What does O(log n) mean exactly?

Question

I am learning about Big O Notation running times and amortized times. I understand the notion of O(n) linear time, meaning that the size of the input affects the g

Answer 1

I can add something interesting, that I read in book by Kormen and etc. a long time ago. Now, imagine a problem, where we have to find a solution in a problem space. This problem space should be finite.

Now, if you can prove, that at every iteration of your algorithm you cut off a fraction of this space, that is no less than some limit, this means that your algorithm is running in O(logN) time.

I should point out, that we are talking here about a relative fraction limit, not the absolute one. The binary search is a classical example. At each step we throw away 1/2 of the problem space. But binary search is not the only such example. Suppose, you proved somehow, that at each step you throw away at least 1/128 of problem space. That means, your program is still running at O(logN) time, although significantly slower than the binary search. This is a very good hint in analyzing of recursive algorithms. It often can be proved that at each step the recursion will not use several variants, and this leads to the cutoff of some fraction in problem space.