[ruggeri]

Strictly speaking, it is not true that shuffling first avoids the worst case. Given a deterministic shuffling function, some permutation of input needs to shuffle to sorted order, which will again trigger the worst case.

Of course, shuffling is still potentially helpful because ascending/descending order is often common in many applications.

[brudgers]

Randomization of the pivots produces O(n log n) average running time with a very low probability of any other running time for any value of n where the worst case running time matters.

[ruggeri]

Even without randomization of pivots expected run time is O(n log n). If the order of the input data is random, I don't believe randomizing pivots changes the distribution of running times.

What changes is that one very common permutation of data (ascending data) does not have O(n * 2) performance.

[brudgers]

There's little justification for expecting data to behave randomly unless our algorithm introduces it.

[mtdewcmu]

Correct. In general, real data will seldom, if ever, look like random data.

[shanusmagnus]

Shuffling with a too-small seed size will render the shuffle susceptible to brute force attacks; but in no way invalidates its use as a dodge for pathological (pre-sorted) arrays submitted to Quicksort in the course of normal usage. There's a better chance of getting eaten by a dinosaur than finding O(N^2) QS performance on shuffled data that comes from anything other than an attacking adversary.

[mtdewcmu]

You're right that shuffling doesn't avoid the worst case, but your explanation isn't quite correct. See my other comment for my explanation.

I'm not sure what you mean by deterministic shuffling. Shuffling is supposed to be random, and therefore non-deterministic. In practice, you'd probably use a pseudorandom function to implement shuffling, but the goal of pseudorandom functions is to be indistinguishable from random functions, so that detail shouldn't matter.