METHOD OF THE MULTIDIMENSIONAL SIEVE IN THE PRACTICAL REALIZATION OF SOME COMBINATORIAL ALGORITHMS

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1 METHOD OF THE MULTIDIMENSIONAL SIEVE IN THE PRACTICAL REALIZATION OF SOME COMBINATORIAL ALGORITHMS Kasii Yakov Yodzhev South-West Uivesity Blagoevgad Aa Geogieva Makovska South-West Uivesity Blagoevgad Abstat Soe diiulties egadig the appliatio o the well-kow sieve ethod ae osideed i the ase whe a patial (poga) ealizatio o seletig eleets havig a patiula popety aog the eleets o a set with a suiietly geat adial ube(adiality). I this pape the poble has bee esolved by usig a odiied vesio o the ethod utilizig ultidiesioal aays. As a theoetial illustatio o the ethod o the ultidiesioal sieve the poble o obtaiig a sigle epesetative o eah equivalee lass with espet to a give elatio o equivalee ad obtaiig the adiality o the espetive ato set is osideed with elevat atheatial poos. Keywods: equivalee lass ato set ultidiesioal aays. INTRODUCTION Let us oside the ollowig lass o pobles ote et i ioatis: A iite set M is give with a aily o its subsets M M I I N suh that M M kowig that o eah I a I eleet x M exists o whih x M is valid whe. Suh a eleet will be eeed to as a "haateisti epesetative o subset M M. We also suppose that o eah I the set is uique o eah o its M haateisti epesetatives x ad that it a be opletely deteied by x. I othe wods we suppose that a eetive algoith exists whih a obtai M whe a abitay haateisti epesetative is give at the iput. Fo exaple let M be the losed iteval [2s] whee s is a atual ube ad let I be the subset o all the pie ubes 2 s ; let us deote by M the set o all ubes o M whih a be divided by. The o eah I thee exists a uique haateisti epesetative o the set M ad it is siply the pie ube. It is possible that M ay otai oe tha oe haateisti epesetative. Fo exaple i is a o-tivial (aely dieet o the elatio equatio) elatio o equivalee i M ad M ae the equivalee lasses with espet to the eah eleet o M is a haateisti epesetative o a etai equivalee lass. A typial poble i opute pogaig is to obtai (at least) oe set M M whih otais a sigle haateisti epesetative o eah subset o M I. As a osequee o this poble ollows the obiatoial poble o idig the adiality N= I o the idex set I. I ase that is a elatio o equivalee the the give poble solves the poble o the adiality o the ato set M. Boolea (o Biay o (01)-atix) is a atix whose eleets ae equal to zeo o oe. Let B be the set o all boolea aties. It is well kow that (1) B 2 Let X Y B. A equivalee elatio is deied as ollows: XY i ad oly i X a be obtaied o Y by a sequetial ovig o the last ow o olu to the ist plae. The goal o this pape is to desibe a eetive algoith o idig the ube o ~ eleets o the ato set B B as well as idig a haateisti epesetative o eah equivalee lass. Hee we will desibe a algoith whih is a odiiatio o the wellkow ethod kow as the "Sieve o Eatosthees" ad whih oveoes soe

2 diiulties whih would ievitably aise with suiietly geat ad i we apply the lassial vesio. The ai diiulty to be oveoe aises o the geat ube o eleets o B with opaatively sall ad aodig to (1). I [9] a algoith is show whih utilizes theoetial gaphial ethods o idig ~ the ato set S S whee S B is a set o all peutatio aties i.e. Boolea aties havig exatly oe 1 o eah ow ad eah olu. B The equivalee lasses o by the equivalee elatio ae alled double oset ( see [4] 1.7 o [6] v. 1h 2 1.1). They ake use o substitutio goups theoy ( see[5] ) ad liea epesetatio o iite goups theoy ( see [3] 44-45). ~ The eleets o the set B B put ay ito patie i the textile tehology [2]. Fo udeied otios ad deiitios we ee to [7] [1] o [8]. STATEMENT 1. Method o the Sieve Let a ado haateisti eleet x be give o the set M M I ad let thee also be give eetive algoiths estiatig the utios (2) Next 1 (x) Next 2 (x) Next k (x) deied i M ad with the help o whih we a obtai the whole set M. By deiitio o eah i=12...k we put: 0 (3) Next i (x) = x (4) Next 1 i (x) = Next i (x) (5) Next t i (x) = Next i (Next t-1 i (x)) whe t 2 t itege Sie M is a iite set the it ollows that o eah i=12...k ad o eah x M thee exists a iiu atual ube (6) i x suh that o eah positive itege ube z ( i x (ix)+z Next i ) Next t i (x) o Next (ix)+z i (x) alls t0 (ix)+z out o the age o M o Next i (x) is ot deied i M. I x is a haateisti epesetative o the set M M the we put (1 x ) t0 (7) M ( x Next t 1 (x) 1 ) (8) M i (x) = ym i 1 ( i y) t0 t Next i ( y) o i= 23 k Sie the uio i the squae bakets o (8) begis at t=0 ad aodig to (3) it is easily see that (9) M 1( x) M 2 ( x)... M k ( x) We suppose that the utios Next i (x) i=12...k ae hose so that goig o the haateisti eleet x M ad usig the oulae (3) (9) we obtai the set M ad we have (10) M = M k The kow ethod o the sieve we a desibe usig the ollowig suaized algoith: Algoith 1 Method o the Sieve 1. I M we itodue a ode (this is always possible sie M is iite) ad we sot it aodig to this ode. Let us deote by #(x) the oseutive ube o x M aodig to this ode. 2. We delae a (oe-diesioal) Boolea aay H with M eleets. The eleets o H will be idexed by the eleets o M i.e. with H[x] we shall deote the eleet o the aay H whih oespods to x o M. (I patie this eas that we have ubeed the eleets o H usig the utio #(x).) 3. Iitially we take all eleets o H to be zeo. Late o i ase we hage a H[x] to 1 the this will ea that we have "ossed out" a x. 4. We delae the oute N whih is iitialized by 0. I ase o oal teiatio o the algoith N will be showig the adiality o the idex set I. 5. We delae the vaiable w ad we take it to be zeo. Vaiable w will "eebe" the oseutive ube o the last oud

3 haateisti eleet o the espetive subset M (The algoith will disove just oe haateisti eleet o eah subset M M.) 6. I a eleet x suh that #(x)>n ad H[x]=0 does ot exist the the algoith teiates. All eleets o H o whih H[x]=0 will oespod to the eleets o the set M whih otais just oe haateisti epesetative o eah subset M M M = M ad whih otais oly suh I eleets. N will be equal to the adiality o M ( o whih it ollows that it will also be equal to the adiality o the idex set I) i.e. to the ube o o-zeo eleets o the aay H. else 7. We id the iiu x M suh that #(x)>w ad H[x]=0. 8. We take w=#(x). 9. We iease N by oe. 10. We obtai the set M 1 aodig to the oula (7). 11. Fo eah i=23...k we obtai the sets M i aodig to the oula (8). Aodig to (10) we have obtaied M = M k 12. Fo eah y M \ { x} we assig H[y] = We etu to poit 6. I patiula i i Algoith 1 the set M is a odeed set o the atual ubes i the iteval [2s] by puttig k=1 ad deiig eusively the utio Next 0 1 (x) = x t ad Next 1 = Next t-1 1 (x) + x o t= the we obtai the well-kow aiet algoith o idig all pie ubes i the iteval [2s] kow by the ae the "Sieve o Eatosthees". A ube o appliatios o the ethod o the sieve o solvig vaious pobles is desibed i [7]. 2. The Method o the Multidiesioal Sieve The ethod desibed i pat 1 has a ube o disadvatages the ai o whih is that it is patially iappliable o pogas whe a suiietly geat ube o eleets is peset i the base set M. This liitatio oes o the axiu itege(a ube witte with a ixed deial poit) whih a be used i the oespodig pogaig evioet. Fo exaple by stadad i the C++ laguage the biggest ube o the type usiged log it is equal to whih i a ube o ases is isuiiet o the peviously deied aay H to be opletely addessed. Fo exaple i the base set M = B the o (1) it ollows that o elatively sall ad the peviously desibed ethod is ipossible to be ealized i this laguage without "speial tiks". I we hoose o exaple betwee all 6 x 6 Boolea aties those whih have a give popety o the lassial sieve ethod it is eessay to delae a oe-dietioal aay with diesios 2 36 whih is sigiiatly geate tha the axiu itege ube whih a be used as a addess o a aay with the widely distibuted taslatos ad pogaig evioets. Hee this will be avoided by usig a ultidiesioal Boolea aay the eleets o whih have a oe-to-oe oespodee to the eleets o the base set with a uh salle age o the idies. Let us deote by Z uv whee u ad v ae atual ubes ad u v the set Z uv ={u u+1 v}. The essee o the ethod whih we ee to as the ultidiesioal sieve ethod is to id suh ubes u 1 u 2... u v 1 v 2. v o whih a oe-to-oe oespodee betwee the base set M ad the Catesia podut Z u. 1 v Z 1 u 2 v... Z 2 u v exists. The i Algoith 1 istead o the oe-diesioal Boolea aay H we will delae ad the go aoud ad wok with a -diesioal Boolea aay W the i-th idex ( 1 i ) o whih will vay o u i to v i. I this way we will edue the ube with whih idexig will be doe. I the exaple whih we oside we pove that this edutio a be sigiiat. With the eleets o the aay W we will eode the eleets o M aodig to the peviously etioed oe-to-oe oespodee. Fo the odeig i W (as i this way aodig to the oe-to-oe oespodee we itodue odeig i M) it sees that the ost atual odeigs theto be a lexiogaphi odeig. As a illustative exaple we will oside the ollowig Poble 1 Let B be the set o all Boolea aties ad let A B B. We will state that A B i B a be obtaied o A by ovig the last ow o olu to the ist plae i the othe ows ae oved oe ow below ad the othe olus - oe olu to the ight.

4 Pove that is a elatio o equivalee. Fid the adiality B o the ato set / B / ad show oe epesetative o eah equivalee lass. The poo that is a elatio o equivalee is tivial ad we will oit it hee. The authos o this pape ae ot ailia with a existig oo oula o idig B. / Hee we have set ouselves the siple task to desibe a algoith o idig B / whe speii ad ae give ad show oe epesetative o eah equivalee lass. The algoith is based o the otes i the begiig o this setio ad o the ollowig two theoes. Theoe 1 Let us deote by P the set (11) P = { } The a oe-to-oe oespodee (bijetio) betwee the eleets o the Catesia podut P P P... P ad the eleets o the set B o all Boolea aties exists. Poo. We oside the iage : P B deied i the ollowig way: I p1 p2... p P the let us deote by z i i=12... the epesetatio o the ube p i i a biay syste ad i less tha digits(0 o 1) ae eessay we ill o the let with isigiiat zeos so that z i will be witte with exatly digits. Sie by deiitio pi P i.e. 0 p i 2 1 this will always be possible. The we o a Boolea atix so that the i-th ow is z i i=12... Appaetly this is a oetly deied iage o P i B It is lea that o dieet -s o P with the help o we will obtai dieet aties o B i.e. is ijetio. Covesely ows o eah Boolea atix a be osideed as atual ubes witte i biay syste by usig exatly digits 0 o 1 evetually with isigiiat zeos i the begiig that is these ubes belog to the set P Theeoe eah Boolea atix oespods to a -tuple o ubes <p 1 p 2 p > that is P is sujetio. Hee is a oe-tooe oespodee(bijetio). It is easy to see the validity o the ollowig stateet whih i at shows the eaig o ou osideatios. Popositio 1 Let us deote by the axiu ube whih is used whe odig the eleets o the set B by eas o the bijetio deied i theoe 1. The o suiietly geat ad the ollowig is valid: (12) ax2 1 B 2 Let a ad b be atual ubes. With a/b we will deote the opeatio "itege ube divisio" o a ad b i.e. i the divisio has a eaide the the atioal pat is ut ad with a%b we will deote the eaide whe dividig a by b. We oside the utio 1 (13) ( a) ( a%2)2 a / 2 whee % ad / ae the deied i the pevious paagaph opeatios with itege ubes. Theoe 2 Let be the deied i the poo o Theoe 1 bijetio ad let the utios : P P be deied i the ollowig ae: o evey (14) p 1 p2... p (15) ( ) p p1 p2... p1 (16) ( ) ( p1) ( p2 )... ( p ) whee the utio (a) is the deied i (13) utio. Let A B be a ado Boolea atix ad let (17) B ( ( 1 ( A))) (18) C ( ( 1 ( A))) The B is obtaied o A by ovig the last ow to the ist plae ad C is obtaied o A by ovig the last olu to the ist plae (espetively the ist ow o olu beoes the seod the seod beoes the thid espetively et.). 1 Poo. Let p 1 p2... p = ( A) P The the ube p i 0 p i 2 1 i= will oespod to the i-th ow o the atix A. The appaetly the atix B ( p p... p )) ( 1 2

5 = p p p p is obtaied o A by ovig the last ow i the plae o the ist oe ad ovig the eaiig ows oe ow below. Let p i P = { } i=1 2. The d i = p i %2 gives us the last digit o the biay epesetatio o the ube p i. I p i is witte i biay o with peisely digits optioally with isigiiat zeos i the begiig the by applyig itege ube divisio o p i to 2 we patially eove the last digit d i ad we ove it to the ist positio i ase we ultiply by 2-1 ad add it to p i /2. This is how by deiitio the utio ( pi ) woks. Hee the atix C ( ( p1 p2... p )) = p1 p2... p is obtaied o the atix A by ovig the last olu to the ist positio ad all the othe olus ae oved oe olu to the ight. Fo the deiitios o the utios (15) ad (16) it is easy to veiy the validity o the ollowig Popositio 2 I by deiitio 0 0 (19) ( ) ( ) k k (20) ( ) ( 1 ( )) k k (21) ( ) ( 1 ( )) whee P k is a positive itege the (22) ( ) ad (23) ( ) o eah P. As a diet osequee o Theoes 1 ad 2 ad Popositio 2 ad thei ostutive poos it ollows that the ollowig algoith ids exatly oe epesetative o eah equivalee lass with espet to the peviously deied i poble 1 elatio o equivalee ad the adiality o the ato set. B / Algoith 2 We id at least oe epesetative o eah equivalee lass ad the adiality o the ato set B / whe ad ae give. 1. We delae the -diesioal Boolea aays W1 ad W2 whih we will be idexed by usig the eleets o the set P i.e. W1[<p 1 p 2...p >] will oespod to the eleet < p 1 p 2... p > P We poeed aalogially with the aay W2. 2. Iitially we take all eleets o W1 ad W2 to be zeos. I W1 we will "eebe" all eleets hose o B (oe o eah equivalee lass) by hagig W1[< p 1 p 2 p >] to oe i we have seleted the eleet (< p 1 p 2...p >) o a epesetative o the espetive equivalee lass. We will hage the eleets o W2 to 1 o eah "ossig out" o a eleet o B i.e. o eah '' P o whih thee exists ' P suh that W1[ '] =1 ad ( '') ( ') o i othe wods by ' ad '' two aties o the sae equivalee lass ae eoded as we have hose ( ') o a epesetative o this equivalee lass. 3. We delae the oute N whih we iitialize by 0. I ase o oal edig o the algoith N will be showig the adiality o the ato set B /. 4. While a zeo eleet exists i W2 do Begiig o Cyle 1 5. We hoose the iiu p P aodig to the lexiogaphi ode o whih W1 [ ]=0. 6. We assig 1 to W1[ ]. 7. We iease N by oe. 8. Fo i vayig o 1 to do Begiig o Cyle 2 9. I plae o the pevious value o i we assig a ew value equal to ( ). 10. Fo j vayig o 1 to do Begiig o Cyle I plae o the pevious value o we assig a ew value equal to i ( ). 12. We assig oe to W2[ ]. Ed o Cyle 3 Ed o Cyle 2 Ed o Cyle 1 REFERENCES [1] Aige M. Cobiatoial Theoy. Spige- Velag1979. [2] BozuovC.W.Thextile Idusty-suvey Ioatio.v.3. MoskowCNII ITEILP 1983.(i Russia) [3] Cutis C. W. I.Raie Repezetatio Theoy o Fiite Goups ad Assoiative AlgebasWiley Itesiee1962.

6 [4] Hall M. The Theoy o Goups New Yok1959. [5] Huppet B. Elishe Guppe. Spige [6] MelikovO. V. V. N. Reesleikov V. A. Roakov L.A.Skoyakov I. P. ShestyakovGeeal Algeba. Moskow Nauka 1990(i Russia). [7] Reigold E.M. J.Nievegeld N. Deo Cobiatoial algoiths. Theoy ad patie. Petie Hall [8] TaakaovV. E. Cobiatoial Tasks ad (01)-aties. Moskow Nauka 1985(i Russia). [9] Yodzhev K. Ya. O a equivalee elatio i the set o the peutatio aties. Disete Matheatis ad Appliatios Reseah i Matheatis ad opute Siees SWU Blagoevgad (BG) Uivesity o Potsda(GER)