Question

I am going through exercies for an exam in algorithm analysis and this is one of them:

Present an algorithm that takes as input a list of n elements (that
are comparable) and sorts them in O(n log m) time, where m is the
number of distinct values in the input list.

I have read about the common sorting algorithms and I really can’t come up with a solution.

Thanks for your help

Answer 1

You can build an augmented balanced binary search tree on the n elements. The augmented info stored at each node would be it’s frequency. You build this structure with n insertions into the tree, the time to do this would be O(n lg m), since there would be only m nodes. Then you do a in-order traversal of this tree: visit the left subtree, then print the element stored at the root f times where f is it’s frequency (this was the augmented info) and finally visit the right subtree. This traversal would take time O(n + m). So, the running time of this simple procedure would be O(n lg m + n + m) = O(n lg m) since m <= n.