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Someone recently claimed that there's an adversarial input for randomized quicksort; he referenced this paper. This defies my intuition because there are results that say that randomized quicksort runs in $O(n \log n)$ time with high probability. (See, for example, the book Randomized Algorithms by Motwani&Raghavan.) I am refering to the version of randomized quicksort that picks the pivot uniformly at random at each recursive call.

So my question is: are there indeed adversarial inputs for randomized quicksort? Please give an intuitive reason as to why such an input can exist.

Update. It seems to me that the adversary in the paper is granted unrealistic capabilities (generate array on the fly), which explains why it can make any quicksort perform poorly. Under reasonable assumptions I would say there is indeed no adversarial input for randomized quicksort.

user12344567
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There are implementations of Quicksort (the partitioning algorithm, specifically) which deal badly with duplicates. No matter how much you randomize -- shuffling the input, random choice of the pivot, random choice of the pivot with sampling, ... -- if all entries are the same (or there are only constantly many distinct values), these bad implementations will result in worst-case behaviour, always.

You can find details e.g. here and in the literature.

Note bene: The reason why many TCSists would claim the opposite with a passion it that they forget that most classical analyses are performed under permutation models. That does not give the whole truth, though, particularly if you use sorting in practice. Duplicates are relevant, for instance if you sort lists of binary values.

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Raphael
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The analysis of randomized quicksort is independent of the input. While there might be better inputs and worse inputs, the upper bound given by the analysis works for every single input.

You can go over the proof in slides of Xi Chen, for example, who proves that whatever the input, randomized quicksort runs in expected time $O(n\log n)$. (I don't know whether your high probability claim is true as well, though it sounds plausible.)

Yuval Filmus
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The mentioned StackOverflow question actually gives the source for this claim.

The source doesn't make this very clear, but the reason for this perceived mismatch between theory and practice is that in theory your comparison function is a mathematical function given ahead of time, but in practice it is a C function that can have side-effects and depend on previous side-effects.

Note that although the adversarial comparison function is non-deterministic in the sense that there is no mathematical function chosen ahead of time it corresponds to, it is still in a sense well-behaved: for any given execution of the program, there exists an order of elements such that the comparison function is compatible with it.

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Rotsor
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The killer adversary referenced builds an input that can make any quicksort go quadratic time provided that the pivot selection process responds the SAME every time the SAME input is used. In reality, a randomized pivot selection would have progressed to a different seed in the random series, so the next time the sort is called it will not cause quadratic time. It will cause quadratic only while it's building the input.

In other words, as soon as the killer input is built, it becomes not a killer. (in the case of random pivot selection).

S0lo
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