Question

Can someone explain to me why this is true. I heard a professor mention this is his lecture

Answer 1

The two notions are orthogonal.

You can have worst case asymptotics. If f(n) denotes the worst case time taken by a given algorithm with input n, you can have eg. f(n) = O(n^3) or other asymptotic upper bounds of the worst case time complexity.

Likewise, you can have g(n) = O(n^2 log n) where g(n) is the average time taken by the same algorithm with (say) uniformly distributed (random) inputs of size n.

Or you can have h(n) = O(n) where h(n) is the average time taken by the same algorithm with particularly distributed random inputs of size n (eg. almost sorted sequences for a sorting algorithm).

Asymptotic notation is a “measure”. You have to specify what you want to count: worst case, best case, average, etc.

Sometimes, you are interested in stating asymptotic lower bounds of (say) the worst case complexity. Then you write f(n) = Omega(n^2) to state that in the worst case, the complexity is at least n^2. The big-Omega notation is opposite to big-O: f = Omega(g) if and only if g = O(f).