Loop detection instead of doubling

I've paged through two problems that were straightforward to solve with loop detection and mistakenly thought "this is doubling".

When reading N vertices ahead in a graph with V vertices

O(VlogN) pre-processing allows O(logN) for this process

Gain if this process is repeated more than V times.

If logN is too large, even this is not possible.

If it has a single starting point, it can be pre-processed with O(V) in the worst case.

What should we do at an arbitrary starting point? If we do it naively, O(V^2)

If N is large enough to enter the loop, subtract the distance to enter the loop and take the remainder by the loop length

O(1)

Comparison

If N is high V at any starting point, doubling is lighter preprocessing.

Loop detection is more advantageous for a single starting point

Loop detection if N is excessively large at any starting point

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