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Lumifer comments on The correct response to uncertainty is *not* half-speed - Less Wrong
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Imagine an undirected graph where each node has a left and right neighbor (so it's an infinitely long chain). You are on a node in this graph, and somewhere to the left or right of you is a hotel (50/50 chance to be in either direction). You don't know how far -- k steps for an arbitrarily large k that an adversary picks after learning how you will look for a hotel.
The solution that takes 1 step left, 2 steps right, 3 steps left, etc. will find the hotel in O(k^2) steps. Is it possible to do better?
Does the solution have to be deterministic? If an adversary places the hotel after looking at my algorithm, my first inclination is introduce uncertainty into my search.
No. But you can't do better than O(k) anyways, and you can do O(k) deterministically.
Randomising the length of the first step will improve on the constant factor by about 2. Similar analysis to the non-adversarial case, and with the same ETA I just added to my earlier comment.
If you make the run lengths increase exponentially instead of linearly then you get O(k) unconditionally.
I know. I was improving the constant factor.
Ah yes, OK, essentially the same randomization helps against an adversary when you're growing exponentially as when you're growing only linearly. Fair enough.