What is the difference between O(1) and Θ(1)?

问题:

I know the definitions of both of them, but what is the reason sometimes I see O(1) and other times Θ(1) written in textbooks?

Thanks.

回答1:

Big-O notation expresses an asymptotic upper bound, whereas Big-Theta notation additionally expresses a asymptotic lower bound. Often, the upper bound is what people are interested in, so they write O(something), even when Theta(something) would also be true. For example, if you wanted to count the number of things that are equal to x in an unsorted list, you might say that it can be done in linear time and is O(n), because what matters to you is that it won't take any longer than that. However, it would also be true that it's Omega(n) and therefore Theta(n), since you have to examine all of the elements in the list - it can't be done in sub-linear time.

回答2:

O(1) and Θ(1) aren't necessarily the same if you are talking about functions over real numbers. For example, consider the function f(n) = 1/n. This function is O(1) because for any n ≥ 1, f(n) ≤ 1. However, it is not Θ(1) for the following reason: one definition of f(n) = Θ(g(n)) is that the limit of |f(n) / g(n)| as n goes to infinity is some finite value L satisfying 0 < L. Plugging in f(n) = 1/n and g(n) = 1, we take the limit of |1/n| as n goes to infinity and get that it's 0. Therefore, f(n) ≠ Θ(1).

Hope this helps!