| Summary: | 碩士 === 國立臺北大學 === 電機工程學系 === 104 === Now the feedback set problem is widely used in many applications, such as circuit design, operating system deadlock recovery, genetic assembly, sort of work, and so on. Since any cycle in any application diagram can cause some troubles. For example, in a topological ordering, if there were a cycle, it would be almost impossible to sort all of nodes in a reasonable order. For a wait-for graph, if there were a cycle, there would be a deadlock. Typically, to break graph cycles is the first step to solve the minimum feedback set problem.
The minimum feedback set problems is a part of the graphic theory. The minimum feedback set problems can be divided into two types. They are minimum feedback arc set and minimal feedback vertex set problems. Given a graph G=(V, E) in the feedback arc set problem, V represents a set of vertices and E represents a set of arcs. Feedback arc set Fa is a subset of E, when Fa is removed from graph G, G is acyclic, and the weight of Fa is minimum. Similarly, feedback vertex set Fv is a subset of V. when Fv is removed from graph G, graph G is also acyclic, and the weight of Fv is minimum.
In 2003, Camil and Finocchi proposed a combination of approximation algorithms to solve the minimum feedback set problem. The proposed algorithms contain two phases. In Phase I, it first searches for a cycle C in the application diagram. Second, find an arc with minimum weight e in C. Third, decrease the weight of all the arcs in C by e. Fourth, an arc whose weight becomes zero is removed. Finally, Phase I is terminated whenever the digraph is acyclic. Otherwise it will continue to repeat. There remains an issue after Phase I operations. The total weights may not be minimal. Hence, there is a need to add any deleted arc back to the digraph and repeat some of Phase I operations for acyclic identification purposes. Whenever the set of removed arcs is a solution to the minimum feedback set problem, Phase II is completed. Unfortunately, there may exist multiple optimal solutions for any minimum feedback set problem, their proposed approximate algorithms can only provide one of optimal solutions.
In this thesis, by using Adleman-Lipton model, newly developed fast molecular solutions in parallel are proposed for minimum feedback set problems on a DNA-based computer. First, a feasible solution space containing all sets by combining different edges (or vertices) in parallelism is constructed. Second, all solutions with cycles are illegal and must be deleted. After all illegal solutions with cycles are deleted, all corresponding weights of all removed edges (or vertices) are added to the end of each corresponding solution. At the same time, the total weight of every solution is calculated. Finally, multiple optimal solutions of minimum feedback set are found. With the help of parallel processing approach, to complete these steps is estimated within O(n*m) time biological operations in the Adleman-Lipton model.
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