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This question and this question got me thinking a little bit. For sorting an array of length $n$ with $k$ unique elements in $O(n + k log k)$, we need to be able to store counts of values in the array. There are some suggestions, but I'm looking for a way to do this in worst case linear time. More specifically:

Given a list $A$ of $n$ elements with $k$ elements distinct, determine a list of tuples $U = {(x_i, c_i)}^k$ of all unique elements $x_i in A$ such that $c_i$ is the count of element $x_i$ in $A$.

Here are some (failed) ideas I've had and have been suggested:

1. 
   Balanced Binary Search Tree - With this it will take $O(log k)$ to insert into the tree and increase values. After inserts we could do a tree traversal in $O(k)$. Thus, total time comes out to $O(n log k)$ which is too slow.


2. 
   Hash Map - With this we can get $O(1)$ expected inserts and thus $O(n)$ expected time. However, this is still not $O(n)$ worst case.


3. 
   Empty Space Mapping - Find the minimum and maximum element in $A$. Allocate (but do not initialize) enough memory to cover this range. Use this memory basically as a hash map and include a random hash so that we don't try to access corrupted memory. This strategy presents issues. (1) It's probabilistic with very very very low probability of failing, but still not guaranteed. Using memory like this limits us to floating-point or integer constraints.


4. 
   Associative Arrays - There are many other associative arrays that can be used, similar to hash maps and BSTs, but I am not finding any that match these constraints.

Maybe there is some obvious method I am missing, but I also think it could be potentially not be possible. What are your thoughts?

This is a nice question.

In the comparison model or, what is more general, the algebraic decision-tree model, the problem of element distinctness has a lower bound of $Theta(nlog n)$ time-complexity in the worst case as said in this Wikipedia article. So there is no algorithm to count distinct elements in linear time in the worst case, even without counting the duplicities.

However, it is not clear whether it can be done in another computational model. It seems unlikely in any reasonable deterministic computational model.

There exist randomized algorithms whose expected running time is $O(n)$; or where the probability that the running time takes longer than $cn$ is exponentially small in $c$.

In particular, randomly choose a 2-universal hash function, then use it to hash all of the elements of the array. This achieves the stated running times, if you choose the length of the output of the 2-universal hash appropriately.

As another example, you can build a randomized algorithm whose worst-case running time is $O(n)$ (it always runs in linear time, no matter what) and has a probability of error of at most $1/2^{100}$. (How? Run the above algorithm, and terminate it if it runs longer than $cn$ steps for some appropriately chosen $c$.) In practice, that's good enough, as the probability that your computer outputs the wrong answer due to a cosmic ray is already much higher than $1/2^{100}$.