What is the worst case complexity for bucket sort?

Question

I just read the Wikipedia page about Bucket sort. In this article they say that the worst case complexity is O(n²). But I thought the worst case complexity was O(n + k) wher

Answer 1

Bucket sort assumes that the input is drawn from a uniform distribution. This implies that a few items fall in each bucket. In turn, this leads to a nice average running time of O(n). Indeed, if the n elements are inserted in each bucket so that O(1) elements fall in each different bucket (insertion requires O(1) per item), then sorting a bucket using insertion sort requires, on average, O(1) as well (this is proved in almost all textbooks on algorithms). Since you must sort n buckets, the average complexity is O(n).

Now, assume that the input is not drawn from a uniform distribution. As already pointed out by @mfrankli, this may lead in the worst case to a situation in which all of the items fall for example all in the first bucket. In this case, insertion sort will require in the worst case O(n^2).

Note that you may use the following trick to maintain the same average O(n) complexity, while providing an O(n log n) complexity in the worst case. Instead of using insertion sort, simply use an algorithm with O(n log n) complexity in the worst case: either merge sort or heap sort (but not quick sort, which achieves O(n log n) only on average).