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ISBN 978-952-5726-09-1 (Print) Proceedings of the Second International Symposium on Networking and Network Security (ISNNS ’10) Jinggangshan, P. R. China, 2-4, April. 2010, pp. 050-053 An Improved Molecular Solution for the Partition Problem Xu Zhou 1 , and ShuangShuang Huang 2 College of Mathematics and Information Engineering, JiaXing University, JiaXing, China Email: { Zhouxu2006@126.com, huangya_0611@163.com} Abstract—Now the algorithms based on DNA computing for partition problem always converted the elements into binary number and then carry on mathematics operations on them. In this paper, our main purpose is to give an improved molecular solution for the partition problem. Our new algorithm does not need mathematics operations. So the algorithm is easier and the number of the operation is reduced. Besides, the time used in the biological experiment is less than before. In order to achieve this, we design a new encoding method. We also design a special parallel searcher to search the legal DNA strands in the solution space. Index Terms—DNA-based computing; NP-complete problem; partition problem Ⅰ. INTRODUCTION Since Adleman worked out an instance of the Hamiltonian path problem in a test tube just by handling DNA strands [1], numerous researchers have explored efficient molecular algorithms for some other famous NP-complete problems: Lipton et al.[2] and Braich et al.[3] solved the 3-SAT problem, Faulhammer et al.[4] solved the knight problem Ouyang et al.[5] and Li et al.[6] solved the maximal clique problem and so on [8-14]. DNA computer’s power of parallel, high density computation by molecules in solution allows itself to solve hard computational problems in polynomial increasing time, while a conventional turing machine needs exponentially increasing time. Through advances in molecular biology, it is now possible to produce roughly 10 18 DNA strands in a test tube, that is to say 10 18 bits of data can be processed in parallel by basic biological operations [15-16]. In this paper, we describe a new algorithm to solve the partition problem [17]. Since Huiqin’s paradigm proposed in 2004 demonstrated the feasibility of applying DNA computer to tackle such an NP-complete problem. Instead of surveying all possible assignment sequences generated in the very beginning, we use the operations of Adleman-Lipton model and the solution space of sticker which is proposed by Chang et al., then apply a new DNA algorithm for partition problem [9,10]. The paper is organized as follows. Section 2 introduces the Chang et al.’s model and the definition of the partition problem. Section 3 introduces the DNA algorithm to solve the partition problem for the sticker solution space. In section 4, the experimental results by © 2010 ACADEMY PUBLISHER AP-PROC-CS-10CN006 50 simulated DNA computing are given. Conclusions and future research work are drawn in Section 5. Ⅱ. DNA MODEL OF COMPUTATION A.The Chang et al.’s model Chang et al. presented the model that took biological operations in the Adleman–Lipton model and the solution space of stickers in the sticker-based model [9,10,18].This model has several advantages from the Adleman–Lipton model and the sticker-based model: 1. The new model has finished all the basic mathematical functions and the number of tubes, the longest length of DNA library strands, the number of DNA library strands and the number of biological operations are polynomial. 2. The basic biological operations in the Adleman– Lipton model had been performed in a fully automated manner in their lab. The full automation manner is essential not only for the speedup of computation but also for error-free computation. 3. Chang and Guo also employed the sticker-based model and the Adleman–Lipton model for dealing with Cook’s theorem, the dominating-set problem, the setsplitting problem and many other NP complete problems for decreasing the error rate of hybridization. If given a tube, one can perform the following operations in Adleman–Lipton model: Extract; Merge, Detect, Discard, Amplify, Append and Read. B. Definition of the partition problem The partition problem is a classical NP-complete problem in computer science [17]. The problem is to decide whether a given multiset of integers can be partitioned into two "halves" which have the same sum. More precisely, given a multiset A of integers, is there a way to partition A into two subsets A 1 and A 2 such that the sums of the numbers in each subset are equal The subsets A 1 and A 2 must form a partition in the sense that they are disjoint and they cover the set A. On the assumption that a finite set A is {a 1 , a 2 …. a m }, where a i is the ith element for 1≤ i ≤m. Also suppose that every element in A is a positive integer. Assume that |A| is the number of element in A and |A| is equal to m.The partition problem is to find a subset S 1 ⊆ A such that the sum of all the elements in S 1 is half of S, where S is the sum of all the elements in A.
Ⅲ. AN IMPROVED DNA ALGORITHM FOR THE PARTITION PROBLEM A. Sticker-based solution space for the partiton problem This algorithm for construction the sticker-based solution space for subsets of a finite set is similar to the Michael’s algorithm at 2005[13]. Being similar to the Knapsack Problem[19],we will construct the solution space for subsets of A. Assume that x 1 ...x n is an n-bit binary number, which is applied to represent n elements. Assume that the sum of all the elements in A is S. Procedure Solution_Generator(T 0 , n) 1: For m = 1 to n 2: Amplify(T 0 , T 1 , T 2 ). 3: Append(T 1 , x 1 m). 4: Append(T 2 , x 0 m). 5: T 01 = ∪ ( T 1 , T 2 ). 6: EndFor EndProcedure From Solution_Generator(T 0 , n), it takes n amplify operations, 2n append operations, n merge operations and three test tubes to construct sticker-based solution space. A n-bit binary number corresponds to an array of input. A value sequence for every bit contains 15 bases. Therefore, the length of a DNA strand, encoding a subset, is 15n bases consisting of the concatenation of one value sequence for each bit. B. Sticker-based solution space for elements of subsets for a finite set An element a i , in A can be converted as a number y i,1 y i,2 …y i,k , for k = a i . Sticker is applied to represent every bit y i,j for 1≤ j ≤ k .For every bit y i, j , two distinct DNA sequence were designed. One corresponds to the value “0” for y i,j and the other corresponds to the value “1” for y i,j . For the sake of convenience in our presentation, assume that y 1 i, j denotes the value of y i, j to be 1 and y 0 i, j denotes the value of y i, j to be 0.The following algorithm is employed to construct sticker-based solution space for elements of 2 n possible subsets for a n-element set A. Assume that S is the sum of all the elements in A. Procedure Make_Value(T 0 , n) 1: For i = 1 to n 2: T 1 = +( T 0 , x i 1 ) and T 2 = -( T 0 , x i 1 ). 3: If (Detect(T 1 )= ‘yes’ ) then 4: For j = 1 to ai 5: Append(T1 , y1i, j). 6: EndFor 7: If (Detect(T 2 )= ‘yes’ ) then 8: For j = 1 to a i 9: Append(T 2 , y 0 i, j). 10: EndFor 11: EndIf 12: T 01 = ∪ ( T 1 , T 2 ). 13: EndFor EndProcedure From Make_Value(T 0 , n), it takes n extract operations, n detect operations, S append operations, n merge operations, and three test tubes to construct sticker-based solution space for elements of 2 n possible subsets to a n-element set A. n-bit binary number corresponds to a subset and the longest bit of binary number, which are used to encode the values of all the elements in A , is S . Assume that a value sequence for every bit contains 15 bases. Therefore, the last length of a DNA strand, encoding elements of 2 n possible subsets to a n-element set A, is 15(n+S). C. The DNA based algorithm to finish the Logic Multiplication operation functions In order to find the legal strands of the partition problem, a parallel searcher is designed. Procedure Parallel_Searcher(T 0, S/2) 1: For i = 0 to S -1 2: t = min{S/2, i}. 3: For j = t down to 0 4: T ON j+1 = + (T j , y 1 i+1) and T j =-(T j , y 1 i+1). 5: T j+1 = ∪ (T j+1 , T ON j+1). 6: EndFor 7: EndFor 8: If (Detect(T s/2+1 ) =“yes” ) 9: Discard(T s/2+1 ). 10: EndIf EndProcedure From Parallel_Searcher (T 01 , S/2), it takes S(S+2)/8 exact operations, S(S+2)/8 merge operations, one detect operation, one discard operation and S/2+2 test tubes where S is the sum of all the element in the set A. The length of a DNA strand has never been changed, is 15(n+S). D. An improved DNA based algorithm for the partition problem Algorithm1. An improved DNA based algorithm for the partition problem 1: Solution_Generator(T 0 , n). 2: Make_Value(T 0 , n). 3: Parallel_Searcher(T 0, S/2). 4: If ( Detect(T S/2 ) =“yes” ) 5: Read(T S/2 ). 6: EndIf Theorem 1: From those steps in Algorithm 1, the improved DNA based algorithm for the partition problem can be solved. Proof: On the execution of Line(1), it calls Solution_Generator(T 0 , n). The algorithm, Solution_ Generator(T 0 , n), is mainly used to construct solution space for 2 n possible subsets of an n-element in the set A. Line (2) calls Make_Value(T 0 , n).The algorithm, Make_ Value(T 0 , n) is employed to construct sticker-based solution space for element of 2 n possible subsets of an n-element in the set A. Line(3) is called Parallel_ Searcher(T 0, S/2). The algorithm, Parallel_ Searcher(T 0, S/2) is mainly used to search the legal DNA strands for the partition problem. 51
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Proceedings The Second Internationa
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Table of Contents Message from the
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Zuming Xiao, Zhan Guo, Bin Tan, and
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Message from the Symposium Chairs T
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Strong earthquake 0.1< M L
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indirect causes, and the logical re
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egression. Granger causality test i
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B. Evaluation We have evaluated thi
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used as a source and neighboring ce
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Figure 3. Drainage networks generat
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Combinational logic unit failures i
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Tabal.1 Combinational fault logic t
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under the endorsement of both the m
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TABLE I. DESCRIPTION OF PROPOSITION
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Figure 2. The process of the invers
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Figure 9. (a)the original image.(b)
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In order to reduce to the number of
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that studying being going to be to
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Reference[10] analyzed the evolutio
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module, communication module, apper
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After the comprehensive performance
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system testing can be seen that the
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III. AUTONOMIC RESOURCE ALLOCATION
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The QoE i (T i ) is the ith user’
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nodes will be formed one cluster, t
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Where n is the number of data point
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Keyboard event Application User mod
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SQL Azure will eventually include a
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The nodes in the suffix tree are dr
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[3] Y. Li, S. M. Chung, and J. D. H
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some drilling fluid produces hydrog
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"normalization", whose membership b
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and minimum structural elements in
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esults of the spatial transform par
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B. Research on protocol actions Com
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ACKNOWLEDGMENT This work is funded
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A. Problem Description In a small b
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[4] Huang, Hung, and J. Y. jen Hsu.
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Further by calculating, the followi
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TABLE II. THE CONCENTRATION BETWEEN
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private key that obtained by using
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ontology, and the domain dictionary
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Eq.4 is NP-hard and can be solved b
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Where: G i is the selection field o
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If there is X j in a generation, wh
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Theorem 2.4([10]). Let L 1 and L 2
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The relation between the lattice im
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Figure 2. Example of single-step de
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evocation and the fourth group stor
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focused mainly on rule-based forms
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Ⅵ. CONCLUSION Figure 6. The Flask
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EDCF in comparison with DCF, has so
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B. Simulation results and analysis
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in routing table. When search resou
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others through the selective emotio
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such as weather information, techno
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the opening window, a related page
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1) Process context storage areas. I
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V. CONCLUSION In this thesis, sCPU-
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main types of horizontal search env
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Enterprise Portal security provides
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() t = [ S () t S () t ] T N S ,...
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[2] Kumaravel, N., and Kavitha, V.,
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output, so BP network has been wide
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input to the artificial neural netw
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(2) In a process (tokens from outsi
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The reduction process consists of t
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all eight normal vectors are identi
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is B object , and the number of emp
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xi, j y' = εα ( (1 + tanh( )) −
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Error 8000 7000 6000 5000 4000 3000
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Architecture (AMBA) a new bus archi
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In order to ensure the smooth proce
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In this paper, we adopt two Sobel o
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four layers.The far right of the gr
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TABLE II. OPERATIONAL EMPLOYEE TABL
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A. Object-relational Type To audit
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to execution time. Also, DBA would
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Liping Chen .......................
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