Lecture Notes 4.B

Recall that the hpp involves nding a path through a

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Unformatted text preview: the directed Hamiltonian Path Problem (HPP). Recall that the HPP involves finding a path through a graph that visits each vertex exactly once. The instance of the HPP that Adleman solved is depicted in Fig. 5.2, with the unique Hamiltonian Path (HP) highlighted by a dashed 232 line. CHAPTER IV. MOLECULAR COMPUTATION 1 2 3 7 6 4 5 Fig. 5.2. Instance of the HPP solved by Adleman Figure IV.6: HPP solved by Adleman. The HP is indicated by the dotted edges. [source: Amos, Fig. 5.2] Adleman’s approach was simple: 1. Generate strands encoding random paths such that the Hamiltonian Path (HP) is represented with high probability. The quantities of in his seminal They did it in a manner similar to the one Adleman usedDNA used far exceeded 1994 paper.”4 those necessary for the small graph under consideration, so it is likely that many strands encoding the HP were present. 2. Remove all strands that do not encode the HP. 3. Check that the remaining strands encode a solution to the HPP. B.1.b Problem Representation The individual steps were implemented as follows: Stage 1: of Adleman’s algorithm is a a distinct 20-mer sequence of ¶1. The heartEach vertex and edge was assignedclever way to encode candidate DNA (Fig. DNA.This implies that strands encoding a HP were of length 140 paths in 5.3a). b.p. Sequences representing edges act as ‘splints’ between strands representing ¶2. their endpoints (Fig. 5.3b). represented by single-stranded 20mers, that Vertices: Vertices were In formal terms, the sequence associated with an edge i → j is the 3’ 10is, of the sequence representing vi followed mer sequences of 20nt (...
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