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49 lines
1.7 KiB
Markdown
49 lines
1.7 KiB
Markdown
# Hamiltonian Path
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**Hamiltonian path** (or **traceable path**) is a path in an
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undirected or directed graph that visits each vertex exactly once.
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A **Hamiltonian cycle** (or **Hamiltonian circuit**) is a
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Hamiltonian path that is a cycle. Determining whether such paths
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and cycles exist in graphs is the **Hamiltonian path problem**.
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One possible Hamiltonian cycle through every vertex of a
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dodecahedron is shown in red – like all platonic solids, the
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dodecahedron is Hamiltonian.
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## Naive Algorithm
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Generate all possible configurations of vertices and print a
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configuration that satisfies the given constraints. There
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will be `n!` (n factorial) configurations.
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```
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while there are untried configurations
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{
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generate the next configuration
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if ( there are edges between two consecutive vertices of this
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configuration and there is an edge from the last vertex to
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the first ).
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{
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print this configuration;
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break;
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}
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}
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```
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## Backtracking Algorithm
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Create an empty path array and add vertex `0` to it. Add other
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vertices, starting from the vertex `1`. Before adding a vertex,
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check for whether it is adjacent to the previously added vertex
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and not already added. If we find such a vertex, we add the
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vertex as part of the solution. If we do not find a vertex
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then we return false.
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## References
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- [Hamiltonian path on Wikipedia](https://en.wikipedia.org/wiki/Hamiltonian_path)
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- [Hamiltonian path on YouTube](https://www.youtube.com/watch?v=dQr4wZCiJJ4&list=PLLXdhg_r2hKA7DPDsunoDZ-Z769jWn4R8)
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- [Hamiltonian cycle on GeeksForGeeks](https://www.geeksforgeeks.org/backtracking-set-7-hamiltonian-cycle/)
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