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

But what exactly is O(log n)

What it means precisely is "as n tends towards infinity, the time tends towards a*log(n) where a is a constant scaling factor".

Or actually, it doesn't quite mean that; more likely it means something like "time divided by a*log(n) tends towards 1".

"Tends towards" has the usual mathematical meaning from 'analysis': for example, that "if you pick any arbitrarily small non-zero constant k, then I can find a corresponding value X such that ((time/(a*log(n))) - 1) is less than k for all values of n greater than X."

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In lay terms, it means that the equation for time may have some other components: e.g. it may have some constant startup time; but these other components pale towards insignificance for large values of n, and the a*log(n) is the dominating term for large n.

Note that if the equation were, for example ...

time(n) = a + blog(n) + cn + dnn

... then this would be O(n squared) because, no matter what the values of the constants a, b, c, and non-zero d, the d*n*n term would always dominate over the others for any sufficiently large value of n.

That's what bit O notation means: it means "what is the order of dominant term for any sufficiently large n".