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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
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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.