Fast matrix multiplication techniques based on the Adleman-Lipton model

On distributed memory electronic computers, the implementation and association of fast parallel matrix multiplication algorithms has yielded astounding results and insights. In this discourse, we use the tools of molecular biology to demonstrate the theoretical encoding of Strassen's fast matrix multiplication algorithm with DNA based on an $n$-moduli set in the residue number system, thereby demonstrating the viability of computational mathematics with DNA. As a result, a general scalable implementation of this model in the DNA computing paradigm is presented and can be generalized to the application of \emph{all} fast matrix multiplication algorithms on a DNA computer. We also discuss the practical capabilities and issues of this scalable implementation. Fast methods of matrix computations with DNA are important because they also allow for the efficient implementation of other algorithms (i.e. inversion, computing determinants, and graph theory) with DNA.Comment: To appear in the International Journal of Computer Engineering Research. Minor changes made to make the preprint as similar as possible to the published versio

A Genetic Algorithm to Solve the Subset Sum Problem based on Parallel Computing

Abstract: The subset sum problem is to find subsets in a given number set, meanwhile number sum of the subset is equal to appointed value. It is a classical NP-complete problem in graph theory. It can be solved by the electronic computer in exponential time. In this paper, we consider a DNA procedure for solving the subset sum problem in the Adleman-Lipton model. The procedure works in O(n) steps for the subset sum problem of an undirected graph with n vertices. The innovation of the procedure is the ingenious choice of the vertices strands' length, which can get the solution of the problem in proper length range and simultaneity simplify the complexity of the computation

Procedures for Logic and Arithmetic Operations with DNA Molecules

In this paper, we consider procedures for logic and arithmetic operations with DNAmolecules. We first show a DNA representation of n binary numbers of m bits, andpropose a procedure to assign the same values for the representation. The representationenables addressing feature, and the procedure is applicable to n binary numbers of mbits in O(1) steps in parallel. Next, we propose a procedure for logic operations. Theprocedure enables any boolean operation whose input and output are defined by a truthtable, and executes different kinds of boolean operations simultaneously for any pairof n binary numbers of m bits in O(1) lab steps using O(mn) DNA strands. Finally,we propose a procedure for additions of pairs of two binary numbers. The procedureexecutes O(n) additions of two m-bit binary numbers in O(1) steps using O(mn) DNAstrands