If I say to you:
"I am thinking of a number between 0 and n, and I will tell you if your guess is high or low", then you will immediately reach for binary search.
What if I remove the upper bound? i.e. I am thinking of a positive integer, and you need to guess it.
One possible method would be for you to guess 2, 4, 8, ..., until you guess 2**k for some k and I say "lower". Then you can apply binary search.
Is there a quicker method?
EDIT:
Clearly, any solution is going to take time proportional to the size of the target number. If I chuck Graham's number through the Ackermann function, we'll be waiting a while whatever strategy you pursue.
I could offer this algorithm too: Guess each integer in turn, starting from 1.
It's guaranteed to finish in a finite amount of time, but yet it's clearly much worse than my "powers of 2" strategy. If I can find a worse algorithm (and know that it is worse), then maybe I could find a better one?
For example, instead of powers of 2, maybe I can use powers of 10. Then I find the upper bound in log_10(n) steps, instead of log_2(n) steps. But I have to then search a bigger space. Say k = ceil(log_10(n)). Then I need log_2(10**k - 10**(k-1)) steps for my binary search, which I guess is about 10+log_2(k). For powers of 2, I have roughly log_2(log_2(n)) steps for my search phase. Which wins?
What if I search upwards using n**n? Or some other sequence? Does the prize go to whoever can find the sequence that grows the fastest? Is this a problem with an answer?
Thank you for your thoughts. And my apologies to those of you suggesting I start at MAX_INT or 2**32-1, since I'm clearly drifting away from the bounds of practicality here.
FINAL EDIT:
Hi all,
Thank you for your responses. I accepted the answer by Norman Ramsey (and commenter onebyone) for what I understood to be the following argument: for a target number n, any strategy must be capable of distinguishing between (at least) the numbers from 0..n, which means you need (at least) O(log(n)) comparisons.
However several of you also pointed out that the problem is not well-defined in the first place, because it's not possible to pick a "random positive integer" under the uniform probability distribution (or, rather, a uniform probability distribution cannot exist over an infinite set). And once I give you a nonuniform distribution, you can split it in half and apply binary search as normal.
This is a problem that I've often pondered as I walk around, so I'm pleased to have two conclusive answers for it.
Worst case, you can find it in time logarithmic in the size of the answer using exactly the methods you describe. You might use Ackermann's function to find an upper bound faster than logarithmic time, but then the binary search between the number guessed and the previous guess will require time logarithmic in the size of the interval, which (if guesses grow very quickly) is close to logarithmic in the size of the answer.
It would be interesting to try to prove that there is no faster algorithm (e.g., O(log log n)), but I have no idea how to do it.