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A Hamiltonian path is a route that passes through every node in a graph exactly once. Finding Hamiltonian paths can overload even the best-known algorithm for the job. Finding Eulerian paths that pass through every edge is computationally simpler.

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Random Walks aim to explore a distribution through random perturbations. This is likely to move into low density regions, which is inefficient. Hamiltonian MC sails through a distribution much more rapidly by incorporating the distribution's structure into proposals.
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