K-sorted Permutations with Weakly Restricted Displacements

Transcription

1 K-sorted Permutatos wth Weakly Restrcted Dsplacemets Tg Kuo Departmet of Marketg Maagemet, Takmg Uversty of Scece ad Techology Tape 5, Tawa, ROC Receved February 0; Revsed 5 Aprl 0 ; Accepted 0 Aprl 0 Abstract. A permutato π = π π π ) of {,,, } s called k-sorted f ad oly f π k, for all (. We propose a algorthm for geeratg the set of all k-sorted permutatos of {,,, } lexcographc order. A verso occurs betwee a par of ( π, π ) f < j but π > π. Let I(, deote the maxmum umber of versos k-sorted permutatos. For k-sorted permutatos wth weakly restrcted dsplacemets,.e., / k, we propose a cocse formula of I(, by usg the geeratg fuctos approach. Key words: k-sorted permutato, geeratg fucto, verso, lexcographc order, ordal represetato j j Itroducto A lear orderg of the elemets of the set of marks {,,,, } s called a permutato. Permutatos are oe of the most mportat combatoral objects computg. May studes have bee doe o algorthms for geeratg permutatos [8, 6]. However, lttle research has bee devoted o a specal kd of permutatos, called k-sorted permutatos. A k-sorted permutato s a class of restrcted permutato that every mark s located at most k dsplacemets from ts rght place. A restrcted permutato s a permutato that the postos of the marks are subject to some restrats. The most wdely kow restrcted permutatos are called deragemets that are permutatos wthout fxed pots, meas o mark s located at ts rght place. I geeral, there are three drectos of research restrcted permutatos: eumeratg, geeratg, ad aalyzg. Eumeratg refers to derve a formula that ca calculate the umber of restrcted permutatos of a class of restrcted permutatos. Geeratg refers to desg a algorthm that ca geerate all restrcted permutatos of a class of restrcted permutatos. Aalyzg refers to aalyze certa property of the restrcted permutato of a class of restrcted permutatos. May studes deal wth the eumerato of restrcted permutatos [, 7, 8, 9]. But, few research o the other two drectos. Bogova et al. [] troduced a problem of quas sortg as the oe of trasferrg a gve permutato to a k-sorted permutato. I that paper, they proposed the bases for the developmet of algorthms for geeratg k-sorted permutatos through elemet comparsos but dd ot propose a algorthm for systematcally geeratg all k-sorted permutatos. Berma [] ad Dutto [] focus o aalyzg k-sorted permutatos. The motvato of ths paper s twofold. Frst, by usg a ordal represetato [9, 0], we propose a algorthm for systematcally geeratg k-sorted permutatos, lexcographc order. That s, geerate all k-sorted permutatos step-by-step creasg order. Secod, as a measure of dsordered, verso of a permutato s a crucal property the study of permutatos [7, ]. Whe a lst of sze s early sorted, a straght serto sort algorthm s hghly effcet sce oly a umber of comparsos equal to the umber of versos the orgal lst, plus at most -, s requred []. So, we aalyze the maxmal umber of versos of k-sorted permutatos. I short, ths study focuses o geeratg ad aalyzg a class of restrcted permutatos, called k-sorted permutatos. Let S deote the set of all permutatos of marks {,,, }. That s, a permutato π = ( ππ π ) belogs to S f ad oly f π {,,, }, for all =,,,, ad π π j, for all j. For a permutatoπ, f t satsfes π = the we say that the mark s located at ts rght place. A k-sorted permutato s a permutato that satsfes the followg defto []. Defto. A permutato π = π π π ) S s called k-sorted f ad oly f (

2 Joural of Computers Vol.5, No., July 0 π k, for all. I other words, every mark s located at most k dsplacemets from ts rght place. Let K deote the set of all k-sorted permutatos of marks {,,, }. It s obvous that 0 k, ad that K S. We propose a cocse formula of the maxmum umber of versos k-sorted permutatos wth weakly restrcted dsplacemets. The ame of k-sorted permutatos wth weakly restrcted dsplacemets was spred by Lehmer []. I that paper, he proposed geeratg fuctos for k-sorted permutatos that s called permutatos wth strogly restrcted dsplacemets, wth k =,, ad. I ths paper, for / k, we call them k-sorted permutatos wth weakly restrcted dsplacemets. Notce that ad k are two postve tegers throughout ths paper. For completeess, let s clude some deftos that are related to t. Defto. I a permutato π = ( ππ π ), a verso occurs betwee a par of ( π, π j ) f < j but π > π j. Defto. Let I( π ) deote the umber of versos of π, the I( π ) s the umber of j s such that < j but π > π j. Defto. Let I(π ) deote the umber of versos of a permutato π = π π π ), the I ( π ) = I ( π ). ( It s well kow that the value of I(π ) s a measure of dsordered a permutato π = ( ππ π ). I ths study, we focus o the maxmum umber of versos K. Defto 5. Let I(, deote the maxmum umber of versos K, that s I (, = max{ I ( π ), for all π K }. It s trval that whe k = 0, for all, the umber of permutatos of K s oe ad I(, s zero. Berma [] gave a value of k as a upper boud of I(,. Dutto [] mproved the upper boud dow to a value of 0.6k by proposg a formula as follows I, = k m{ f ( t ), f ( )}, wth ( t + 8k + 8k + t =, t =, m =, m m ad the fucto f(t) s defed as follows + ( ) ( ) t f t = t k k + +. () t t Here, x (read the floor of x ) stads for the greatest teger that less tha or equal to x, ad x (read the celg of x ) the least teger that greater tha or equal to x. Dutto s soluto s a more or less sophstcated approach. The frst cotrbuto of ths paper s that we propose a algorthm for geeratg the set of all k-sorted permutatos of marks {,,, } lexcographc order. The secod cotrbuto of ths paper s that, by usg the geeratg fuctos approach, for k-sorted permutatos of marks {,,, } wth weakly restrcted dsplacemets, we propose a cocse formula of the maxmum umber of versos K as follows ( ) I (, = k k( k + ). () The rest of ths paper s orgazed as follows. I Secto, we dscuss represetato schemes of permutatos. I Secto, we propose a algorthm to geerate, lexcographc order, K ad to compute I(,. I Secto, we preset several recurreces of I(, s that are used the followg secto. I Secto 5, we derve a cocse formula of I(, by usg the geeratg fucto approach. Coclusos are summarzed Secto 6. Ordal Represetato Scheme I combatorcs ad mathematcs, several represetato schemes have bee used for permutato, such as twole form [6], cycle otato [6], permutato matrx [], verso vector [5], verso table [7]. From a

3 Kuo: K-sorted Permutatos wth Weakly Restrcted Dsplacemets dfferet operatoal pot of vew, we proposed a ew represetato scheme of permutato that s called ordal represetato [9, 0]. Now, let s take a brefly look at t. Defto 6. For a permutato π the form of ordal represetato, that s [D D - D ], t belogs to S f ad oly f D, for all =,,,. Here, [D D - D ] s called the ordal dgts of a permutato π. 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 of marks {,,, }. At the begg of wthdrawg, there are choces we ca choose to be the frst compoet of π. That s why we have D. Oce we have chose a tem as the frst compoet of π, there are choces left the ordered tem set. So, we have D. I the ed, oly oe choce left, so we have D. I other words, the compoet π + of π s determed by D. Furthermore, the value of D s oe plus the umber of tems that are less tha π + ad to the rght of t. Sce each permutato π S correspods uquely to a teger q the rage of [0,! ], we have the followg theorem. Theorem. I S, there s a oe-to-oe correspodece betwee [D D - D ] ad π π ). ( π Proof. Clearly, t s easy to covert a teger q to ts factoral represetato []. Frst, we dvde the teger q by ( )! ad set the quotet to C -, the the remader s dvded by ( )! ad the quotet s set to C -, ad so o. That s, ay teger q betwee 0 ad! ca be represeted as q = C ( )! + C ( )! + + C! + C 0 0!. () Here, the followg costrats 0 C j j, for all j = 0,,,, are mposed to esure uqueess. These C j s are called the factoral dgts of teger q []. By Defto 6 we kow that D, for all =,,,. Hece, we have a oe-to-oe correspodece betwee D ad C j as follows: D = C j +, where = j +, for all =,,,. Thus, f we orderg all permutatos S lexcographc order, for example whe = 7, the we ca use the ordal dgts [ ] to represet the frst (. e., 0 th ) permutato π = ( 5 6 7) ad [7 6 5 ] to the last (. e., 509 th ) permutato π = (7 6 5 ), respectvely. It s easy to see that π = D for all permutatos S. Algorthm Although may algorthms have bee doe for geeratg S ad varous permutato problems [8, 6], we kow of o publshed algorthms for geeratg K lexcographc order. By usg ordal represetato, we have proposed a method for geeratg S lexcographc order [9, 0]. I ths study, we exted that method to geerate K lexcographc order ad to compute I(,. The computato s based o a terestg property that the umber of versos of a permutato π s equal to the summato of π ' s ordal dgts mus. Ths property s demostrated the followg theorem. Lemma. For a permutato π the form of ordal represetato, π = [D D - D ], we have D = I ( π ), for all =,,,. + + Proof. By Defto, we kow that I( π ) s the umber of j s such that < j but π > π j. From the meag of ordal dgts metoed above, we kow that the value of D s oe plus the umber of tems that are less π ad to the rght of t. I other words, we have tha + That s, D = I( π ). + + D I( π ) +. = + 5

4 Joural of Computers Vol.5, No., July 0 Theorem. For a permutato π the form of ordal represetato, π = [D D - D ], we have = I (π ) D. Proof. By Defto ad Lemma, we have I( π ) = I( π ) = D. Therefore, by usg ordal represetato scheme, we ca systematcally geerate the whole K lexcographc order, ad by usg Theorem, we ca compute I (π ) drectly ad mmedately. Note that, by Deftos ad, totally! comparsos are eeded to compute I (π ). These two tasks ca be descrbed as the followg algorthm. Algorthm. Geerate K lexcographc order ad compute I(,. Iput: ad k. Output: K ad I(,. Beg I(, = 0 For D = To For D - = To For D = to Let tem set A = {,,, } For = To Retreve the D th tem of A If the D th tem satsfes Defto the Let π + = the D th tem of A Delete the D th tem of A Goto Next Else Select case Case Goto Next D Case Goto Next D - Case Goto Next D Ed Select Edf Next Output π = π π π ) ad Compute I(π ) ( If I(π ) > I(, The Let I(, = I(π ) Next D Next D - Next D Output I(, Ed By usg ths algorthm, for k = ad 0, we fd the umbers of K are 6,,, 7, 7, 00, 9, ad 77, respectvely, ad lst some of them Table. All of these umbers are same as those umbers descrbed [8]. I Table, we also lst ther correspodg I(π ) ad I (,. By usg ths algorthm, for 9 ad k, we fd ther I(, s. The accordg to these I(, s, Secto, we preset several recurreces of I(, s that are further used for dervg a cocse formula of I(, for k-sorted permutatos wth weakly restrcted dsplacemets. 6

5 Kuo: K-sorted Permutatos wth Weakly Restrcted Dsplacemets = = = 5 π = π π π ) ( Table. K, for k = ad 5. π π π π π I (π ) I (, 7

6 Joural of Computers Vol.5, No., July 0 Recurreces of I(, s I ths secto, we preset some examples of I(, s Table, ad propose several recurreces that arse aturally from Table. By carefully observg the umbers Table, t s ot hard to come up wth the followg recurreces that are correspodg to the dagoals (wth gray color) of Table. For coveece, wth teger a, we use the a th dagoal of Table to stad for those umbers that are preseted the a th dagoal of Table. For example, the th dagoal of Table stads for {,, 6, 0, 5,, 8, 6, 5, 55, 66, 78, 9, 05, 0, 6, 5, 7, 90}, ad the th dagoal of Table stads for {,, 8,, 9, 6,,, 5, 6, 76, 89, 0,8,, 5, 69,88, 08}. I case of the th dagoal of Table,.e., k =, we have the followg recurrece I (, ) = I (, ) +, for, () wth I (,0) = 0. I case of the th dagoal of Table,.e., k =, we have the followg recurrece I (, ) = I (, ) +, for, (5) wth I (,) =. I case of the th dagoal of Table,.e., k =, we have the followg recurrece I (, ) = I (, ) +, for 6, (6) wth I ( 5,) =. Now, t s ot dffcult to geeralze these recurreces as follows. I case of the a th dagoal of Table,.e., k = a, we have the followg recurrece I (, a) = I (, a ) +, for a, (7) wth I ( a, a ) = ( a ). (8) Although we have foud these recurreces, we are ot satsfed yet. Our goal s to fd a cocse formula of I(, that ca gve us a aswer quckly. Ths goal s acheved ext secto. Table. I(,, for 0 ad k. \k A Formula of I(, I ths secto, we derve a cocse formula of I(, for k-sorted permutatos wth weakly restrcted dsplacemets,.e., / k, by usg the geeratg fucto approach. Usually, a geeratg fucto s a power seres. The two most commo types of geeratg fuctos are ordal geeratg fuctos (ogf) ad expoetal geeratg fuctos (egf) []. I ths paper, we adopt ogf ad the result s the followg theorem. Theorem. For / k, wth, the maxmum umber of versos k-sorted permutatos of marks {,,, } s ( ) I (, = k k( k + ). (9) Proof. It s easy to come up wth the formula case of k =,.e., the frst dagoal of Table. By Defto, t s reasoable to locate the larger umbers as left as possble ad locate the smaller umbers as 8

7 Kuo: K-sorted Permutatos wth Weakly Restrcted Dsplacemets rght as possble. Sce all permutatos S are ( ) -sorted, the maxmum umber of versos occurs the last permutato,.e., π = ( ). Thus, ( ) I (, ) =. (0) Deftely, the detty (0) s a specal case of the detty (9). Now, we start by dscussg the secod dagoal of Table. Frst, let A( deote the ogf of the sequece (a, a, a 5,, a, ) the secod dagoal of Table,.e., k =, as follows 0 A ( = ax + ax + a5x + + a x +. The, we ca rewrte the correspodg recurrece (5) as a = a +, for, wth a =. () To fd A(, we multply both sdes of the recurrece () by x ad sum over, the we have = a x = = x a + = ( ) x. That s, x ( A( a) x = A( x + x x. Sce a =, we obta x A( ( x x ) = x x. Thus, we have + x x A( =. () I order to obta a explct formula for the sequece (a, a, a 5,, a, ), we have to expad A( a seres of partal fracto. Fortuately, t s easy to expad A( as follows. x A( = = + + ( ) x ( ) x ( ) x = 0 = = x. () = 0 Here, the coeffcet of x 0 equals to, s the value of a,.e., I(, ); the coeffcet of x equals to, s the value of a,.e., I(, ); the coeffcet of x equals to 8, s the value of a 5,.e., I(5, ); ad so o. Smlarly, let B( deote the ogf of the sequece (b 5, b 6, b 7,, b, ) the thrd dagoal of Table,.e., k =, as follows 0 B ( x ) = b 5 x + b 6 x + b 7 x + + b x +. The correspodg recurrece (6) ca be rewrtte as b = b +, for 6, wth b 5 =. As before, we have x B( =. () By expadg B( a form of partal fracto, we obta a explct formula for the sequece (b 5, b 6, b 7,, b, ) as B x ( ) = x. (5) = 0 Here, the coeffcet of x 0 equals to, s the value of b 5,.e., I(5, ); the coeffcet of x equals to 9, s the value of b 6,.e., I(6, ); the coeffcet of x equals to 5, s the value of b 7,.e., I(7, ), ad so o. Hece, geeral, let F( deote the ogf of the a th dagoal of Table,.e., k = a, we have a a F x + ( ) + ( ) ( ) = x. (6) = 0 Here, the coeffcet of x s the value of = 0 5 9

8 Joural of Computers Vol.5, No., July 0 I ( a +, a + ). So, we have = a +, that s = a +. Thus, by replacg a by k, we have = k +. Fally, by replacg a by k, ad by k +, we have ( ) I (, = k k( k + ). For example, for = 9, what s the maxmum umber of versos of 6-sorted permutatos? Frst, sce a = k =, we kow that I(9,6) s the thrd dagoal of Table. Secod, sce the frst umber the thrd dagoal of Table s the coeffcet of x 0. So, by = a + =, or by = k a + =, we kow that I(9,6) s the ffth umber the thrd dagoal of Table, that s 0. By (6), the coeffcet of x s the value of I(9,6), we have, + ( ) + ( ) I (9,6) = = 0. Alteratvely, by (9), we also have 9 (9 ) I ( 9,6) = (6 + ) = 0. 6 Coclusos The algorthm we proposed s easy to mplemet wthout ay preprocessg ad adg by auxlary data structures. It s qute sut for geeratg the set of all k-sorted permutatos of marks {,,, } lexcographc order. We derve a cocse formula of I(, for permutatos wth weakly restrcted dsplacemets,.e., / k, by usg the geeratg fucto approach. The geeratg fucto approach s a powerful ad elegat way aalytc combatorcs [5].The beauty of the geeratg fucto approach les ot the result tself, but rather ts wde applcablty. Our results ca be extet to develop the formula of the dstrbuto of versos K. Ackowledgemet The author s very grateful to aoymous referees for makg a umber of helpful suggestos. Refereces [] K. A. Berma, J. L. Paul, Fudametals of Sequetal ad Parallel Algorthms, PWS Publshg Co., Bosto, MA., 997. [] G. Bogova, F. Lucco, ad L. Pagl, The Problem of Quas Sortg, Calcolo, Vol. 6, No., pp. 5-0.,979. [] T. H. Corme, C. E. Leserso, R. L. Rvest, ad C. Ste, Itroducto to Algorthms, Secod Edto, The MIT Press, 00. [] R. D. Dutto, Iversos k-sorted Permutatos, Dscrete Appled Mathematcs, 87, pp , 998. [5] P. Flajolet, ad R. Sedgewck. Aalytc Combatorcs. Cambrdge Uversty Press, 009. [6] D. E. Kuth, The Art of Computer Programmg, Volume : Fudametal Algorthms, Secod Edto, Addso- Wesley., 97. [7] D. E. Kuth, The Art of Computer Programmg, Volume : Sortg ad Searchg, Secod Edto, Addso-Wesley., 998. [8] D. E. Kuth, The Art of Computer Programmg, Volume, Fasccle : Geeratg all Tuples ad Permutato, Addso-Wesley.,

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