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 fltering out oF the solution molecules that are not part oF the solution. 2. A solution can be treated mathematically as a fnite bag or multi-set oF molecules, and fltering operations can be treated as operations to produce multi- sets From multi-sets. 3. Initial multi-set: Typically, For a problem oF size n ,s t r ing so Fs ize O ( n )arerequ ired . Should contain enough strings to include many copies all possible so- lutions. ThereFore, For an exponential problem, we will have O ( k n )str ings . 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 )andtwoo F its vertices v in ,v out 2 V , whether there is a HP From v in to v out ,tha t 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. Specifcally, 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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Lecture Notes 4.B - B FILTERING MODELS B 231 Filtering...

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