A DNA procedure for solving the shortest path problem☆
Introduction
In recent works for high performance computing, computation with DNA molecules, i.e., DNA computing, has considerable attention as one of non-silicon based computing. Watson-Crick complementarity and massive parallelism are two important features of DNA. Using the features, one can solve an NP-complete problem, which usually needs exponential time on a silicon based computer, in a polynomial number of steps with DNA molecules, e.g., Adleman [1] for Hamiltonian path problem – the first work for DNA computing, Lipton [11] for satisfiability (SAT) problem (the first NP-complete problem), Ouyang et al. [13] for the maximal clique problem, etc. Meanwhile, procedures for primitive operations, such as logic or arithmetic operations, have been also proposed so as to apply DNA computing on a wide range of problems [2], [3], [4], [6], [7], [8], [16], [17]. However, most of the previous works in DNA computing do not require the consideration of the representation of numerical data in DNA strands. In fact, many practical applications in the real world involve edge-weighted graph problems such as shortest path problem, the travelling-salesman problem, etc. Therefore, representation of numerical data in DNA strands is an important issue toward expanding the capability of DNA computing to solve numerical optimization problems. There have been some previous works to represent the numerical data with DNA. Narayanan et al. [12] presented a conceptual encoding method that represents costs with the lengths of DNA strands. Shin et al. [15] proposed a method for representing the real numbers in fixed-length DNA strands by varying the number of hydrogen bonds. Yamamura et al. [18] proposed a concentration control method which encoded the numerical data by means of the concentrations of DNA strands. Lee et al. [9] introduced a novel encoding method that utilizes a temperature gradient to design the sequences so that the DNA strands for higher-cost values have higher melting temperatures than those for lower-cost values.
In this paper, a DNA procedure is presented for figuring out solutions of the shortest path problem: for an edge-weighted graph G = (V, E) find a path starting and ending at the specified vertices such that the total weights on the path is smallest. For instance, the edge-weighted graph G in Fig. 1 defines such a problem. We assume that the starting and ending vertices are 1 and 7, respectively. It is easy to see that the path 1 → 2 → 3 → 4 → 6 → 7 with total weights 8 is a solution to the shortest path problem for graph G in Fig. 1. We encode the numerical data by means of the lengths of DNA strands, the same way as that in [12]. A DNA procedure is formally presented by means of the DNA operations proposed by Adleman [1] and Lipton [11]. Since the shortest path discussed in the present paper is not required to go through each vertex, e.g., the solution for the graph G in Fig. 1 does not go through the vertex 5, we use the append operation in the design of data pool so that the shortest path can be found out by comparing the lengths of the DNA strands.
The rest of this paper is organized as follows. In Section 2, the Adleman–Lipton model is introduced in detail. Section 3 introduces a DNA algorithm for solving the shortest path problem and the complexity of the proposed algorithm is described. We give conclusions in Section 4.
Section snippets
The Adleman–Lipton model
Bio-molecular computers work at the molecular level. Because biological and mathematical operations have some similarities, DNA, the genetic material that encodes for living organisms, is stable and predictable in its reactions and can be used to encode information for mathematical systems.
A DNA (deoxyribonucleic acid) is a polymer, which is strung together from monomers called deoxyribonucleotides [14]. Distinct nucleotides are detected only with their bases. Those bases are, respectively,
DNA algorithm for the shortest path problem
Let G = (V, E) be an edge-weighted graph with the set of vertices V = {vk ∣ k = 1, 2, … , n} and the set of edges E = {ei,j ∣ for some 1 ⩽ i, j ⩽ n, i ≠ j}. Note that both ei,j and ej,i are in E if the vertices vi and vj are connected by an edge. Without loss of generality, we assume that v1 and vn are the starting and ending vertices, respectively. Let ∣E∣ = s. Then .
In the following, we use the symbols X, #, Ak and Bk (k = 1, 2, … , n) to denote distinct DNA singled strands for which , where ∥ · ∥
Conclusions
As the first work for DNA computing, Adleman [1] presented an idea to demonstrate that deoxyribonucleic acid (DNA) strands can be applied to solving the Hamiltonian path NP-complete problem of size n in O(n) steps using DNA molecules. Adleman’s work shows that one can solve an NP-complete problem, which usually needs exponential time on a silicon based computer, in a polynomial number of steps with DNA molecules. From then on, Lipton [11] demonstrated that Adleman’s experiment could be used to
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2015, BioSystemsCitation Excerpt :Some typical DNA computing models, such as Adleman-Lipton model (Adleman, 1994; Lipton, 1995), the sticker model (Roweis et al., 1998), the restriction enzyme model (Ouyang et al., 1997), the self-assembly model (Winfree et al., 1998), the hairpin model (Sakamoto et al., 2000) and the surface-based model (Xiao et al., 2005), have already been established. Based on these models, lots of papers have occurred for designing DNA procedures and algorithms to solve various NP-complete problems (Li et al., 2006; Xiao et al., 2006; Wang et al., 2006, 2008, 2010, 2012, 2013; Guo et al., 2005; Chang et al., 2012, 2008; Han, 2008; Liu et al., 2010). In order to fully understand the power of biological computation, it is worthwhile to try to solve more kinds of computationally intractable problems with the aid of DNA biologic operations.
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2009, International Journal on Artificial Intelligence ToolsSolving the Family Traveling Salesperson Problem in the Adleman-Lipton Model Based on DNA Computing
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Supported by Bio-X DNA Computer Consortium No. 03DZ14025.
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Supported by National Science Foundation of China #10371043 and Shanghai Priority Academic Discipline.