Efficient algorithm to find first available name

Question

I have an array containing names of items. I want to give the user the option to create items without specifying their name, so my program will have to supply a unique defau

Answer 1

Yes, there is.

First sort the array. Then run through it and return the first element whose value is not equal to its index (plus 1). The sort is O(n log n), the final step is O(n), so the entire thing is O(n log n).

If you put all items into a hash table, you can do it in O(n) at the cost of some space and an additional O(1) step at creation of new items. Since each element needs to be visited, O(n) is clearly optimal.

I'd be interested to see if there's an O(n) way to do this, without using any "persistent" data structure like the hash table. (And assuming unbounded integers, otherwise a bucket sort could be used as an O(n) sorting algorithm).