + λ ε ) s luded the shema S [,)] Therefore, by Defto ad, λ s a rtal value of the moomal wth respet to the varable ad the moomal has deepto about the varable Utl ow, the theorem s proved

8 IJCSNS Iteratoal Joural of Computer See ad Networ Seurty, VOL6 No3B, Marh A proedure for omputg the deeptve degree of a futo Now we a gve a proedure for omputg the deeptve degree of a futo at the odto of that all the value of A[] are ow ad the best strg of f s L The orete steps are as follows: Frst, let dd represets the deeptve degree of f, set dd= Compute partbest [] for =,, The, for every {, }, ompute the umber of varables whh the orrespodg moomal has deepto about, f the umber s larger tha dd, hage the value of dd by the umber I the fal, the value of dd s the deeptve degree of f Proedure desrbes the orete steps a smlar omputer program laguage Proedure beg dd=; for to do partbest[] ; for to - do for to do f =) ad A[]>partbest[])) do partbest[] A[]; for to - do f A[]>) do beg flag=; for to do f partbest[] A[ -]-A[]) do flag flag+; fflag>dd) dd flag; ed returdd); ed Remar: By Lemma 5, we a smplfy Proedure we a ompute deeptve degree of moomals aordg to the sequee of moomal degrees from bg to small If the moomal degree of a moomal s less tha the urret value of dd, we eed t ompute the deeptve degree of the moomal for whh has o fluee o the deeptve degree of the futo 5 A fast algorthm for omputg the deeptve degree of a futo Seto 3 ad 4 have prepared for a fast algorthm to ompute the deeptve degree of a futo whh has oly oe best strg uder the odto of that the futo value of every put s ow Now we desrbe the fast algorthm Algorthm : Iput: the futo value of every put of a futo f ) Output: the deeptve degree of the futo f ) Step : verfy whether f ) has oly best strg ad the best strg s L If f ) has more tha oe best strg, we have to termate the algorthm If the best strg of f ) s C {, }, whh C s t equal tol ad C C = L, by Defto 4 ad Lemma 4, we wll ompute the deeptve degree of f C) stead of the deeptve degree of f ) the otug part of the algorthm Step: get polyomal oeffets A[] of the orrespodg futo by usg Proedure Step3: get the deeptve degree of the orrespodg futo by usg Proedure Se the spae omplety of the algorthm s maly for storg futo values of f, polyomal oeffets A[] ad other termedate results, the algorthm eeds O ) storage O the other had, the tme omplety of the algorthm a be get as follows: Se Step eeds O ) tme, Step eeds O 3 ) tme ad Step eeds O ) tme, so the algorthm eeds O 3 ) tme

9 8 IJCSNS Iteratoal Joural of Computer See ad Networ Seurty, VOL6 No3B, Marh 6 6 Summary I [], we preseted a ew defto for the deeptve degree of a futo, but t s very omplated to ompute the deeptve degree of a futo aordg to the ew defto I ths paper we preset a fast algorthm for omputg the deeptve degree of a futo We dsuss theoretal foudatos of the fast algorthm ad how to get the polyomal represetato of a futo quly uder the odto of that the futo value of every put s ow We prove a fast deso theorem of whether a moomal has deepto about a varable t, whh maes omputg the deeptve degree of a futo easer I the fal, we desrbes the fast algorthm ad aalyses the omplety of the algorthm The fast algorthm s helpful to dsussg the usefuless of the ew defto of deepto Whether the ew defto a play a mportat role the oeto betwee the GA performae ad the struture of a gve futo deserves further osderato The relato betwee the ew defto ad the old oes should also be tae to aout for future researh Referees LI Yuqag The Deeptve Degree of the Objetve Futo Foudatos Of Geet Algorthms 5 FOGA 5), Azu- Waamatsu Cty, Japa,Jauary 5 Revsed Seleted Paper Leture Notes Computer See 3469:3-34 Sprger Verlag, 5 A D Bethe: Geet Algorthm as Futo Optmzatos Dotoral Dssertato, Uversty of Mhga Dssertato abstrats Iteratoal 49), 353B Uversty Mroflms No86 98) 3 Goldberg, D E: Smple geet algorthms ad the mmal deeptve problem I L DavsEd), Geet algorthms ad smulated aealg pp74-88) Lodo: Ptma Y Davdor: Epstass varae: a vewpot o GA-hardess I GJE Rawls, edtor, Foudatos of Geet Algorthms, pages 3-35 Morga Kaufma, 99 5 Leps, G E ad Vose, M D: Represetatoal Issues Geet Optmzato Joural of Epermetal ad Theoretal Artfal Itellgee 99):-5 6 Whtley, L D: Fudametal Prples of Deepto Geet Searh I G Rawls ED) Foudatos of Geet Algorthms Sa Mateo, CA:Morga Kaufma99) 7 J J Grefestette: Deepto osdered harmful : L Darrell WhtleyEd), FOGA-, Morga Kaufma, 993, pp J J Grefestette ad J E Baer: How Geet Algorthms Wor: A Crtal Loo at Implt Parallelsm Proeedgs of the Thrd Iteratoal Coferee o Geet Algorthms, J D Shaffer, Edtor, Sa Mateo, CA, Morga Kaufma, M Mthell, S Forrest, ad J H Hollad: The Royal Road Geet Algorthms: Ftess Ladsapes ad GA Performae Proeedgs of the Frst Europea Coferee ad Artfal Lfe, Cambrdge, MA, MIT Press/Braford Boos, 99 Stephae Forrest ad Melae Mthell: What Maes a Problem Hard for Geet Algorthm? Some Aomalous Results the Eplaato Mahe Learg, 3, pages 85-39, 993 CRReeves ad C C Wrght : Epstass geet algorthms: a epermetal desg perspetve I L J Eshelma Ed) 995)Proeedgs of the 6 th Iteratoal Coferee o Geet Algorthms, Morga Kaufma, Sa Mateo, CA,7-4 B Lebla ad E Lutto: Btwse regularty ad GA-hardess ICEC 98, May 5 Ahorage, Alasa, Chryssomalaeos ad CRStephes What Bass for Geet Dyams? AtaPhysSlov 4 4 CRStephes The reormalzato group ad the dyams of geet systems AtaPhysSlov 5:55-54,3 5 Mar Shoeauer Evolutoary Algorthms for Parameter Optmzato theory ad Prate Letures of 4 Iteratoal Worshop o Nature Ispred Computato ad Applatos Ot5-9,4 Hefe,Cha 6 CRStephes, ad AZamora EC theory: A ufed vewpot I Er CatuPaz; edtor, GECCO3 pp394-4 Berl, Germay,3