Does this imply Hamiltonian path cannot be decided in nondeterministic logspace?


Suppose I nondeterministically walk around in a graph with n vertices.

When looking for a Hamiltonian path, at some point I’ve walked n/2 vertices.

There are (n choose n/2) different combinations of vertices I could have walked (meaning the unordered set, not the ordered walk).

Each of those states must be distinct from one another.

If not, then, depending on the remaining n/2 vertices, I would decide the wrong answer.

Therefore, midway through my processing, at n/2, I need (n choose n/2) different states. That is too big for logspace.

Therefore you cannot decide a Hamiltonian path by nondeterministically walking around.

Does this imply Hamiltonian path cannot be decided in nondeterministic logspace – at least by “walking around”?