Fast parallel molecular solution to the
dominating-set problem on massively
parallel bio-computing
Minyi Guo
a,*, Michael (Shan-Hui) Ho
b,1,
Weng-Long Chang
b,1aDepartment of Computer Software, The University of Aizu, Aizu-Wakamatsu City,
Fukushima 965-8580, Japan
bDepartment of Information Management, Southern Taiwan University of Technology,
Tainan County, 710, R.O.C. Taiwan
Received 5 April 2004; revised 1 July 2004; accepted 15 July 2004 Available online 25 September 2004
Abstract
This paper shows how to use DNA strands to construct solution space of molecules for the dominating-set problem and how to apply biological operations to solve the problem from the solution space of molecules. In order to achieve this, we have proposed some DNA based par-allel algorithms using the operations in Adleman–Lipton model, together with the analysis of the computational complexity for DNA parallel algorithms.
2004 Elsevier B.V. All rights reserved.
Keywords: Parallel biological computing; DNA-based computing; NP-complete problem; Dominating-set problem; DNA based parallel algorithms
0167-8191/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.parco.2004.07.014
*
Corresponding author. Tel.: 0242 37 2557; fax: 0242 37 2744.
E-mail addresses:minyi@u-aizu.ac.jp(M. Guo),michael@mail.stut.edu.tw(Michael (Shan-Hui) Ho),
changwl@mail.stut.edu.tw(W.-L. Chang).
1Tel.: +886 6 2533131x4300; fax: +886 6 2541621.
1. Introduction
Through advances in molecular biology[1,2], it is now possible to produce 1018or more DNA strands in tube. Those 1018or more DNA strands can also be applied for representing 1018or more bits of information. Biological operations can be used to simultaneously operate 1018or more bits of information. Or we can say that 1018or more data processors can be executed in parallel. Hence, it becomes obvious that biological computing can provide a very huge parallelism for dealing with problems in the real world. Especially, the problems from the NP-complete class are well-known to be exponentially more difficult than evaluating determinants whose entries are merely numerical. It is difficult to solve these kinds of problems even if very mas-sive supercomputers are used when the problem sizes become large.
On the other hand, DNA computers have the full potential of the high perform-ance computing technology. One test tube can be viewed as a processing unit like standard computer architecture. Furthermore, DNA algorithms using biological operations have natural parallelism because DNA strands are separated (melted, an-nealed) in test tubes in parallel.
Feynman[29]first proposed molecular computation in 1961, but his idea was not implemented by experiment for a few decades. In 1994 Adleman [2]succeeded to solve an instance of the Hamiltonian path problem in a test tube, just by handling DNA strands. Lipton[3]demonstrated that the Adleman techniques could be used to solve the satisfiability problem (the first NP-complete problem). Adleman and coworkers[14]proposed sticker for enhancing the Adleman–Lipton model.
In this paper, we use sticker to construct a solution space of DNA for the domi-nating-set problem. Simultaneously, we also apply DNA operations in the Adleman– Lipton model to develop a DNA algorithm. The results of the proposed algorithm show that the dominating-set problem is resolved with biological operations in the Adleman–Lipton model for solution space of sticker. Furthermore, this work pre-sents clear evidence of the ability of DNA based computing to solve NP-complete problems.
The paper is organized as follows. Section 2 introduces the Adleman–Lipton model in detail then this model is compared with other models. Section 3 introduces a DNA algorithm for solving the dominating-set problem for the solution space of sticker. In Section 4, the experimental results of simulated DNA computing are given. Conclusions and future researching works are drawn in Section 5.
2. DNA model of computation 2.1. The Adleman–Lipton model
It is cited from[16]that DNA (DeoxyriboNucleic Acid) is the molecule that plays the main role in DNA based computing. In the biochemical world of large and small molecules, polymers, and monomers, DNA is a polymer, which is strung together from monomers called deoxyriboNucleotides. The monomers used for the
construc-tion of DNA are deoxyribonucleotides, where each deoxyribonucleotide contains three components: sugar, phosphate group, and nitrogenous base. Sugar has five car-bon atoms—for the sake of reference there is a fixed numbering to them. The base also has carbons, so to avoid confusion the carbons of the sugar are numbered 10 to 50 (rather than 1–5). The phosphate group is attached to the 50 carbon, and the base is attached to the 10carbon. Within the sugar structure there is a hydroxyl group attached to the 30carbon.
Due to[1,16], distinct nucleotides are detected only from their bases, which come in two types: purines and pyrimidines. Purines include adenineand guanine, abbrevi-ated A and G. Pyrimidines contain cytosine and thymine, abbreviabbrevi-ated C and T. Since nucleotides are only distinguished from their bases, they are simply represented as A, G, C, or T, depending upon the type of base they have. The structure of a nucleotide, cited in[16], is simplified inFig. 1. In this figure, B is one of the four possible bases (A, G, C, or T), P is the phosphate group, and the rest (of the ‘‘stick’’) is the sugar base (with its carbons enumerated 10 through 50).
It was indicated from[1,11,16]that nucleotides could link together in two ways. Firstly the 50-phosphate group of one nucleotide is joined with the 30-hydroxyl group of another forming a phosphodiester bond. The resulting molecule has the 50 -phos-phate group of one nucleotide, denoted as 50end, and the 30-OH group of the other nucleotide is available for bonding, denoted as 30end. This gives the molecule direc-tion, and we can talk about the direction of 50end to the 30end or 30end to the 50 end. The second way is that the base of one nucleotide interacts with the base of an-other to form a hydrogen bond. This bonding is based on pairing: A and T can pair together, and C and G can pair together—no other pairings are possible. This pair-ing principle is called the Watson–Crick complementarity (named after J.D. Watson and F.H.C. Crick who deduced the famous double helix structure of DNA in 1953, and won the Nobel Prize for its discovery).
According to[1,11,16], a DNA strand is essentially a sequence (polymer) of four types of nucleotides detected by one of the bases they contain. Two single strands of DNA under appropriate conditions can form a double strand, if the respective bases are the Watson–Crick complements of each other—A matches T, and C matches G; also the 30end matches the 50end. The length of a single stranded DNA is the num-ber of nucleotides comprising a single strand. Thus, if a single stranded DNA in-cludes 20 nucleotides, we can say that it is a 20 mer (it is a polymer containing 20 monomers). The length of a double stranded DNA (where each nucleotide is base paired) is counted in the number of base pairs. Thus if we make a double stranded DNA from a single stranded 20 mer, then the length of the double stranded DNA is 20 base pairs, also written 20 bp. (For more discussion of the relevant biological background refer to[1,11,16].)
In the Adleman–Lipton model[2,3], splints were used to construct the correspond-ing edges of a particular graph of paths, which represented all possible binary num-bers. As it stands, their construction indiscriminately builds all splints that lead to a complete graph. This is to say that hybridization has a higher probability of errors. Hence, Adleman and coworkers[14]proposed the sticker-based model, which was an abstract of molecular computing based on DNA with a random access memory as well as a new form of encoding the information
The DNA operations in the Adleman–Lipton model[2,3,11,12]are described be-low. These operations will be used for figuring out solutions of the dominating-set problem.
The Adleman–Lipton model:
A (test) tube is a set of molecules of DNA (i.e. a multi-set of finite strings over the alphabet {A, C, G, T}). Given a tube, one can perform the following operations: 1. Extract. Given a tube P and a short single strand of DNA, S, produces two tubes
+(P, S) and (P, S), where +(P, S ) is all of the molecules of DNA in P which contain the strand S as a sub-strand and (P, S) is all of the molecules of DNA in P which do not contain the short strand S.
2. Merge. Given tubes P1 and P2 yield [(P1, P2), where [(P1, P2) = P1[ P2. This operation is to pour two tubes into one, with no change in the individual strands. 3. Detect. Given a tube P, we have ÔyesÕ if P includes at least one DNA molecule, and
we have ÔnoÕ if it contains none.
4. Discard. Given a tube P, the operation will discard the tube P.
5. Read. Given a tube P, the operation is used to describe a single molecule, which is contained in the tube P. Even if P contains many different molecules each encod-ing a different set of bases, the operation can give an explicit description of exactly one of them.
2.2. Comparison of the Adleman–Lipton model with other models
Techniques in the Adleman–Lipton model could be used to solve the NP-com-plete Hamiltonian path problem and satisfiability (SAT) problem by linearly increas-ing time and exponentially increasincreas-ing volumes of DNA [2,3]. Quyang et al. [4]
showed that restriction enzymes could be used to solve the NP-complete clique prob-lem (MCP). The maximum number of vertices that can be processed is limited to 27 because the size of the pool with the size of the problem exponentially increases[4]. Arito et al. [5] described new molecular experimental techniques for searching a Hamiltonian path. Morimoto et al. [6] offered a solid-phase method to finding a
Hamiltonian path. Narayanan and Zorbala [7] proved that the Adleman–Lipton
model was extended towards solving the traveling salesman problem. Shin et al.
[8] presented an encoding scheme that applies fixed-length codes for representing integer and real values. Their method could also be employed towards solving the traveling salesman problem. Amos[13]proposed a parallel filtering model for resolv-ing the Hamiltonian path problem, the sub-graph isomorphism problem, the
3-ver-tex-colorability problem, the clique problem and the independent-set problem. In
our previous work, Chang and Guo [17–20,25] proved how the DNA operations
for solution space of splint in the Adleman–Lipton model could be employed for developing DNA algorithms to resolve the dominating-set problem, the vertex cover problem, the clique problem, the independent-set problem, the three-dimensional matching problem, the set-packing problem, the set cover problem and the problem of exact cover by 3-sets.
Roweis et al.[14]proposed sticker-based model to enhance the Adleman–Lipton model. Their model could be used for determining solutions to the set cover problem. Perez-Jimenez et al. [15] employed sticker-based model [14] to resolve knapsack problems. Fu [21]proposed new algorithms to resolve 3-SAT, 3-Coloring and the independent set. In our previous work, Chang et al.[26–28]also employed the stick-er-based model and the Adleman–Lipton model for dealing with the subset-sum problem, CookÕs theorem[9,10]and the set-splitting problem for decreasing the error rate of hybridization.
3. Using sticker for solving the dominating-set problem in the Adleman–Lipton model 3.1. Definition of the dominating-set problem
Mathematically, a dominating set of a graph G = (V, E), where V is the set of the vertex and E is the set of the edge, is a subset V1 V of vertices such that for all u2 V V1
there is a v2 V1
for which (u, v)2 E[9,10]. The dominating-set problem is to find a minimum size dominating set in G. This has been proved to be a NP-com-plete problem[10].
The dominating-set problem asks: Given a network consisting of n vertices and m edges, how many vertices are in a minimum size dominating set? The graph includes three vertices and two edges as shown inFig. 1, where each circle in the figure rep-resents a vertex and the arc connecting two circles reprep-resents an edge. The minimum size dominating set for the graph inFig. 2is {v1}. Hence, the size of the dominating-set problem in this graph is one. It is indicated from[10]that finding a minimum-size dominating-set is a NP-complete problem, so it can be formulated as a ‘‘search’’ problem. The dominating set problem is widely used in network routing, town
simulation for Step 4(a) generated a resulted value, yes, since the tube T1 is not empty. Therefore Step 4(b) of simulation, the minimum-size dominating set from the tube T1was shown inTable 10.
5. Conclusions and future work
The present method for solving the dominating-set problem is based on biological operations in the Adleman–Lipton model and the solution space of stickers in the sticker-based model and thus is similar to the proposed method based on the solu-tion space of splints to solving the same problem [17]. The proposed algorithm has three advantages from the Adleman–Lipton model and the sticker-based model. First, the proposed algorithm actually has a lower rate of errors for hybridization because we modified the Adleman program to generate good DNA sequences for constructing the solution space of stickers to the dominating-set problem. Only sim-ple and fast biological operations in the Adleman–Lipton model were employed to solve the problem. Secondly, those biological operations in the Adleman–Lipton model had been performed in a fully automated manner in their lab. The full auto-mation manner is essential not only for the speedup of computation but also for error-free computation. Thirdly, in the proposed algorithm the number of tubes, the longest length of DNA library strands and the number of DNA library strands, respectively, are O(n), O(15· n) and O(2n
) strands. This implies that the proposed algorithm can be easily performed in a fully automated manner in a lab. Further-more, the present algorithm generates 2nlibrary strands, which satisfies the seven constraints in Section 3.2, which corresponds to 2npossible dominating sets. This al-lows the present algorithm to be applied to a larger instance of the dominating-set problem.
Currently, there are lots of NP-complete problems that cannot be solved because it is very difficult to support basic biological operations using mathematical opera-tions. We are not sure whether molecular computing can be applied to dealing with every NP-complete problem. Therefore, in the future, our main work is to solve other NP-complete problems that were unresolved with the Adleman–Lipton model and the sticker model.
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