Counting sort on non-integers - why not possible?

Question

What is it about the structure of counting sort that only makes it work on integers?

Surely strings can be counted?

''' allocate an array Count[0..k] ; initialize each array cell to zero ; THEN '''
for each input item x:
    Count[key(x)] = Count[key(x)] + 1
total = 0
for i = 0, 1, ... k:
    c = Count[i]
    Count[i] = total
    total = total + c

''' allocate an output array Output[0..n-1] ; THEN '''
for each input item x:
    store x in Output[Count[key(x)]]
    Count[key(x)] = Count[key(x)] + 1
return Output

Where above does the linear time break down if you try to use counting sort on strings instead (assuming you have strings of fixed length)?

Answer 1

If you naively apply the algorithm you need an encoding $f$ of strings of length $m$ into integers such that $s_1≤s_2$ iff $f(s_1)≤f(s_2)$. The bound on these integers is necessarily at least $c^m$ where $c$ is the number of possible characters.

You then have a complexity in $O(nc^m)$ where $n$ is the number of elements to be sorted. This could be interesting when $c^m$ is not too big compared to the usual bound $\log n$, which is possible, in very peculiar cases.

In general, one would prefer the bound $O(n\log n)$ which does not depend on the length of your strings and in general much faster than $O(nc^m)$. To give a formal comparison between them, the former would be $O(s\log s)$ and the latter $O(sc^s)$ where $s$ is the size of the input.

However what is possible if you know that you will have a number of different strings no greater than some $k$, and that there will be a lot of occurrences of the same strings, is to build a correspondence between these strings and $\{1,\dots,k\}$ on which you will apply the counting sort, thus obtaining a complexity of $O(kn)$ i.e. linear (since $k$ is constant). But in this case the counting sort is not the only one reaching a linear complexity, for example all algorithms using balanced trees can be adapted to be linear, even insertion sort which is quadratic in the worst case, becomes linear.