A New Bit Wise Technique for 3-Partitioning Algorithm

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1 Special Issue of Iteratioal Joural of Computer Applicatios ( ) o Optimizatio ad O-chip Commuicatio, No.1. Feb.2012, ww.ijcaolie.org A New Bit Wise Techique for 3-Partitioig Algorithm Rajumar Jai Idia Istitute of Techology, Idore, Survey No.113/2-B, Opp. to Veteriary College AB- Road, Village Haria Khedi, Mhow (M.P.) Naredra S. Chaudhari Idia Istitute of Techology, Idore, Survey No.113/2-B,Opp. to Veteriary College AB- Road, Village Haria Khedi, Mhow (M.P.) ABSTRACT From last five decades peoples are worig o Partitioig problem. Partitioig is a fudametal problem with applicatios i several fields of study. The 3-partitio problem is oe of the most famous strogly NP-complete combiatorial problems i computer sciece [1]. Most of the existig partitioig algorithms are heuristics i ature Geeral Terms Partitioig, 3-Partitioig, NP-Complete Problem, Stirlig umber, combiatorial optimizatio, Patter Matchig. Keywords 3-BitPartitio, Rule of Belogigess, Optimal Partitioig techique. 1. INTRODUCTION A Partitio of a set U is a subdivisio of the set ito subsets that are disjoit ad exhaustive, i.e. every elemet of U must belog to oe ad oly oe of the subsets. The subsets A i i the partitio are called cells. Thus { A 1, A2,..., Ar } is a partitio of U if two coditios are satisfied: (1) A 1 Aj if i j (the cells are disjoit) ad (2) A1 A2... Ar U (the cells are exhaustive). I computer sciece, the partitio problem is a NP-complete problem [2] ad it is also NP-Hard to fid good approximate solutios for this problem [3]. 1.1 Combiatorial problems Combiatorics is, loosely, the sciece of coutig. This is the area of mathematics which deals with the study of families of sets (usually fiite) with certai characteristic arragemets of their elemets or subsets, ad as what combiatios are possible, ad how may there are. Combiatorics is a brach of mathematics cocerig the study of fiite or coutable discrete structures. Combiatorial Aalysis is a brach of mathematics which teaches to exhibit all the possible ways i which a give umber of thigs may be associated ad mixed together; it is to esure to eumerate all collectio or arragemet of the thigs. Combiatorics is cocered with arragemets of the objects of a set ito patters satisfyig specified rules. Combiatorics is a brach of mathematics cocerig the study of fiite or coutable discrete structures. Aspects of combiatorics iclude coutig the structures of a give id ad size, decidig whe certai criteria ca be met, ad costructig ad aalyzig objects meetig the criteria, fidig "largest", "smallest", or "optimal" objects. Combiatorics has may applicatios i optimizatio, computer sciece ad i statistical physics. 1.2 Combiatorial optimizatio problems I applied mathematics ad theoretical computer sciece, combiatorial optimizatio is a topic that cosists of fidig a optimal object from a fiite set of objects. I may such problems, exhaustive search is ot feasible. It operates o the domai of those optimizatio problems, i which the set of feasible solutios is discrete or ca be reduced to discrete, ad i which the goal is to fid the best solutio. Some commo problems ivolvig combiatorial optimizatio are the travelig salesma problem, miimum spaig tree problem ad Cluster Aalysis Combiatorial optimizatio is a subset of mathematical optimizatio that is related to operatios research, algorithm theory, ad computatioal complexity theory. It has importat applicatios i several fields, icludig artificial itelligece, machie learig, mathematics, ad software egieerig. The optimizatio versio ass for the "best" partitio, ad ca be stated as: Fid a partitio ito two subsets S1, S2 such that Max(Sum(S 1 ), Sum(S 2 )) is miimized. 1.3 Applicatio of Partitioig Applicatio of Partitioig problem [4, 5] is very broad. Followig are the importat applicatios of partitioig problem: 1. I circuit ad i VLSI desig where graph or etwor decompositios are of great iterest. 2. I parallel processig 3. I combiatorial optimizatio such as simulated aealig, mea field aealig, Tabu Search, geetic algorithms, ad stochastic evolutio 4. I may DB applicatio, icludig histogram-based selectivity estimatio, load balacig ad costructio of idex structures. 5. The schedulig of jobs to processors so as to miimize some cost fuctio is a importat problem i may areas of computer sciece. 6. Optimal geeralized Bloc Distributio i parallel Computig 7. I cluster Aalysis 2. 3-PARTITION PROBLEM The 3-Partitio problem [8] is oe of the most famous strogly NP-complete combiatorial problems i computer sciece. I this variatio of 3-partitio problem, set S must be partitioed ito S /3 triples each with the same sum. The problem is to decide whether a give multi-set of itegers ca be partitioed ito triples that all have the same sum. I the 3-Partitio Problem, we are give a positive iteger b ad a set N = { 1, 2, 3,, } of = 3m elemets, each havig a positive iteger size a j, such that. 1

2 Special Issue of Iteratioal Joural of Computer Applicatios ( ) o Optimizatio ad O-chip Commuicatio, No.1. Feb.2012, ww.ijcaolie.org a j mb j 1 The problem is to determie whether a partitio of N ito m subsets, each cotaiig exactly three elemets from N ad such that the sum of the sizes i each subset is b, exists. The solutio is yes if such a partitio exists, ad o otherwise. Garey ad Johso (1975) [1] origially proved that 3- partitio to be NP-complete. 2.1 Complexity of Partitioig Problem Zucerma shows that all of the problems (21 Problems) listed by Karp s [9] are NP-Complete ad have a versio that's hard to approximate. Partitioig Problem is oe of the problems i the listig of Karp s 21 Problems. These versios are obtaied from the origial problems by addig essetially the same, simple costrait. These problems are absurdly hard to approximate. Karp also shows that oe caot eve approximate ( ) log of the magitude of these problems to withi a costat factor, where ( ) log deotes the iterated logarithm, uless NP is recogized by slightly super polyomial radomized machies. 2.2 Stirlig umbers I mathematics, Stirlig umbers [7,8] arise i a variety of combiatorics problems. They are amed after James Stirlig, who itroduced them i the 18 th cetury. Two differet sets of umbers bear this ame: the Stirlig umbers of the first id ad the Stirlig umbers of the secod id Stirlig umbers of the first id They commoly occur i combiatorics, where they appear i the study of permutatios. Stirlig umbers of the first id [7, 8] are writte with a small s. The Stirlig Numbers of the First Kid ca be defied as s(, ) ways of partitioig a set of elemets ito disjoit cycles. Stirlig umbers of the first id couts the umber of ways to arrage objects ito cycles istead of subsets. Stirlig umbers of the first id are represeted as it Meas cycle s. Cycles are cyclic arragemets, therefore {a, b, c, d} = {b, c, d, a} = {c, d, a, b} = {d, a, b, c} But {a, b, c, d} {a, b, d, c} There are eleve differet ways to mae two cycles from four elemets (see table 1). Table 1: Number of Combiatio geerated for Stirlig umbers of the first id s(4,2) No. of Combiatio Partitio 1 1 {a, b, c}, {d} 2 {a, b, d}, {c} 3 {a, c, d}, {b} 4 {b, c d}, {a} 5 {a, c, b}, {d} 6 {a, d, b}, {c} 7 {a, d, c}, {b} 8 {a, b, c}, {d} 9 {a, b}, { c, d} 10 {a, c}, { b, d} 11 {a, d}, { b, c} Recursive relatio for Stirlig umbers of the first id is 1 Where s, 1 For > 0, with the iitial coditios ad 0 for Stirlig umber of the secod id Stirlig umbers of the secod id [7, 8] occur i the field of mathematics called Combiatorics ad i the study of partitios. I mathematics, particularly i combiatorics, a Stirlig umber of the secod id is the umber of ways to partitio a set of objects ito o-empty subsets ad is deoted by S(,). Stirlig umbers of the secod id are writte with a Capital S. Recursive relatio for Stirlig umbers of the secod id is 1 Where s, 1 for > 0, with the iitial coditios 1 1 Whe = ad 1 ad for 0 Example: I how may ways we ca partitio 4 objects ito 2 classes. Let us Cosider a set U, such that U = { a, b, c d } U = 4. 2

3 Special Issue of Iteratioal Joural of Computer Applicatios ( ) o Optimizatio ad O-chip Commuicatio, No.1. Feb.2012, ww.ijcaolie.org The umber of ways i which we ca partitio 4 objects ito 2 classes is deoted by 4 2 ad is called a Stirlig umber of the secod id. 4 = 7, see Table 2 for details 2 Table 2: Number of Combiatio geerated for Stirlig umbers of the secod id S (4, 2) No. of Partitio Partitio 1 Partitio 2 1 a, b c, d 2 a, c b, d 3 a, d b, c 4 a b, c, d 5 b a, c,d 6 c a, b, d 7 d a, b, c Bell Number[4,5] The Bell umbers describe the umber of ways a set with elemets ca be partitioed ito disjoit, o-empty subsets. Startig with B 0 = B 1 = 1 bell umbers are: 1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, , , , I geeral, B is the umber of partitios of a set of size. A partitio of a set S is defied as a set of oempty, pair-wise disjoit subsets of S whose uio is S. For example, B 3 = 5 because the 3-elemet set {a, b, c} ca be partitioed i 5 distict ways: The set {a, b, c} ca be partitioed i the followig ways( see Table 3). Table 3: Number of partitio geerated for B 3 Other way of Represetatio of B 3 So, B 3 = 5 No. of Partitio Partitios 1 { {a}, {b}, {c} } 2 { {a, b}, {c} } 3 { {a, c}, {b} } 4 { {b, c}, {a} } 5 { {a, b, c} } Fig 1: Number of partitio geerated for B 3 Bell umber satisfies the recurrece formula B B 1 0 Each Bell umber is a sum of Stirlig umbers of the secod id B 1 secod id For Example: Where = S (, ) B is a Stirlig umbers of the 3 B 3 = S (3, 1) + S (3, 2) + S (3, 3) B 3 = B 3 = 5 3. PROPOSED APPROACH FOR 3- PARTITIONING - BIT WISE REPRESENTATION OF PARTITION I this method Partitios are represeted i the form of bits. For the 3 partitioig of -etities 3 bits are required. Each of the bits represets oe partitio. Whe 3-partitioig is represeted i bit form, a specific patter is geerated for 3- Partitio for ay value of. I this patter first cosecutive bits represet 1 st Partitio, ext cosecutive bits represet 2 d Partitio ad last cosecutive bits represet 3 rd Partitio. 3.1 Rule of Belogigess If a etity O i belogs to Partitio C i the the i th value of O i is 1 otherwise it is zero. Let us cosider a set of 5 etities U = {a, b, c, d, e}, Let us a tae a sample P 1 = {a, b, c}, P 2 = {e} & P 3 = {d} Above partitio ca be represeted i bit format as Partitio 1 Partitio 2 Partitio 3 a b c d e a b c d e a b c d e Fig 2 : Shows bit wise represetatio of a partitio sample So from figure 2 P 1 = {a, b, c} = 11100, P 2 = {e} = 00001, P 3 = {d} = Rule-Set for 3-BitPartitio Total umber of 1 s i 3 bits is always equal to.(so rest 2 bits are always zero) Amog the three cluster, o the same bit positio exactly oe cluster cotai value 1 rest will be zero. I geeral for etities A i Bi Ci 1 Where 1 i Where is ex or operator. Each partitio cotais at least a sigle 1 ad maximum -2 1 s. Decimal value of secod partitio is always less the first partitio. Decimal value of third partitio is always less the secod partitio. Decimal value of first combiatio (tuple) of first cluster always starts with the value of power(2,)-4 3

4 Special Issue of Iteratioal Joural of Computer Applicatios ( ) o Optimizatio ad O-chip Commuicatio, No.1. Feb.2012, ww.ijcaolie.org Decimal value of last combiatio (tuple) of third cluster always is power (2, -2)-1. Let us geerate a patter Usig Rule of Belogigess(sectio 3.1) ad usig Rule-Set for 3-BitPartitio(Sectio 3.2) for 5 etities for 3-partitioig. Geerated patter will be as show i Table 4 (Appedix A). A uique patter is geerated for every value which is differet from other value of. So usig above rule sets 3-partitioal bit patters ca be geerated for ay positive iteger value. 3.3 Algorithm 3-BitPartitio Iput: No of etities to be partitioed () Output: All possible combiatio of 3 partitios oly i the bit patter. 3-BitPartitio (it ) 1. Read 2. Iitialize i = pow(2,) for a = pow (2, ) - 4 to 1 //geerate I partitio. 4. { 5. j = Complemet(a) & i //geeratig II partitio 6. if (j i) 7. brea; 8. for b = j to 1 9. { 10. if ( o overlappig of bits betwee(a & b) ad uio of (a & b) is ot equal to i) 11. c = (complemet of ( bitwise or of a & b )) bitwise & with i) // geeratig III partitio 12. if ( value of third partitio is less the first ad secod partitio) 13. { 14. prit( dectobi(a), a, dectobi(b), b, dectobi(c), c)) 15. } 16. } 17. } Algorithm: Char * dectobi(it) Iput: Taes a Decimal umber. output: A biary strig char * dectobi (it deco) { bio[0]='\0'; // bio is biary strig holds biary o while (deco!=0) { if(deco%2==1) apped 1 to strig bio else apped 1 to the strig bio deco =deco/2; } retur bio; } 3.4 Descriptio of Algorithm Above 3-BitPartitio algorithm geerated 3 partitios i the bit format. 3- Partitios are geerated i the 3 bits. Lie o. 2 geerates first cell (colum) of partitio of a particular tuple (Costrait o. 1). Lie o. 7 checs complemet of a is greater the a, if yes the it decremets the value of i. This step removes ay repetitio of tuples. I lie o. 8 loop fid out possible value of secod cell (colum). Lie o 10 checs whether there is ay clash of bits i first cell ad i secod cell (chec for Rule o. 2). Lie o. 10 also checs third cell should cotai at least a sigle 1.Lie o. 11 geerates the third partitio Lie o 12 checs value of third partitio is less the first ad secod partitio. Subroutie dectobi(it) traslates decimal umber ito a biary strig which represets a partitio. 3.5 Characteristics ad Validatio of Algorithm Proposed algorithm is a ew approach for the geeratio ad represetatio of 3-Partito. Bit approach is far better tha other approach. Bit operatios are used to geerate partitio ad to chec for the costraits. So reduces time complexity. I this algorithm o duplicates tuples are geerated. The beauty of this algorithm is that if the first cell (colum) of ay tuple is ot accordig to costraits the it does ot geerate secod cell (colum) ad similarly if the geerated secod cell clashes with ay with the first cell the it does ot geerate third cell (colum) of the tuple. So it reduces overall ruig time of the algorithm. Algorithm is validated by the umber of combiatios geerated are equal to the Stirlig umber of the secod id. This algorithm is very basic algorithm which ca be applied i ay of the applicatio which requires 3-Partitioig. The most beautiful thig is that ruig time ca be further reduced by applyig Costraits. Costraits will be specific to applicatio. 4. CONCLUSION Bit represetatio of 3-Partitio provides a ew dimesio to solve may of the partitioig algorithms. Proposed algorithm uses bit operator to geerate partitio. Proposed algorithm is basic 3-Paritioig algorithm which ca be applied i ay of the applicatio which requires 3-Partitioig. Applyig Costraits specific to applicatio o this algorithm ca further reduce the time complexity. It is the optimized partitioig algorithm as it does t geerates duplicate tuples ad geerate oly 3-paritito tuples. 5. REFERENCES [1] M.R.Garey ad D.S. Johso, Computers ad Itractability: A guide to the theory of NP-Completeess, W.H. Freema ad Co., New Yor, [2] Hayes Bria, The Easiest Hard Problem, America Scietist, May-Jue [3] Zucerma, D, NP-complete problems have a versio that's hard to approximate, Structure i Complexity Theory Coferece, 1993., Proceedigs of the Eighth Aual, IEEE, 1993, pp [4] C. Dig, H. Xiaofeg, Z. Hogyua, G. Mig, ad H. Simo, A Mi-Max Cut Algorithm for Graph Partitioig ad Data Clusterig, Proc. IEEE Iteratioal Coferece Data Miig, pp ,

5 Special Issue of Iteratioal Joural of Computer Applicatios ( ) o Optimizatio ad O-chip Commuicatio, No.1. Feb.2012, ww.ijcaolie.org [5] Y. G. Saab, A Effective Multilevel Algorithm for Bisectig Graphs ad Hypergraphs,IEEE Trasactio, Jue 2004, pp [6] M. Dell Amico, S. Martello, Reductio of 3-Partitio Problem, Joural of Combiatorial Optimizatio, 1999, Vol. 3, No. 1, [7] R.L. Graham, D.E. Kuth, Ore Patashi, Cocrete Mathematics-A Foudatio for Computer Sciece, 2 d Editio, Pearso Educatio [8] J.H. Va-Lit, R.M. Wilso, A course i Combiatorics 2 d Editio, Cambridge Uiversity Press, 2001 pp [9] M. Karp, Reducibility Amog Combiatorial Problems I R.E. Miller ad J.W. Thatcher, eds., Complexity of Computer Computatios, 1972, pp