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Depending on how the pivot is picked, Quicksort can actually be implemented in O(n lg n) worst-case time.

EDIT: I was going to link a proof for this but it's surprisingly hard to find. IIRC, the idea is to use the median of medians algorithm ([1]) to pick the median for the pivot, and deal with values equal to the pivot by alternatingly placing them in the left and right partition, or alternatively just keep them in a third partition in the middle.

[1] https://en.wikipedia.org/wiki/Median_of_medians

That's incorrect. The /mathematical/ definition means that it's the upper bound. I.e. yes, the set of all functions of O(N) is a subset of the set of all functions O(N^2). So you could say Quicksort is O(N^100) and still be technically correct.

However, the big-O notation does NOT specify anything about worst/avg/best-case complexity of a given algorithm. That should still be defined in the analysis.

You mixed up those two slightly different concepts.

Must have been a while, I thought you went with average, but I guess its been 20 years. I mostly lived in optimizing cache since college.