algorithm - Meaning of average complexity when using Big-O notation

Question

While answering to this question a debate began in comments about complexity of QuickSort. What I remember from my university time is that QuickSort is O(n^2) in worst case, O(n log(n)) in average case and O(n log(n)) (but with tighter bound) in best case.

What I need is a correct mathematical explanation of the meaning of average complexity to explain clearly what it is about to someone who believe the big-O notation can only be used for worst-case.

What I remember if that to define average complexity you should consider complexity of algorithm for all possible inputs, count how many degenerating and normal cases. If the number of degenerating cases divided by n tend towards 0 when n get big, then you can speak of average complexity of the overall function for normal cases.

Is this definition right or is definition of average complexity different ? And if it's correct can someone state it more rigorously than I ?

Answer 1

You're right.

Big O (big Theta etc.) is used to measure functions. When you write f=O(g) it doesn't matter what f and g mean. They could be average time complexity, worst time complexity, space complexities, denote distribution of primes etc.

Worst-case complexity is a function that takes size n, and tells you what is maximum number of steps of an algorithm given input of size n.

Average-case complexity is a function that takes size n, and tells you what is expected number of steps of an algorithm given input of size n.

As you see worst-case and average-case complexity are functions, so you can use big O to express their growth.