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Lecture Notes 4.B - B FILTERING MODELS B 231 Filtering...

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B. FILTERING MODELS 231 B Filtering models 1. Filtering models: Operate by filtering out of the solution molecules that are not part of the solution. 2. A solution can be treated mathematically as a finite bag or multi-set of molecules, and filtering operations can be treated as operations to produce multi- sets from multi-sets. 3. Initial multi-set: Typically, for a problem of size n , strings of size O ( n ) are required. Should contain enough strings to include many copies all possible so- lutions. Therefore, for an exponential problem, we will have O ( k n ) strings. 4. This is essentially a brute-force method. B.1 Adleman: HPP B.1.a Review of HPP 1. Hamiltonian Path Problem (HPP): The Hamiltonian Path Prob- lem is to determine, for a given directed graph G = ( V, E ) and two of its vertices v in , v out 2 V , whether there is a HP from v in to v out , that is, a path that goes through each vertex exactly once. 2. NP-complete: HPP is an NP-complete problem. 3. We will see that for Adleman’s algorithm the number of algorithm steps is linear in problem size. 4. Laboratory demonstration: Leonard Adleman gave a laboratory demonstration of the procedure in 1994 (for n = 7). (By the way, he is the “A” of “RSA.”) 5. “In 2002, he and his research group managed to solve a ‘nontriv- ial’ problem using DNA computation. Specifically, they solved a 20- variable SAT problem having more than 1 million potential solutions.
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232 CHAPTER IV. MOLECULAR COMPUTATION 7 1 2 3 4 5 6 Figure IV.6: HPP solved by Adleman. The HP is indicated by the dotted edges. [source: Amos, Fig. 5.2] They did it in a manner similar to the one Adleman used in his seminal 1994 paper.” 4 B.1.b Problem Representation 1. The heart of Adleman’s algorithm is a clever way to encode candidate paths in DNA. 2. Vertices: Vertices were represented by single-stranded 20mers, that is, sequences of 20nt (nucleotides).
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