A numerical analysis of Quicksort: How many cases are bad cases? [pdf]
arxiv.orgIf you're going to post articles from arxiv, please post the top-level description so I can read the abstract before being forced to download a PDF document.
> An attempt to solve this problem has been randomization as shown already in Hoare’s first articles on Quicksort [1]. On the other hand, this does not change the statistical probability for bad cases.
I really like the randomization solution. Worst case? What worst case?
Well, the worst case is still there. You're just going to have a very hard time crafting an input that triggers it.
I remember, at uni, in the second semester, we had an assignment to specifically craft a sequence of n numbers 1..n that would trigger the worst case (#permutations) in a quicksort algorithm that always takes the first number in the array as the pivot. It was a nightmare. I spent days on it, and I was barely able to scrape together a notation to specify such a series. Turns out the sample solutions were like "The appropriate sequence is not easy to write. The sequence must be designed such that every chosen pivot halves the area that will be stored." Well, yeah.
EDIT: I wrote worst case #permutations. That's the best case #comparisons. So, no, I don't mean a sorted sequence ;)
If quicksort always takes the first element in the array as the pivot, then an array that is already sorted is the worst case.
Doesn't a sorted array trigger the worst case in that case? Either ascending or decending. It's one of the reasons not to choose the first element as the pivot, iirc.
A much more interesting case is the median-of-three pivot. (Namely, median of first / middle / last elements). You can still do it, however.
What do you mean by worst case #permutations?
If Quicksort worst case performance is a problem then use Introsort (https://en.wikipedia.org/wiki/Introsort), which the paper fails to mention.
It strikes me that, if three way partitioning is useful only if you are dealing with a limited set of values in relation to n, you could hash or insertion sort into an array the values for the first pass of the algorithm over the whole array (stopping if the count of uniques got bigger then some value based on n), and then decide to threeway partition if you hadn't stopped.
I feel like the recursive median of medians throws away a great deal of information, given all the comparisons you make. At the very least, for the final step, you know where the high and low values go, and could easily place them in the appropriate sides of the array.
I love quicksort, and enjoyed the paper. Yes, I'm a nerd. Thanks for asking.