# Worst case analysis of bucket sort using insertion sort for the buckets

Suppose I am using the Bucket-Sort algorithm, and on each bucket/list I sort with insertion sort (replace nextSort with insertion sort in the wikipedia pseudocode).

In the worst case, this would imply that we would have $O(n^2)$ performance, because if every element was in one bucket, then we would have to use insertion sort on $n$ elements which is $O(n^2)$.

So the first thing that comes to mind to fix the worst case running time is to not use insertion-sort, because it is $O(n^2)$. Instead we could use merge-sort or heap-sort m, because the worst case running time for both of those algorithms is $O(n\log n)$. However, if we use merge-sort and heap-sort, do they preserve the expected linear running-time of bucket-sort?

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The use of $O$ weakens your statements; I guess you want to use $\Theta$? – Raphael Feb 18 '13 at 11:36