International Journal of Advanced Computer Science, Vol. 1, No. 3, Pp. 106-109, Sep. 2011.

Manuscript Received: 25, Aug., 2011 Revised: 13, Sep., 2011 Accepted: 25, Sep., 2011 Published: 30, Sep., 2011 Keywords DNA Algorithm, Isomorphic, graph, Adjacency matrix.

Abstract Finding the isomorphic graph is the problem that have algorithms with the complexity time. For this in general, because of classification for algorithms time complexity, this solution stay in NP-complete group. In this essay tried to show the new style for DNA algorithms, until decrease the solution of the problem in vast. This style heel show to exploit of graphs Adjacency matrix representation, to investigate the isomorphic of graphs in proximity matrix by the method of using molecule model and DNA operation to be investigate and determine this isomorphic.

1. Introduction In the recent decade, the collection of complexity

problem that didn’t solve with general silicon computers solved with a lot of problems. It was the preface, for the researchers until they find the new and optimum solution for these problems. This style that created with the inspiration of nature, with decrease of complexity, by this problems, trying to solve them. Among all can name, base on calculation for DNA [1]. However the accounting styles don’t noted outspread yet[1]. But the collection of this problems like Boolean satisfiability problem (SAT) [3,4,5,6], Maximum click , Hamiltonian path problem ,Tree coloring , and etc solved by that[7].

The computers that is operate on the basis of this style, famous to Biomolecule processor that remanded like previous accounting machines models, for example, Turning machine [8], Splicing system [9] and Boolean circuit [10] and etc. in 1994, Adleman, after to spend period study in genetic sciences[11] with the description of new idea to raise the suggestion of constructing the processor machine, that with the use of the DNA molecule operation solve the Hamiltons path problem in the group of NP-complete[2,7,11]. This style was the first solution that Adleman succeed to show with the using of DNA molecules This work was supported by the Islamic Azad University, lashtenesha –Zibakenar branch, Iran I. Mehdi Eatemadi, Department of Electronics, lashtenesha–Zibakenar branch, Islamic Azad University, Iran (�Mehdi_eatemadi@gmail.com ).Tel: +989112366173 II. Ali Etemadi Department of Electronics, Chabahar University (AliEtemadi.ai@yahoo.com) III. Mohammad-Mehdi Ebadzadeh, Amir Kabir University of Technology (Ebadzadeh@gmail.com)

generally this style cause creating new way for researcher until they solve another problems with this style in recent years.

In this essay we use DNA to show new solution for isomorphic graph. For this with using of sticker model suggestion, solution is planning. In the continuation of essay after the second, third and fourth part to allocate to problem theoretical description in the fifth parts, suggestion algorithms to show and at the end to show adding and concluding.

2. Graphs Two graphs G1(V,E), G2(V,E) that each of them

containing collection of vertices and the collection of edge describe like below:

In theory graphs each edge to consider like path between two vertex that can describe non direction or directional. In this essay graphs are non directional. In non directional graphs each Edge is one, two disorderly of vertices. If to show the collection of graphs edge with double collection of vertices can use of one matrix, to show graphs. Meantime square matrix n*n (n is the number graphs vertices) have build of their communication. This matrix shows all the communicative between vertices that created with edge and to unit equal entry between vertices, their name is no adjacency matrix[12].

3. Isomorphic Graph when we say two graph are isomorphic, first, the number

of their vertices are similar to each other. Second like the number of their vertices and edge, are able to equivalent as in exchange for each vertices and edge in the first graph, one vertex and edge with the same benefits to exist in the second graph. This problem in the finding of isomorphic graph cause to contrast each vertex of first graph with another vertices of second graph. As require; cause to increase of complexity until n!. This complexity is in the worst case and show with O(n!). In the showing of graphs with adjacency matrix, instead of searching each first graphs vertex, in the collection of vertices of adjacency matrix in the kind of second graph of adjacency matrix (in exchange of n, the situation of label allocation). As if the

Finding the isomorphic graph with the use of algorithms based on DNA

Mehdi Eatemadi, Ali Etemadi, & Mohammad-Mehdi Ebadzadeh

Eatemadi et al.: Finding the isomorphic graph with the use of algorithms based on DNA.

International Journal Publishers Group (IJPG) ©

107

conclusion of the search was positive two graph are isomorphic if not they are un isomorphic graph[13].

4. Sticker Models This is model is famous as memory in DNA computer

[14] with notice to its ideal construction to stop binary memory. Each strand build of smaller and similar cell. As the duty of each cell is the protection of situation or one. As molecules in one cell have the one situation that one supplement sticker of that cell in the connect with cell nuclear basis is stay. Assign the area of molecules that strand is two string. The position is one and in another area the situation is zero. Function of usable in this model consist of[14,15,21]:

For changing Bit to 1 refer to K address, Set (N,K). For changing Bit to 0 refer to K address, Clear (N,K). For staying content N+ ,N- in N, Merge (N,N+,N-). For copy N in {N.,..Nn}, Copy(N, {N.,..,Nn}). For to be separately N contents base on amount Bit kth in the causes N in two tube N+ and N-,Separate (N,K,N+,N-). For remove Ni in N, Remove(N,Ni).

5. Suggested Algorithem

G1 and G2 are two graphs that we consider their isomorphic must be investigated the two first condition for starting the problem is two graphs must be non directional. the number of their vertices are equal. With notice to second condition, the label that is used in the G1 graphs vertices collection as G2 graphs vertices collection. Now with use of adjacency matrix, we have two square matrix for showing the graphs with the n*n distance,

Fig. 1 G1 and G2 adjacency matrices .

Now for accidental label, the first graph vertex with available quantity in V(G1), for each label belonging to one vertex for the label of second graph with the use of label belong in V(G1) and we are facing with allocation label to nvertex that equal with permutation of object to n place and mean[22]:

In investigating adjacency matrices refer to n! type and this the position the be obtain in the second graph. Tow joint property is obvious. matrices are reflective, The

collection of row and column in this matrices are fixed. Just their place in matrices are different. The first quality for graphs is their non directional reason that adjacency matrix will be matching for, this condition in graph theory is providing[12]. In fact the second quality, represent another meaning of permutation, that each label with notice to that vertex is allocate to that. Row (or column) are dislocated. Matrix row (or column) in surface row (or column) in the witness (or length). In fact n place in row (or column) of allocation, is the amount of n row (or column) of subject as if the position of n in allocation of this row (or column) is to n place matrix in the row (or column) that equal (the same n case) in allocation label in n vertex. For reaching to the answer we follow 4 step:

Step1 The structure of the graphs representation model. Step2 Present all of the space case in the first step. Step3 Separate the kind of correct case. Step4 Assessment and announce the answer.

In first step we describe DNA molecule for representation problem in Sticker model. In that we consider one strand of molecule that consist of n2 (n is the number of vertices of each graph) of cell as if each of 12 A,T,C,G nucleotide column that makes like below.

Fig. 2 An example of sticker.

This branch that describe in Figure2 separate to n part and each part to n cell. As if each cell is equivalent one entry of adjacency matrix can protect zero or one. So each part of that according with one row of matrix.

1th bit 2th bit (n2-1) th bit n2 th bit

Fig. 3 An example of strand.

In second step we product all of the case of zero or one to DNA molecules cells and this mean, product different 2(n*n) that product by below algorithm.

Function Production (N0 , N+ , N-)N+ = N-={}

K=1 For i=1 to n2

Copy(N0 , N+ , N-)Set(N+,K)

Clear(N-,K) K=K+1 Merge(N0 , N+ , N-)

End For N+ = N-={} Return(N0)End

Fig. 4 Product all of the condition of 0 or 1 to DNA molecules.

In the end of algorithm, tube N0, is containing all of the case of allocation 0 or 1 to 2(n*n) of different place of

International Journal of Advanced Computer Science, Vol. 1, No. 3, Pp. 106-109, Sep. 2011.

International Journal Publishers Group (IJPG) ©

108

molecule. We should extract, molecules that be elected n!different case of matrix between this case. For this purpose we obtain first graph adjacency matrix that we accidentally label it and determine the molecules like it. Now with notice to n! collection of adjacency matrix that from permutation is in the n! label in second graph have two important quality that mentioned before so, we accidentally, label second graph and obtain adjacency matrix, now with notice to second quality (the amount of matrix row and column in n! different case is fixed.) The collection H so its factor equal G2 graph row. That we describe like below:

So, this matrix that is the conclusion of n permutation of H collection to n place of adjacency matrix is the right answers, if it be symmetrical. With obtain the first condition, that determine of H collection permutation, then the symmetrical in obtain matrix is sure. Because H from non directional graph so that displacement factor according to second quality don’t exterminate matrix symmetrical, now with using of permutation algorithm, the molecule that is the conclusion of n permutation is number of H collection to n adjacency matrix place is separate below:

Function Selection1(N0)

For j=1 to n Copy(N0,{N1,…,Nn}) For i=Row1,…,Rown and all k >j ii=Index(i) In parallel do remove(Nii ,{Rowj!=i,Rowk=i) Merge({N1,…,Nn},N0)

End For Return(N0)End

Fig. 5 The DNA computing algorithm to solve the Isomorphic graphs.

After algorithm performance, the molecules that is in tube, is carefully like n! case that is in adjacency matrix (the carefully analysis of algorithm is in [7] reference) for determine the answer, we should search different n! molecule that obtain from n allocation label of V(G1)collection to n vertex of G2 graph. So, if molecules is in the tube, two graph one isomorphic if not are un isomorphic, with finding this answer, the solution of this question be finish.

6. Errors

Generally the DNA operation have less error[16]. But below cases for decreasing this quantity is effective, Useful use of material sequences[16,17], using of protocols[16] that decrease one after another repeated Nucleotide until less of 4 and preventation of creating second base can be the element for decreasing the error[17,18,19,20].

7. Conclusion In this essay with using of DNA algorithms. To show the solution for finding isomorphic graph that is in NP-complete group. so we decrease the complex with the space representation of question with using of DNA structure, base on sticker model and with suggested model that is replacing adjacency matrix and doing three main steps in production answer space correct answers, we reach to correct answer. In study, time complex, algorithm like DNA branch is determining the quality. In worse case the complex from O(n!) (n is the number graphs vertex) to O(n2)(n is the number of molecule bit) is decrease. And this conclusion in contrast with first solution is ideal decrease.

References [1] C.N. Yang & C.B. Yang, “A DNA solution of SAT problem

by a modified sticker model,” (2005) Elsevier , BioSystems,Vol. 81, Pp. 1–9.

[2] W. Li & Y. Ding, “A microfluidic systems-based DNA algorithm for solving special 0-1 integer programming problem,” (2007) Elsevier, Applied Mathematics and Computation, Vol. 185, Pp. 1160-1170.

[3] Corman, T.H., Leiserson, C.E., Rivest, R.L, & Stein, C., Introduction to Algorithms, Boston, second ed. MIT Press, MA, 2001.

[4] Chen, K., Ramachandran, V., Condon, A., & Rozenberg, G., DNA Computing, Springer, pp. 199–208, 2001.

[5] D. Faullhammer, A.R. Cukras, R.J. Lipton & L.F. Landweber, “Molecular computation: RNA solutions to chess problems,” (2000) Proc. Natl. Acad. Sci. U.S.A. 97, Pp. 1385–1389.

[6] H. Yoshida & A. Suyama, “DNA based computers”, In: E.Winfree, D.K. Gifford, (Eds.), (1999) DIMACS Series in Discrete Mathematics and Theoretical Computer Science, American Mathematical Society, Providence, RI, Vol. 54. Pp. 9–20.

[7] Amos M., Theoretical and Experimental DNA Computation, Springer-Verlag Berlin Heidelberg, 2005.

[8] P. Rothemund, “A DNA and restriction enzyme implementation, of turing machines,” (1996) In: L. Landweber, E. Baum, (Eds.) DNA Base Computers, American Mathematical Society Providence, Vol. 27, Pp. 75–119.

[9] G. Paun, G. Rozenberg & A. Salomaa, DNA Computing, New Computing Paradigms. Springer-Verlag, Berlin, 1998.

[10] M. Amos & P. Dunne, “DNA Simulation of Boolean Circuits,” (1997) Tech. Re CTAG-97009, Department of Computer Science, Uni versity of Liverpool.

[11] M. Adleman, “Computing with DNA, The manipulation of DNA to solve mathematical problems is redefining what is meant by computation,” (1998) Scientific American Computing with DNA.

[12] Van Dam, E.R., Haemers, W.H., & Koolen Cospectral J.H., Graphs and the generalized adjacency matrix, Elsevier, Linear Algebra and its Applications, Vol. 423, Pp. 33–41, 2007.

[13] Cormen, T. H., Leiserson, C.E., Rivest, R. L., & Stein, C., Introduction to Algorithms, Third Edition, Hardcover, Sep. 30, 2009.

Eatemadi et al.: Finding the isomorphic graph with the use of algorithms based on DNA.

International Journal Publishers Group (IJPG) ©

109

[14] Roweis, S., Winfree, E., Burgoyne, R., Chelyapov, N.V., Goodman, M.F., Rothemund, P.W.K. & Adleman, L.M., A Sticker Based Model for DNA Computation, Univercity Of Southern California, May 1996.

[15] Ignatova, Z., Martínez-Pérez, I., & Zimmermann Karl-Heinz, DNA Computing Models, Springer, edition 1,July 2, 2008.

[16] H. Ahrabian, M. Ganjtabesh and A. N. Dalini, “DNA algorithm for an unbounded fan-in Boolean circuit,” (2005) BioSystems Vol. 82, Pp. 52–60.

[17] R. Deaton, M. Garzon, R. Murphy, J. Rose, D.Franceschetti and Jr. Stevens, “Reliability and efficiency of a DNA-based computation,” (1998) Phys. Rev. Lett. Vol. 80, Pp. 417–420.

[18] M. Nilsson, H. Malmgren, M. Samiotaki, M. Kwiatkowski, B. Chowdhary and U. Landegren, “Padlock probes: circularizing oligonucleotides for localized DNA detection,” (1994) Science Vol. 265, Pp. 2085–2088.

[19] R.Braich, N.Chelyapov ,C. Johnson, P. Rothemund and L. Adleman, “Solution of a 20-variable 3-sat problem on a DNA computer,” (2002) Science, Vol. 296, Pp. 499–502.

[20] M. Amos, A. Gibbons and D. Hodgson, “Error-resistant Implementation of DNA Computation” Draft Jan 1996.

[21] L. Adleman, “On Constructing a Molecular Computer,” Draft Jan, 11, 1995.

Mehdi Eatemadi received the M.Sc. degree in Electronic Engineering from the University of Arak, Arak, Iran, in 2005. His research interests are computer vision, image Processing, speech processing, DNA computer, neural networks. He is a Professor

of Electronic science at the Islamic Azad University, lashtenesha, Iran.

Ali Etemadi received the M.Sc. degree in Artificial Intelligence (Biocomputing) from the University of Qazvin, Qazvin, Iran, in 2010.His research interests are brain technology, DNA computer, neural networks, genetic algorithms, fuzzy logic

and computer graphic. He is a Professor of computer science at the International University of chabahar , Iran.

Mohammad-Mehdi Ebadzadeh is aAssociate Professor with the Department of Computer Engineering, University of Amir Kabir, Tehran, Iran. His current research interests include Artificial Immune Systems, Evolutionary Algorithms, Neural Network, Artificial Life, Fuzzy Systems,

Data Mining, Multi Agent Systems, Ants Colony, Robotic and Medical Image Processing.