The function remdup (also called mkset in some functional languages) removes duplicates from a given list. The following definition of remdup is standard and leads to a quadratic time algorithm

The operator (—) used in the last expression subtracts one list from another; its definition is

Defined in this way remdup x returns the list of distinct elements of x in the order in which they first appear in x. In other words, the position of remdup JC as a subsequence of x is lexically the smallest among all possible solutions.

Now let us change the problem and ask that remdup simply return the lexically least solution. Note the subtle difference between this version and the previous one: before, it was the position of the subsequence that was lexically the least, now it is the subsequence itself. To make the distinction clear, consider x = [1,4,2,4,3]. With the original definition remdup x returns [1,4,2,3]; the lexically least subsequence, however, is [1,2,4,3].

The question we are interested in is this: can we also find a quadratic time algorithm for the new problem? The answer turns out to be yes, but justifying it requires a bit of work.

The problem of computing the smallest natural number not contained in a given set of natural numbers has a number of practical applications. Typically, the given set represents the indices of a class of objects ‘in use’ and it is required to find a ‘free’ object with smallest index. Our purpose in this article is to derive a linear-time functional program for the problem. There is an easy solution if arrays capable of being accessed and updated in constant time are available, but we aim for an algorithm that employs only standard lists. Noteworthy is the fact that, although an algorithm using lists is the result, the derivation is carried out almost entirely in the world of sets.