Find an algorithm for sorting integers with time complexity O(n + k*log(k))
Here is a possible solution:
Using a hash table, count the number of unique values and the number of duplicates of each value. This should have a complexity of O(n).
Enumerate the hashtable, storing the unique values into a temporary array. Complexity is O(k).
Sort this array with a standard algorithm such as mergesort: complexity is O(k.log(k)).
Create the resulting array by replicating the elements of the sorted array of unique values each the number of times stored in the hash table. complexity is O(n) + O(k).
Combined complexity is O(n + k.log(k)).
For example, if k is a small constant, sorting an array of n values converges toward linear time as n becomes larger and larger.
If during the first phase, where k is computed incrementally, it appears that k is not significantly smaller than n, drop the hash table and just sort the original array with a standard algorithm.