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) where k are the number of buckets. This is how I calculate this complexity:

1. Add the element to the bucket. Using a linked list this is O(1)
2. Going through the list and put the elements in the correct bucket = O(n)
3. Merging the buckets = O(k)
4. O(1) * O(n) + O(k) = O(n + k)

Am I missing something?

Answer 1

What if the algorithm decides that every element belongs in the same bucket? In that case, the linked list in that bucket needs to be traversed every time an element is added. That takes 1 step, then 2, then 3, 4, 5... n . Thus the time is the sum of all of the numbers from 1 to n which is (n^2 + n)/2, which is O(n^2).

Of course, this is "worst case" (all the elements in one bucket) - the algorithm to calculate which bucket to place an element is generally designed to avoid this behavior.