Multiset Permutations in Lexicographic Order

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1 Webste: ISSN , ISO 9001:2008 Certfed Joural, Volume 4, Issue 1, Jauary 2014 Multset Permutatos Lexcographc Order Tg Kuo Departmet of Marketg Maagemet, Takmg Uversty of Scece ad Techology, Tape, Tawa Abstract I a prevous work [12], we proposed a method for geeratg permutatos lexcographc order. I ths study, we exted t to geerate multset permutatos. A multset s a collecto of tems that are ot ecessarly dstct. The gudele of the exteso s to skp, as soo as possble, those partally-formed permutatos that are less tha or equal to the latest geerated elgble permutato. Multset permutato ca be appled combatos geerato, sce a combato of q tems out of tems s a specal case of multset permutatos that cota q 1s ad -q 0s. Keywords multset permutato, lexcographc order, rakg, urakg, ordal represetato I. INTRODUCTION Permutatos are oe of the most mportat combatoral objects computg. I ths study, we focus multset permutato. A multset s a set that each tem the set has a multplcty whch specfes how may tmes the tem repeats. For coveece, wthout loss of geeralty, we use otato S m, to deote the set of all permutatos of a tems multset { 1 d 1, 2 d 2,, m d m } that s composed of m dstct, but ot ecessary successve, tegers d 1,,d m. Here, the multplctes of the m dstct tegers d 1,,d m are 1, 2,, m, respectvely, ad satsfy the followg costrat: m 1, for all 1. 1 For example, the multset {1, 1, 1, 2, 2, 3, 4, 4} ca be expressed as {3 1, 2 2, 1 3, 2 4}. The multplctes of the tems 1, 2, 3, ad 4 are 3, 2, 1, 2 respectvely. That s, a permutato 1 2 belogs to the S m, f ad oly f d,, d }, for all 1,,, { 1 m 2 ad the multplctes of those tems satsfy 1. Moreover, there s o two permutatos ad both belog to the S m, such that a, b, total umber of S m, s a for all 1,,. Clearly, the b 442! 2... m 1! 2!... 1 m Obvously, a set s a specal multset that the multplcty of each tem s oe. May methods have bee proposed o multset permutato [2, 3, 6, 10, 11, 17, 19]. Although these methods have ther ow characterstcs ad merts, there s a commo feature they share that oe of them works lexcographc order. The order of a lst of permutatos s determed by the method used to geerate them. However, f such a order wthout ay specfc characterstc that ca be utlzed the the permutato geerato method s ot good eough. I cotrast, there s a ature order of all permutatos called lexcographc, or alphabetcal, order [18]. I the proper sese of the word, a lst of permutatos s lexcographc order f these permutatos are sorted as they would appear a dctoary. Strctly speakg, f the tems gog through permutatos are ordered by a precedece relato <, the permutato precedes a.! a, 1 a,2 a, permutato f ad oly f, for some >= 1, we have a, b, b b, 1 b,2 b, a, j b, j 3 for all j < ad [15]. For example, the lexcographc order of sx permutatos of three dstct tems {1 2 3} s < < < < < Besdes, there s a kd of reverse lexcographc orderg [18] or called reverse colex order [9], the result of readg the lexcographc sequece backwards ad the permutatos from rght to left that s also of some terest. It s worthy to meto that, passg, our method ca be used to geerate multset permutatos both Lexcographc order ad Reverse Lexcographc order. I Table 1, we lst all 12 permutatos of a four tems multset { } Lexcographc order ad Reverse Lexcographc order respectvely.

2 Webste: ISSN , ISO 9001:2008 Certfed Joural, Volume 4, Issue 1, Jauary 2014 Table 1: Lexcographc order ad Reverse Lexcographc order of S 3,4 {1 1, 2 2, 1 3}. Lexcographc order Reverse Lexcographc order The reaso why we cocetrate ths study o lexcographc order ca be see the followg remark. Furthermore, the cotext of a backtrack search for all solutos to some problems, geerato of solutos lexcographc order mght be preferred o aesthetc grouds, ad has at least two practcal advatages, amely 1 Whe a subset of solutos has bee geerated, t s mmedately clear whch permutatos have bee rejected up to the most recetly geerated soluto. 2 It s easer to verfy whether a partcular permutato s preset the complete lst of solutos f that lt s lexcographc order [7]. Sce lexcographc order s a atural ad smple order, we belef that t should be easy mapulated by a computer program. Why do t we desg a permutato geerato method that ca full-utlze ths trsc order? Ths drves us to coduct ths study. 443 I the rest of ths paper, we wll propose a smple ad flexble method for geeratg multset permutatos lexcographc order. I Secto II, we frst revew a ew represetato scheme that s coceptually easy to uderstad ad mplemet. I Secto III, the rakg ad urakg algorthms are proposed. Example ad results are preseted Secto IV. Fally, dscusso ad coclusos are summarzed Secto V. II. REPRESENTATION SCHEMES Represetato schemes are of cetral terest scetfc research. Not oly because that they provde us a way to realze the cocept dscussed, but also because that they eable us to mapulate the objects whch they represet. I combatorcs ad mathematcs, several represetato schemes have bee used for permutato. Such as: two-le form [9], cycle otato [9], permutato matrx [4], verso vector [16], verso table [8], -ary p-umber [14], ad a p-sequece [1]. Each oe of these represetato schemes metoed above has ts ow characterstcs ad operatoal meag. However, a good represetato scheme of permutato should ot oly be used for geeratg all permutatos but also should have a property that t ca be easly mapulated by smple arthmetc operatos drectly. Moreover, t should be flexble for dfferet types of permutato problems. From a dfferet operatoal pot of vew, we proposed a ew represetato scheme of a permutato called ordal represetato that meets these goals [12]. Now, let us gve a quck revew o t. Defto 1: For a permutato the form of ordal represetato, that s [D D -1 D1 ], t belogs to S f ad oly f 1 D j j, for all j 1, 2,,. Here, we use otato S to deote the set of all permutatos of a tems set ad [D D -1 D1 ] s called ordal dgts of. The meag of ordal dgts s easy to uderstad, f we mage that a permutato s the result of a successve wthdrawg of tems dvdually, oe after the other wthout replacemet, from a ordered tem set {1, 2,, }. At the begg of wthdrawg, there are choces we ca choose to be the frst compoet of. That s why the equalty 1 D holds. Oce we have chose a tem as the frst compoet of, there are 1 choces left the ordered tem set. So, we have D

3 Webste: ISSN , ISO 9001:2008 Certfed Joural, Volume 4, Issue 1, Jauary 2014 Fally, oly oe choce s left, so t s 1 D1 1. I other words, the compoet j 1 of s determed by D j. Itrscally, the value of D j s oe plus the umber of tems that are less tha j 1 ad to the rght of t. Sce each permutato S correspods uquely to a teger q the rage of [0,! 1 ], we have the followg theorem. Theorem 1: I S, there s a oe-to-oe correspodece betwee [D D -1 D1 ] ad. Proof: Clearly, t s easy to covert ay teger q betwee 0 ad! 1 to ts factoral represetato [13]. Frst, we dvde q by 1! ad set the quotet to be C -1, the the remader s dvded by 2! ad the quotet s set to be C -2, ad so o. That s, ay teger q betwee 0 ad! 1 ca be represeted as q C 1 1! C 2 2! C1 1! C 0 0!. 3 Here, the followg costrats 0 C, for all 0, 1,, 1, are mposed to esure uqueess. These C s are called factoral dgts of teger q [13]. Thus, we have 1 C 1 1, for all 0, 1,, 1, By Defto 6, we kow that 1 D j j, for all j 1, 2,,. Ad from operatoal pot of vew, both factoral dgts ad ordal dgts are lexcographc. That s, Ca, 1Ca, 2 Ca, 0 precedes Cb, 1Cb, 2 Cb, 0 f ad oly f, for some k 0, we have C a, j Cb, j for all j k ad C C. Smlarly, D D D a, k b, k a, a, 1 a, 1 precedes Db, Db, 1 Db, 1 f ad oly f, for some k 1, we have D a, j Db, j for all j k ad Da, k Db, k. Hece, we have a oe-to-oe correspodece betwee D j ad C as follows: D C 1, where j 1, for all j 1, 2,,. j Thus, f we order all permutatos of S lexcographc order the we ca, for example = 7, use the ordal dgts [ ] to represet the frst. e., 0 th permutato π = , [ ] to 536 th permutato π = , ad [ ] to the last. e., 5039 th permutato π = , respectvely. It s easy to see that D = π 1 for all permutatos of S. I ths paper we use the term rakg to refer to covertg each permutato the S to ts ordal dgts uquely, ad urakg meas to covert ordal dgts to ts correspodg permutato uquely. I ext secto, we wll descrbe how to geeratg multset permutatos lexcographc order by usg ordal represetato scheme. III. ALGORITHMS I geeral, whe we meto a method of permutato geerato t s evtable to talkg about the rakg ad urakg algorthm. A rakg algorthm coverts each permutato S m, of a tems multset { 1 d 1, 2 d 2,, m d m } to a teger the rage of uquely.! 0,! 2!... 1 m 1! I cotrast, the correspodg urakg algorthm coverts a teger the rage of 4 to oe permutato S m, uquely. Kuth metoed a recurrece formula of rakg for permutatos of a multset. However, there s o urakg formula has bee proposed up to date. Now, let us tur to the ma subject of ths paper: geerate S m, lexcographc order. By usg ordal represetato scheme, we ca easly hadle multset permutato. Frst of all, we costruct a multset that s a ordered lst composed of all tems whch are to be arraged. Thus, gve a ordal represetato [D D - 1 D 1 ] of a permutato, we ca geerate the permutato th as follows. For each D,, 1,,1, output the D tem of the multset ad mmedately delete t from the multset. For example, f we wat to geerate the 10 th permutato S 3,4 of a multset { 1 1, 2 2, 1 3}, we talze the multset to be { }. Sce the ordal dgts of the 10 th permutato are [ ], we frst output the 4 th tem, here s 3, of the multset ad delete t from the multset

4 Webste: ISSN , ISO 9001:2008 Certfed Joural, Volume 4, Issue 1, Jauary 2014 After the frst step, the multset becomes {1 2 2}. Next, we output the 1 th tem, here s 1, of the multset ad delete t from the multset. After ths secod step, the multset becomes {2 2}. By followg the same process, the 10 th permutato S 3,4 of a multset { 1 1, 2 2, 1 3} we fally obta Ths process s descrbed Algorthm 1 as follows. Algorthm 1: Urakg the ordal dgts [D D -1 D 1 ] to a permutato the S m,. Iput: ; a multset M = { 1 d 1, 2 d 2,, m d m } ; [D D -1 D 1 ] Output: Beg For j = To 1 Retreve the D j th tem of M Let π +1-j = D j th tem of M Delete D j th tem of M Output Ed Oce we have the Algorthm 1, we ca easly geerate ay permutato correspodg to a teger k the rage of 4. Naturally, we ca systematcally geerate the whole S m, lexcographc order. Sce a multset s a collecto of tems wth repettos, the major problem of geeratg the whole S m, s how to avod geeratg permutatos that have bee geerated. Straghtforwardly, we ca treat all tems of a multset as dstct ad geerate permutatos the same maer as metoed Algorthm 1 but skp those permutatos that are geerated already. However, the check of duplcato s a bottleeck. Fortuately, sce we follow the lexcographc order, aturally, the gudele o the check of duplcato s to skp those permutatos that are less tha or equal to the latest geerated elgble permutato. Ths task ca be doe by Algorthm 2 as follows. Algorthm 2: Geerate S m, lexcographc order. Iput: ; a multset M = { 1 d 1, 2 d 2,, m d m }. 445 Output: All permutatos 1 2 belog to the S m,. Beg For j = 1 To talze the latest geerated elgble permutato π Let π j = 1 For D = 1 To For D -1 = 1 To 1 For D 2 = 1 to 2 For D 1 = 1 to 1 Let multset M = { 1 d 1, 2 d 2,, m d m } Ok = 0 For j = To 1 Retreve the D j th tem of M If Ok = 1 the Let π +1-j= D j th tem of M Delete D j th tem of M Goto Else f D j th tem of M > π +1-j the Else Select Case j Case 1 Goto Next D 2 Case else Let π +1-j = D j th tem of M Delete D j th tem of M Ok = 1 Goto f D j th tem of M = π +1-j the Let π +1-j = D j th tem of M Delete D j th tem of M Goto

5 Webste: ISSN , ISO 9001:2008 Certfed Joural, Volume 4, Issue 1, Jauary 2014 Else Select Case j Case Goto Next D Case -1 Case 3 Case 2 Ed Select Edf Ed Select Edf Edf Goto Next D -1 Goto Next D 3 Goto Next D 2 Output Next D 1 Next D 2 Next D -1 That s, we drectly embed the coverso of all tegers the rage of 4 to D j s the algorthm. Ths s why we use the lower ad upper bouds of D j s each ested For loop statemet of Algorthm 2. Cosequetly, t ca effectvely be used to hadle permutato geerato eve for a bg. However, those permutato geerato methods whch have to drectly deal wth a teger the rage of 4 are uable to hadle a bg because of the lmtato of computer hardware. Now, let us tur to the rakg algorthm. O the other had, order to covert each permutato the S m, to ts ordal dgts uquely, we desg Algorthm 3 as follows. Algorthm 3: Rakg a permutato 1 2 S m, to ts ordal dgts [D D -1 D 1 ]. Iput: Output: [D D -1 D 1 ] Beg Let multset M = { 1 d 1, 2 d 2,, m d m } For j = To 2 Let D j = r, f π -j+1 = r th tem of M search Delete r th tem of M Let D 1 =1 Ed By usg a bary Next D Ed Orgally, order to geerate the whole S m, lexcographc order we eed to coverts every teger k the rage of 4 to correspodg permutato the S m, uquely. Wth a slghtly dfferet, we do ot covert teger k but covert ordal dgts D j s to correspodg permutato. I other words, we omt both operatos of covertg a teger k to factoral dgts C s ad covertg C s to ordal dgts D j s. IV. EXAMPLE AND RESULTS For example, the multset {0, 0, 1, 1, 1, 1, 1, 1, 1} ca be expressed as {2 0, 7 1}. Totally, there s permutatos, but actually oly 36 permutatos are uquely. By usg a Acer otebook wth a Itel Core 2 CPU 1.83GHz ad a VBA program executed uder the Mcrosoft Excel evromet, we geerate the 36 permutatos lexcographcally ad totally escape duplcated permutatos oe secod. Table 2 shows the 36 permutatos Lexcographc order. 446

6 Webste: ISSN , ISO 9001:2008 Certfed Joural, Volume 4, Issue 1, Jauary 2014 Table 2 Lexcographc order of S 2,9 {2 0, 7 1} V. DISCUSSION AND CONCLUSIONS Multset permutato ca be appled combatos geerato, sce a combato of q tems out of a tems set s a specal case of multset permutatos that cota q 1s ad -q 0s. For example, a combato of 4 tems out of 8 tems ca be descrbed as to stad for we pck up the frst tem, the 4th tem, the 5th tem, ad the last tem. Obvously, ths s a permutato of a multset {4 0, 4 1}. The algorthms descrbed above are easy to mplemet by ay computer programmg laguage especally those provde data structures ad fudametal operatos that support us to drectly mapulate a set of tems. By usg the ordal represetato, our ew method s ot restrcted to umber the tems from 1 to, successvely. 447

7 Webste: ISSN , ISO 9001:2008 Certfed Joural, Volume 4, Issue 1, Jauary 2014 I other words, wthout ay ad of remappg, we ca drectly geerate the permutatos of dstct tems that are umbered, for example, by 3, 5, 8, 12, 18, ad 31, or eve wth o-umeral marks, provded there exst a predefed order amog these marks. It s terestg to ote that f we reverse all ested For loops the Algorthm 2 from upper boud dow to lower boud by step -1 ad set π j be the D j th tem of the tem set, the Algorthm 2 ca geerate the whole S m, Reverse Lexcographc order. I cocluso, the ew method s coceptually easy to uderstad ad mplemet ad s wellsuted to a wde varety of permutato problems. Therefore, we ted to cotue pursug ths le of study related topcs. REFERENCES [1] B. Bauslaugh ad F. Ruskey, Geeratg Alteratg Permutatos Lexcographcally, BIT 1990, pp [2] P. Bratley, Permutatos wth Repettos Algorthm 306, Comm. ACM, Vol. 10, No , pp [3] P. J. Chase, Permutatos of a Set wth Repettos Algorthm 383, Comm. ACM, Vol. 13, No , pp [4] T. H. Corme, C. E. Leserso, R. L. Rvest, ad C. Ste, Itroducto to Algorthms, 2 d Ed., The MIT Press, [5] J. R. Howell, Geerato of Permutatos by Addto, Math. Comp., Vol. 16, No , pp [6] T. C. Hu ad B. N. Te, Geeratg Permutatos wth Nodstct Items, The Amer. Math. Mothly, Vol. 83, No , pp [7] R. W. Irvg, Permutato Backtrack Lexcographc Order, The Computer J., , pp [8] D. E. Kuth, The Art of Computer Programmg, Volume 3: Sortg ad Searchg, Secod Edto, Addso-Wesley, [9] D. E. Kuth, The Art of Computer Programmg, Volume 4, Fasccle 2: Geeratg all tuples ad permutatos, Addso-Wesley, [10] J. F. Korsh ad S. Lpschutz, Geeratg Multset Permutatos Costat Tme, J. Algorthms, , pp [11] J. F. Korsh ad P. S. LaFollette, Loopless Arary Geerato of Multset Permutatos, The Comput. J. Vol. 47, No , pp [12] T. Kuo, A New Method for Geeratg Permutatos Lexcographc Order, Joural of Scece ad Egeerg Techology, Vol. 5, No , pp [13] D. H. Lehmaer, The Mache Tools of Combatorcs, Appled Combatoral Mathematcs E. F. Beckebach, Ed., pp. 5~31, Wley, New York, [14] D. Pager, A Number System for the Permutatos, Comm. ACM, Vol. 13, No , p [15] J. P. N. Phlps, Permutato of the Elemets of a Vector Lexcographc Order Algorthm 28, The Comput. J. Vol. 10, No , pp [16] E. M. Regold, J. Nevergelt, ad N. Deo, Combatoral Algorthms: Theory ad Practce, Pretce-Hall, Ic., [17] T. W. Sag, Permutatos of a Set wth Repettos, Comm. ACM, Vol. 7, No , p [18] R. Sedgewck, Permutato Geerato Methods, Computg Survey, Vol. 9, No , pp [19] L. Ye, A Note o Multset Permutatos, SIAM J. Dscrete Math.., Vol. 7, No , pp