[asrp]

This is a good video but it gets the definition of omega and theta wrong, or at least their definition is very non-standard.

By default, O, omega and theta all refer to the worst case running time.

    - O(n) means worst_case_run_time <= C n
    - omega(n) means worst_case_run_time >= C n
    - theta(n) means C1 n <= worst_case_run_time <= C2 n

If you want to talk about something other than worst case, you usually just say it in words, like "the average complexity of this algorithm is O(n^2)".

(I can't remember when someone has ever looked at "best case".)

To rephrase

- "This algorithm is O(something)" means I have an upper bound on the running time on the worst inputs.

- "This algorithm is omega(something)" means I have a lower bound on the running time on the worst inputs.

- "This algorithm is theta(something)" means I have both an upper and lower bound on the running time on the worst inputs (that differ only by a constant multiple.

In particular, a lower bound does not mean "best case" (they aren't even in the same part of the sentence).

(There's also some other detail from the video like the bit representation of string length that needs a footnote but that's much less important, I think.)

[j2kun]

Big-O typically refers to worst case when people discuss algorithms, but its definition has nothing to do with runtime, or even algorithms, at all. It was invented by mathematicians decades before computers existed.

It's also more nuanced than the inequality in your correction. It's a statement about limits of functions.

[asrp]

Thanks for the clarification. I never really know how much (accuracy) to put in a post. I'm assuming that part of it is what turned some folk away from learning complexity in the first place. (I have intuition but not a firm grasp of what turns people away either.)

The implicit default of "worst-case running time" (in the context of algorithm) may actually be the source of errors like the one made in the video. For what its worth, I actually think the lim sup definition is easier because its fewer things to memorize.

[pmattos]

Nice overview ;) Minor typo in theta(n): the bounds should be C1 n and C2 n.

[asrp]

Thanks, fixed!