Finding repeating signed integers with O(n) in time and O(1) in space

Question

(This is a generalization of: Finding duplicates in O(n) time and O(1) space)

Problem: Write a C++ or C function with time and space complexities of O(n) and O(1) respec

Answer 1

I doubt this is possible. Assuming there is a solution, let's see how it works. I'll try to be as general as I can and show that it can't work... So, how does it work?

Without losing generality we could say we process the array k times, where k is fixed. The solution should also work when there are m duplicates, with m >> k. Thus, in at least one of the passes, we should be able to output x duplicates, where x grows when m grows. To do so, some useful information has been computed in a previous pass and stored in the O(1) storage. (The array itself can't be used, this would give O(n) storage.)

The problem: we have O(1) of information, when we walk over the array we have to identify x numbers(to output them). We need a O(1) storage than can tell us in O(1) time, if an element is in it. Or said in a different way, we need a data structure to store n booleans (of wich x are true) that uses O(1) space, and takes O(1) time to query.

Does this data structure exists? If not, then we can't find all duplicates in an array with O(n) time and O(1) space (or there is some fancy algorithm that works in a completely different manner???).