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Hamiltonian path
This document discusses the Hamiltonian path problem in graph theory. A Hamiltonian path visits each vertex in a graph exactly once. The Hamiltonian path problem is determining if a Hamiltonian path exists in a given graph. It is computationally difficult to solve and several algorithms have been developed, including brute force search, dynamic programming, and Monte Carlo algorithms. Unconventional models of computing like DNA computers have also been used to attempt solving the Hamiltonian path problem by exploiting parallel chemical reactions.
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Introduces the concept of Hamiltonian path, illustrated with a puzzle involving red dots.
Presents an engaging challenge to draw a Hamiltonian path through 20 red dots.
Introduces the Icosian game, created by William Rowan Hamilton, which concerns Hamiltonian paths.
Defines Hamiltonian paths, cycles, connected graphs, and introduces Hamiltonian graphs.
Describes the problems relating to determining the existence of Hamiltonian paths and cycles in graphs.
Discusses various algorithms, including brute-force, dynamic programming, and others for solving Hamiltonian path problems.
Explores unconventional computing methods, like DNA computing, to solve Hamiltonian path problems.
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Hamiltonian path
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- 2. A game….. The figurehas 20 red dots. Can you draw a path that visits each vertex (Dots) exactly once???? Try it out….
- 3. Solution!!! Icosian game developedby William Rowan Hamilton
- 4. Hamiltonian path • Inthe mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle in a dodecahedron
- 5. Some definitions…. • AHamiltonian path or traceable path is a path that visits each vertex exactly once. • A graph that contains a Hamiltonian path is called a traceable graph. • A graph is Hamiltonian-connected if for every pair of vertices there is a Hamiltonian path between the two vertices. • A Hamiltonian cycle, Hamiltonian circuit, vertex tour or graph cycle is a cycle that visits each vertex exactly once (except for the vertex that is both the start and end, which is visited twice). • A graph that contains a Hamiltonian cycle is called a Hamiltonian graph.
- 6. Hamiltonian path problem •Hamiltonian path problem and the Hamiltonian cycle problem are problems of determining whether a Hamiltonian path (a path in an undirected or directed graph that visits each vertex exactly once) or a Hamiltonian cycle exists in a given graph (whether directed or undirected).
- 7. Algorithms for solvingthe problem • Brute-force search algorithm There are n! different sequences of vertices that might be Hamiltonian paths in a given n-vertex graph (and are, in a complete graph), so a brute force search algorithm that tests all possible sequences would be very slow. There are several faster approaches. • Inclusion–exclusion principle • Dynamic programming algorithm • Frank Rubin method • Monte Carlo algorithm
- 8. Usingunconventional models ofcomputing • Because of the difficulty of solving the Hamiltonian path and cycle problems on conventional computers, they have also been studied in unconventional models of computing. For instance, Leonard Adleman showed that the Hamiltonian path problem may be solved using a DNA computer. • Exploiting the parallelism inherent in chemical reactions, the problem may be solved using a number of chemical reaction steps linear in the number of vertices of the graph; however, it requires a factorial number of DNA molecules to participate in the reaction. ag1805x