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IJCSIS Vol. 9 No. 4, April 20:lSSN 1947-5500

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Computer Scien& Informatia

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DCSIS Vot 9, No. 4, April2011 Edition

ISSN 1947-5500 © DCSIS, USA.

Abstracts Indexed by (among others):

IJCSIS EDITORIAL BOARD

Dr. M. Emre Celebi,Assistant Professor, Department of Computer Science, Louisiana State Universityin Shreveport, USA

Dr. Yong LiSchool of Electronic and Information Engineering, Beijing Jiaotong University,P. R. Chin~

Prof. Hamid Reza NajiDepartment of Computer Enigneering, Shahid Beheshti University, Tehran, Iran

Dr. Sanjay JasolaProfessor and Dean, School of Infonnation and Communication Technology,Gautam Buddha University

Dr Riktesh SrivastavaAssistant Professor, Information Systems, Skyline University College, UniversityCity of Sharjah, Sharjah, PO 1797, UAE

Dr. Siddhivinayak KulkarniUniversity of Ballarat, Ballarat, Victoria, Australia

Professor (Dr) Mokhtar BeldjehemSainte-Anne University, Halifax, NS, canada

Dr. Alex Pappachen lames, (Research Fellow)Queensland Micro-nanotechnology center, Griffith University, Australia

Dr. T.e. Manjunath,ATRIA Institute of Tech, India.

TABLE OF CONTENTS

1. Paper 28031141: Dynamic Rough Sets Features Reduction (pp. 1-10)

Walid MOUDANI (1), Ahmad SHAHIN (1), Fad; SHAKIK (1), and Felix Mora-Camino (2)(1) Lebanese University, Faculty cf Business, Dept. a/Business Information System, Lebanon(2) Air Transportation Department, ENAC, 31055 Toulouse, France

2. Paper 30031145: A Study on the Performance of Classtcal Clustering Algorithms with UncertainMoving Object Data Sets (pp. 11-16)

Angeline Christobel . Y, College 0/ Computer Studies, AMA International University, Salmabad, Kingdomof BahrainDr. Sivaprakasam, Department a/Computer Science, Sri Vasavi College, Erode, India

3. Paper 14031106: Bijection and Isomorphism on Grapb of Sn(I23; 132) from One of(n -1) LengthBinary Strings (pp. 17-20)

A. Juama, A.B. MutiaraFaculty a/Computer Science and Information Technology, Gunodarma University, JL Margonda RayaNo.100, Depok 16424, Indonesia

4. Paper 28031140: An Investigation ofQoS in Ubiquitous Network Environments (pp. 21-30)

Aaqif 'Afzaal Abbasi, Mureed Hussain

5. Paper 31031181: Information Agents in Database Systems a5 a New Paradigm for SoftwareDeveloping Process (pp. 31-34)

Eva Cipi, Department 0/ informatics engineering, University of Vlora, Vlora, Albania,Betim Cico, Department 0/ informatics engineering, Polytechnic University of Tirana, Tirana; Albania

6. Paper 28031142: Determination of the Traveling Speed of a Moving Object of a Video UsingBackground Extraction and Region Based Segmentation (pp. 35-39)

Md. Shafiul Azam, Lecturer, Dept. a/Computer Science and Engineering, Pabna Science and TechnologyUniversity, Pabna, Bangladesh.Md. Rashedul Islam, Senior Lecturer, Dept. of Computer Science and Engineering, Leading University,Sylhet, BangladeshMd. Omar Faruqe, Lecturer, Dept. a/Computer Science and Engineering, Rajshahi University, Rajshahi,Bangladesh

7. Paper 14031105: An introduction to Biometrics (pp. 40-47)

Sarah BENZIANE, Institut of-maintenance and-industrial security, University of'Oran, AlgeriaAbdelkader BENYE1TOU, Department a/Computer Science, Faculty a/Science, University of Science &Technology Mohamed Boudiaf of Oran, Algeria

8. Paper 14031107: Score-Level Fusion for Efficient Multimodal Person Identification using Face andSpeech (pp. 48-53)

Hanaa S. AIi, Mahmoud l. Abdalla,Faculty a/Engineering, Zagazig University, Zagazig, Egypt

9. Paper 17021102: Access Control Via Biometric Authentication System (pp. 54-63)

Okumbor Anthony N., Computer Centre, Delta State Polytechnic, Otefe-Oghara; NigeriaS. C. Chiemeke (Ph.D), Associate Professor Computer Science, University ofBenin, Benin City, Nigeria

10. Paper 22031123: A middleware platform for Pervasive Environment (pp. 64-73)

Vasanthi. R. Research Scholar, Computer Science and Engineering, Anna University of Technology,Coimbatore, Tamilnadu , IndiaDr. RS.D. Wahidabanu, Research Supervisor, Anna University of Technology, Coimbatore, Tamilnadu;India

11. Paper 22031129: Watermarking Social Networking Relational Data using Non-numeric Attribute(pp. 74-77)

Rajneeshkaur Bedi , Dr. V.M. Wadhai , Rekha Sugandhi ,Atul MirofkarComputer Engineering Department, Pune University, MIT College of Engineering, Pune, India

12. Paper 28031138: Internet Adoption in Indonesian Education: Are Female Teachers Able to Useand Anxious of Internet? (pp. 78-87)

Farida I, Sri Wulan Windu Ratih 1, Betty Yudha Sulistiowati 3, Budi Hermana 4

1.2.3 Faculty if Computer Science and Information Technology, 4 Faculty of Economics, GunadarmaUniversity, JL Margonda Raya No.I 00, Depok City, West Java, Indonesia

13. Paper 22031130: Synthesis of Linear Antenna Array using Genetic Algorithm to Maximizt:Sidelebe Level Reduction (pp. 88-93)

T. S. Jeyali Laseetha I, Professor, Department Of Electronics And Communication Engineering, HolycrossEngineering College, Anm University Of Technology, Tirunelveli, Tamil Nadu, IndiaDr. (Mrs.) RSukanesh 1, Professor, Department Of Electronics And Communication EngineeringThiagarajar College Of Engineering, Madurai, Tamil Nodu, India

14. Paper 31031153: An Efficient Constrained K-Means Clustering using Self Organizing Map (pp.94-99)

M. Sakthi 1 and Dr. Antony Selvadoss Thanamani 1

1Research Scholar 1Associate Professor and Head,Department if Computer Science, NGM College, Pollachi, Tamilnadu

15.Paper 31031163: Applying and Analyzing Security using Images: Steganography v.s. Steganalysis(pp. 100-105)

Nighat Mir, Computer Science Department, Effat University, Jeddah, Saudi ArabiaAsrar Qadi, Wissal Dandachi , Computer Science Department, Effat University, Jeddah, Saudi Arabia

16. Paper 31031182: An Overview and Study of Security issues & Challenges in Mobile Ad-hocNetworks (pp. 106-111)

Umesh Kumar Singh, Institute of Computer Science, Vikram University Ujjain INDIA-4560IOShivlal Mewada; Institute of Computer Science, Vikram University UjjainINDIA-45601OLokesh laddhani, Institute of Computer Science, Vikram University Ujjain INDIA-45601OKamal Bunkar, Institute of Computer Science, Vikram University Ujjain INDIA-45601O

17. Paper 31031165: An Intelligent Agent Based Text-Mining System: Presenting Concept throughDesign Approach (pp. 112-117)

Kaustubh S. Rami, Ranjeetsingh S. Suryawanshi, Professor Devendra M. ThakoreBharati Vidyapeeth Deemed University, College of Engineering, Pune - 411043.

18. Paper 31031151: Temperature Measurement of Dynamic Object (pp. 118-122)

Varsha Khare, Shivajirao S.Jondhle Polytechnic, Asangaon, Maharashtra IndiaMrs. Rodge M.P., H.OD.-Shivajirao SJondhle College of Engineering & Technology, AsangaonMaharashtra India

19. Paper 28031139: Dynamic Slicing of Aspect Oriented Programs using AODG (pp. 123-126)

Sk Riazur Raheman, Dept of MCA, REC,Bhubaneswar, Orissa, IndiaAbhishek Ray, School of Technology, KllT University, Orissa, IndiaSasmita Pradhan, Dept ofMCA, REC, Bhubaneswar, Orissa, India

20. Paper 24031134: Qualitative Analysis of Hardware Description Languages: VHDL and Verilog(pp. 127-135)

R Uma, Department of Electronics and Communication Engineering, Rajiv Gandhi College of Engineeringand Technology, Poadicherry, IndiaR Sharmila, Electronics and Communication Engineering, Rajiv Gandhi College of Engineering andTechnology Puducherry, India

21. Paper 22031131: Data Mining: A prediction for performance improvement using classification(pp. 136-140)

Brijesh Kumar Bhardwaj, Research Scholar, Singhaniya University, Rajasthan; IndiaSaurabh Pal, Dept. of Computer Applications, VBS Purvanchal University, Jaunpur (UP) - 224001, India

22. Paper 22031125: ASIP Design Space Exploration: Survey and Issues (pp. 141-145)

Deepak Gour, Assistant Professor - Dept. ofCSE, Sir Padampat Singhania University, Udaipur, IndiaDr. M. K. Jain, Assistant Professor - Dept. of CS, Mohan Lal Sukhadia University, Udaipur, India

23. Paper 20031119: POur-NIR: Modified Node Importance Representative for Clustering ofCategorical Data (pp. 146-150)

S. Viswanadha Raju, N. Sudhakar Reddy, H. Venkateswara Red4y, G. Sreenivasulu, C. Nageswarakaju

24. Paper 21041117: Packet Forwarding Encouragement Scheme in a Wireless Sensor Network (pp.151-156)

Praveen Kaushik, Department ofCSE, MANIT, Bhopal, IndiaJyoti Singhai, Department ofECE, MANIT, Bhopal, India

25. Paper 18031113: A Multi-criteria Decision Model for EOL Computers in Reverse Logistics (pp.157-161)

K. ArunVasantha Geethan , Department of Mechanical Engineering, Sathyabama University, Chennai.IndiaDr. S. Jose, Loyola-ICAM College of Engineering & Technology ,Chennoi. IndiaR Devisree, Cognizant TechnologySolutions, Chennai. IndiaS. Godwin Bamabas, St.Joseph/s College of Engineering, Chennai. India

26. Paper 12031101: Implementation of Direct Processor Access in Transient Faulty Nodes (pp. 162-166)

P. S. Balamurugan, B. E., M. E., Research Scholar, Anna university, CoimbatoreDr. K. Thanushkodi, B. E., M. Se (Engg),Ph. D, Director, Akshaya College of Engineering and Technology,Coimbatore

(1JCSIS) International Journal of Computer Science and Information Security,Vol. 9, No. 4, April 2011

Bijection and Isomorphism on Graph of Sn(123,132)from One of(n-l) Length Binary Strings

A. Juama', A.B. Mutiara'Faculty of Computer Science and Information Technology, Gunodarma University

JI. Margonda Raya No. LOO,Depok 16424, Indonesia1.2 {ajuarna,amutiara}@staff.gwuulanna.ac.id

rtrad-Simion and Sdunidt showed in 1985 that thedinaIity 01' the set S,,(I23,132llength 11permutations avoidmgpatterns 123 and 132, is r- , but in the other side r-1 is thedinality of the set B.1 = {O,I}·1orIength (11-1)binary strinp.eoretically,it must exist a bijection between S,,(I23,132) andI. In thls paper we give a COIL'Jtructivebijection between B.1

18,,(123,132);we show that it is actually an isomorphism andstrate this by constructing a Gray code for S,,(I23,132) from aown similar result for B•..1• As we noted that an isomorphismt'cen two combinatorial classes is a closeness preservingfctionbetween those classes, that is, two objects in a class are~ if and only if their baaces by this bijection are also closed.en, as in this paper, doseness is expressed in terms ofnming distance. Isomorphism allows us to find out someperties of a cornbinatorial class X (or for the graph induced byclass Xl if those properties are found in the pre image of the,bmatorial class X; some mentioned properties are,iltonian path, graph diameter, eIhaustive and randomeration, and ranking and Wlranking algorithms.

KeywOl'fls..pattertUfUtg penrultatimu; bUuuy striItgs,stnu:tive bijectiott; HammiIIg disttuu:e; combUuztoritd

ism.

I. INTRODUCTION

In this paper an element denotes a member of a list or set,a term denotes a term in a string or sequence. Let x = XI X2

andy =YIY2 ... Y,. be two strings of same length. We say X

Y arepiecewise comparison if XI ~ Xj whenever YI ~ Yj. Letthe set of all non-negative integers less than or equal to

Wedenote by S,. the set of all permutations of [n] and itsIinality is obviously nL Let 1( E S,. and r E Sk be twonutations, k ~ n. We say 1( contains r if there exists kgers I ~ i1 < i2 ... ik ~ n such that subsequence Xi ••• Xi is

I k

ewise comparison to T, in such context r is usually called am. We say that 1! avoids T, or 1( is t-avoiding, if such

tequence does not exist. The set of all r-avoidingnutations in S,. is denoted by S,.( r) and sJ r) is its~ty. For an arbitrary finite collection of patterns T, we1C avoids T if 1!avoids any t E Sk; the corresponding subset~ is denoted by SJ1) while sJ1) is its cardinality. Forhples, let T= {l23,231,1324} is a set of patterns. Clearlyrutation 1234567 j! S~1) since it contains 123,rutation 652341 ~ S'-.1) since it contain 234 which isewisecomparison to 123 (and also 231 and 341 which are

piecewise comparison to 231), while permutation 4321 E S4(1)since it not contain any subsequence which is piecewisecomparison to any pattern of T. Also si123) = 5 becauseSi123) = {132, 213, 231, 312, 321}.

Fundamental questions about pattern-avoiding permutationsproblems are:

I. to determine sJ1) viewed as a function of n for given T,2. to find an explicit bijection (a one-to-one and onto

correspondence) between SJ1) and SJT,) if S,.(1) =sJT,), and

3. to fmd relations between SJ1) and other combinatorialstructures.By determining sJ1) we mean finding explicit formula, or

ordinary or exponential generating functions. From theseresearches, a number of enumerative results have been proved,new bijections found, and connections to other fieldsestablished.

Problems of pattern avoiding permutations appeared for thefirst time when Knuth [5], in his text book, posed a sortingproblem using single stack. This problem actually is the 312-patterns avoiding permutations. In the other section of hisbook, he showed that the cardinality of all three-length-patterns-avoiding permutations is the Catalan numbers.Investigations on problems of pattern avoiding permutationsthen become wider to some set of patterns of length three, four,five, and so on, some combinations of these patterns,generalized patterns, and permutations avoiding some patternswhile in the same time containing exactly a numbers of otherpatterns.

Pattern avoiding permutations have been proved as usefullanguage in a variety of seemingly unrelated problems, fromtheory of Kazhdan-Lusztig polynomials, to singularities ofSchubert varieties, to Chebyshev polynomials, to rookpolynomials for a rectangular board, to various sortingalgorithms, sorting stacks and sortable permutations [4],statistic permutation [6], also in practical application such as oncryptanalysis (see [7] for example).

The first systematic study of patterns avoiding permutationsundertaken in 1985 when Simion and Schmidt [9] solved theproblem with patterns come from every subset of S3. The ideaof this paper is the following propositions,

Proposition 1 (see [9]) The number of (123,132)-avoidingpermutations in S,., n ~ I is s,.(l23,132) = 211-1.

17 http://sites.google.comlsite/ijcsis/ISSN 1947·5500

f. Let K e 8,,(123,132). If x;. = n then K= (n-IXn-2) ...ln.= n then KI> ~ > ...> 4.1 in order to avoid 123; on the

~ band, in order to avoid 132, "i> (n-k) if i < k. Hence, .Cj

.j for I ~ i ~ k-I, while ""'14t1".x;., must be a (123,132}idingpermutation in 8,..k. Thus, sl(123,132) = I, and for n >In(123,132) = 1+ Lk:~k(123,132). The solution for thisurence relation is: s••(123,132) = 2,..1.0

The cardinality of set 8,,(123,132), as stated by Simion-midt, is the number of elements of B,..(, the set of all binaryags having length (n-I) without any restriction. This paperes (in the next section) constructive bijection between B,..I8,,(123,132). Then, in section 3 we show that this bijectionctually isomorphism. Remark that is not always the case: action between combinatorial classes may magnify theroo: between two consecutive objects. This result allows us, nstruct in section 4 a Gray code for 8,,(123,132). In the

part some concluding remarks are given.

IT. CONSTRUCllVEBIJECTIONBETWEENBn-I AND8n(123,132)

Simion and Schmidt proved that cardinality of set23,132) is 2".1,but the 2"'1 is also cardinality of B".(, set ofbinarystrings of length n-I. Theoretically it must be exists action between 8,,(123,132) and B,..I; here we construct suchijection.

The general pattern of K E 8,.(123,132), as is mentioned insition I, can be described as three parts as,

K = .1CI%2 "'%k-l%k%k+I"'%n-IKn

ch (2) (3)

%j = n, ~ = n-I, ..., Xi.1= Xi.2= I , (eventually empty)

;q=n,

I"'x;. E 8,..i123,132) (also, eventually empty)

example, Figure I is the matrix representation ofmutation6573421 E 8JCI23,132).

•• • •• • •I'C 1. s= 6573421 E 8,(123,132) consist ofthrec part as is mentioned byqoticethat the third part is an element of 8.(123,132), the first stage in the!cationof It= 6573421 as element of 8,(123,132) recursively using (1).

(LlCSIS) Intemational Joumal of ComputerScienceand Informal/on Security,Vol. 9, No. -I, April 2011

For example, Figure I is the matrix representation ofpermutation 6573421 e 8~123,132).

If we trace the terms of Kin (I) from the left to the right, atfirst we will fmd Kl as the second largest term in K(after n). Ifwe remove Kt, then ~ again will be the second largest, and sountil4.I' Next, Xi= n is the largest term of 1C. This tracing andinterpretation is similar for the third part of K until one placebefore the largest term.

Now, we associate K E 8,.(123,132) to s, a binary string oflength (n-I), and assign the largest of Kwhenever we find 1 insand assign the second largest of Kwhenever we find 0 in s. It iseasy to see that this construction is a bijection, so we get thefollowing proposition:

Proposition 2 For each n ~ I, there exists a constructivebijection betweenB".1 and 8,.(123,132).Proof. Let 3 = SI32... s••e B,..I' We construct its correspondingXE 8,,(123,132) by determining 1Cj. I ~ i <n, as follows: if X,»{I, 2, ..., n} - {%h 1liz, ... , 4d, then set:

{kuge3t element in Xi if Si = Ix-

i - second largest element in Xi if Si = 0(2)

and x;. is the single element in XII' For examples, 0000 E B4produces 43215 e 8s(l23,132), 10110 E Bs will produce645312 E 8~123,132), and 010110 E B6 will produce 6745312E ~123,132). 0

Table I shows the set B4 together with its image, the set85(123,132).

(1)

TABLE!. lID: LIST B4 AND ITS IMAGE, 8,(123,132), BY BIJECTION (2).

nmk s, S,(113,132)1 0000 4321S2 0001 432513 0011 43S214 0010 435125 0110 453126 0111 4S3217 0101 4S2318 0100 4S2139 1100 5421310 1101 5423111 1111 S432112 1110 5431213 1010 5341214 1011 53421IS 1001 5324116 1000 53214

Ill. ISOMORPHISMBETWEENBn-I AND8n(123,132)

A graph associated with a combinatorial class is a graphwhere objects of the class act as vertices of the related graph.Two vertices of this graph are connected (or adjacent) if theassociated two combinatorial objects are closed, that is fulfill apredetermined condition(s), usually in the term of Hamming

18 http://sites.google.comlsite/ijcsis/ISSN 1947-5500

Fees. Two graphs G and H are said to be isomorphic ifis a bijection qJ such that (u,v) is an edge in G if and only

9(u), 9(v» is an edge inH.

Before exploring the graph associated with thesbiaatorial classes B".I and S,,(123,132) and showing the~orphism between the two graph, we define the closeness\>ertiesof two elements of B".I and S,,(123,132) and then

a theorem concerning the isomorphism.

NnitioD 1Two binary strings B,..I are closed if they differ in a singleposition.Two permutations in S,,(123,132) are closed if they differby a transposition of two terms.

rem 1The bijection (2) is a combinatorial isomorphism,is, two binary strings in B,..I are closed if and only if theires in S,,(123, 132) under this bijection are closed.

/Of. Let x and x' be two elements of B,..I which differ atltioni, and also, without loss of generality, let x, = 1, and:

X = xl"xl-llO ...Olxj+I...x,..1x = xl"xI-IO<L.Olxj+I...x,..1

the contiguous sequence of Os: X/+I= X/+I= ... = Xl-I = °tuallyempty.• If Xj until'%"'l is °then A;, = (m-I) for xand m for 1C'.

• Let m be the largest element in Xl as is mentioned in(2). Let ;r; 1C' E S,,(123,132) the images of x and x' bythe bijection (2), clearly J!i= m, Ri+1= (m-2), and so on,while XI' = (m-I). XI+I'= (m-2), and so on. Then theshapes of xand xare:

X = XI JIl_Im(m-2) ... (m-j+i+l) (m-l) ~I'" A;,.1,t;,

!c'= 8) Ri-I(m-I) (m-2) ... (m:i+i+l)m ~I'" 41;r;.

casefor x, = ° is similar. 0

Since(3) is cyclic, we can draw an (n-l)-cube graph of B,..Ialsowe can find at least a Hamiltonian cycle in the graph.since(2) is an isomorphism, we also can draw a congruentof S,,(123,132) and also can fmd the Hamiltonian cycle.2 shows the two graphs for n = 4 together with one of

Hamiltonian path.

:!lll

Hr'-;·I(t~:I. t;l?jI = ~iG(lhl)

2. Isomorphism between graph 83 and graph S4(123,132). Thisalso shows a Hamiltonian cycle in each graph, as is indicated by the. Notice that the Hamihonian path in S4(123,132) is the isomorphic

ofthe path in 83

(UCSIS) International JQUTnQ/ ofC omputer Science and lnfonrtalion Security,Vol. 9, No. 4, April 2011

IV. GRAY CODE FOR Sn(123,132) AND TIlE HAMMING

DISTANCES

A binary string is a string over a binary alphabet, {O,l}.The set of binary strings of length p codes the set of non-negative integers over closed interval [0, 2l'-I]. For example,set of all 3 length binary strings is {OOO,001, 010, 011, 100,101, 110, Ill} and represents set of all non-negative integersless than or equal to 7, the all non-negative integers over theclosed interval [0, 23_1].

A Gray code for binary strings is a listing of all p length pbinary strings so that successive strings (including the first andlast) differ in exactly one bit position [8]. The simple and best-known example of Gray code for binary strings is binaryreflected Gray code which can be described the followingrecursive definition:

{&

B = -P O.Bp_1 Dl.Bp-1

p=O

p~1(3)

where e is empty string, a .B is the list obtained byconcatenation a to each_string of B , D is concatenationoperator of two lists, and B is the list obtained by reversing B.Fist(Bp) = if since it is constructed by recursivelyconcatenation 0 to e and so on inp times, while Last(Bp) ':. 1crisince it just concatenation 1 to First(Bp-I) and since Last( Bp) =First(Bp). For examples, BI = to, I}, ~= tOO,01, 11, IO}, andB3 = {OOO,001, 011, 010,110, Ill, 101, lOO}.

Since the first and last elements of Bp also differ in one bitposition, the code is in fact a cycle. Generating of (3) can beimplemented efficiently as a loop free algorithm (1]. Note that,since a binary Gray code is a cycle, it can be viewed as aHamilton cycle in the n-cube.

Existence of at least a Hamiltonian cycle in the graph ofS,,(123,132), as is showed in the last part of the previoussection, is an indication that there is at least a Gray code forS,,(123,132). Since there is a bijection between B".I andS,,(123,132), here we construct a Gray code for S,,(123,132). Byconsidering bijection (2), Gray code Bp (3) is transformed intofollowing Gray code for S,,(123,132):

{

{I}

Sn(l23,132) = (n -1)~S:_1(123,132) 0

n ,Sn-l(123,132)

n =1

(4)

n~2

where S:_1(123,132) is S".I(123,132) after replacing (n-l) withn. This replacement is taken place since 0, which is the prefixto the first part of (3), is associated to (n-l), the second largestelement as is mentioned in (2). Hence (n-l) must be prefix tothe second part of (4). For examples, SiI23,132) = {12, 21},S3(123,132) = 2·{13, 31} 0 3·{l2, 21} = {213, 231, 321, 312}.Table 1. shows the list of B4 together with its image, the list ofSs(123,132).

19 http://sites.google.com/site/ijcsis/ISSN 1947-5500

The recursively properties of (4) imply First(S,.(123,132» =Xn-2)...ln. In the other hand, since Last(S" _ 1(123,132» =

rt(S••.I(123,132», so Last(S" (I23,132»must be n{n-I}(n-I(n-I).

sitiOD J. The Hamming distance between twosecutive elements of S,.(123,132) is 2 and, except betweenfirst and the last, the two different terms are adjacent.of. For n = 2 the Hamming distance is between 12 and 21ch is 2. For n > 2, Hamming distance between twosecutiveelements of S,.(123,132), except between the firstlast elements, is determined recursively by the distance insmaller list, and so OD, and finally by the distance in23,132) which is 2. Concatenating (n-I) and n,~tively to the two parts of (4), of course will not changeHammingdistance values in each part. Also, replacing (n-

vith n in S:'I(123,132) will not change the Hammingmce between each its two consecutive elements. So we

must to check the Hamming distance betweent«n-I).S:'I(123,132» and First(n.SIt-I (I 23,132», asew:It«n -I ).S:'I (123,132»

= (n-I).Last(S:'1 (123,132»= (n-l)·n·Last(Slt-2(l23,132»

,t(n·S ••..1(123,132»

= n .First(Slt-l (l 23,132»= n·(n-I)·Last(S"_2(123,132»

Irly the Hamming distance between

1(123,132» and Firsttn-S It-l(123,132»cent.o

Last(n-I)·is 2 and

Hamming distance between the first and the lastent of Si(123,132) is also 2, but the two terms are parted2) other terms since the first element is the image of 0,..1,

cly (n-IXn-2)...ln, while the last is the image of lon-2,

lyn(n-2Xn-3)...1(n-3).

V. CONCLUDING REMARKs

somorphism between graph of B,..I and graph ofb,132) is more simple than isomorphism between graph ofand graph of S,,(123,132,213), where F"_l is the set ofY stringsoflength (n-I) having no 2 consecutive Is. Theructive bijection between F,,_l and S,.(123,132,213)ed by Simion-Schmidt [9]. There is no Hamiltonian cycles case, while Hamming distance between two consecutivemtsof S,,(123,132,213), a Gray code for S,.(123,132,213),~2, as is showed by Juarna-Vajnovszki [3, 2].

(LlCSIS) Intemational Joumo/ o/Compuler Science and InfomtaJion Security,Vol. 9, No . .I, April 2011

REFERENCES

[1) J.R. Bitncr, G. Ehrlich, &DdE.M Rcingold. Efficient genc:ntion of thebilwy reflected Gray code, Communicatio" c(theACM, 19(9):517-521,2008.

(2) A. Juama and V. VajDovazlci. Combinatorial homorphiam BctwCQlFibcmacci Claaaca. Joama! of Discrete Mathematical Sciences andCryptography, D(2), 2008.

(3) Aac:p Juama and Vincent VajnOVllzki. Isomorphism bctwCCIIclaaaescounted by Fibonacci numbers. WordJ 2005, pages 51-62, 2005.UQAM - Canada.

(4) Scrgcy Kitaev and Toufik Mamour. A Survey on Certain PatternProblems. Tcclmical report, University ofKentuclty, 2003.

[5] Donald E. Knuth. The Art c( Programming, volume I. Addiaon Wesley,Reading Masaachusctla, 1973.

[6] M Bamabci, F. Bonctti, and M Silimbani. The Dcaccnt StatUtic on123-Avoiding PmnutatiOlUl. Semmaire Lothari71gien de Combinatoire,(63), 2010.

(1) Nioolaa T. CourtiJ, Grcgory V. Brad, Shaun V. Ault Statistics ofRandom Permutation and Ihe Cryptanalysis of Periodic Block Ciphers.J.Math. Crypt., (2): 1-20, 2008.

(8) Carla Savage. A Survey of Combinatorial Gray Code. SIAM Review,:60S-629,1997.

[9] Rodica Simion &DdFrank W. Schmidt. Restricted Pmnutationa. Europ.J. Combinatoeics, (6):383-406, 1985.

AT.ITHORSPROFILEA. Juarna is a combinatorlist at Faculty of Computer Science and

Information Technology, Gunadarma Univenity, Indonesia. He got hiaPh.D dual degree in Combinatorica from Universitc de Bourgogne-France under supervising of Prof. Vincent Vajoovszki and fromGunadanna Univenity under supervising of Prof. Bclawati Widjaja.Some of his papcn were presented in lOIDeconference such aa Worda-2OOS,CANT-2006, GASCom-2006, and some othc:n are published insome journals or research reports such as CDMfCS-242 (2004),CDMfCS-276 (2006), The Computer Journal 60(5)-2007, Taru-DMSC11(2)-2008.

A.B. Mutiara is a Professor of Computer Science. He is also Dean of Facultyof Computer Science and Information Technology, GunadannaUniversity, Indonesia.

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