/= is Common Lisp’s not-equals function (similar to the `!=`

operator from C). In Common Lisp /= is restricted to *numbers* and it is in fact an error to call it with any other type, but not a lot of people know that. It looks like an innocuous enough function, until one considers that like a lot of Lisp functions it can take any number of arguments:

(/= x1 x2 x3 ... xn)

The semantics of /= are to return True if and only if no two numbers are the same value. Note that (/= 1 2 1) is False; it is insufficient to merely consider adjacent pairs, if *any* pair have the same value then the function should return False.

Contrast this with the semantics of the function for equality, =, which returns True if and only if *all* numbers have the same value. This *can* be implemented by checking adjacent pairs, or comparing each argument against the first, a straightforward O(n) algorithm.

So the puzzle is, how do you implement /=? Given a list of numbers we want to return False if any pair have the same value, returning True otherwise. It’s most interesting if we consider the complexity and therefore the case when we have large numbers of arguments.

Common Lisp implements a wide range of number types: unbounded integers (bignums), unbounded rationals, finite precision floating-point types, and complex versions of all of those. /= compares numbers *by value* so the integer 0, the floating-point number 0d0, and the complex floating-point number #C(0d0 0d0) are all equal. Note that any real number can be considered as being equal in value to a complex number with a 0 imaginary part; also note that any floating-point number can be converted to a rational having exactly the same value (see my blog article on the subject). That means that for simplification we might want to convert all input numbers to the type `(complex rational)`

before proceeding. Since that’s obviously a bounded time operation it doesn’t affect the complexity of whatever algorithm we come up with. That cuts down on the number of different types of pair-wise comparison operators that we might need.

The first, obviously naïve, approach would be to compare all possible pairs. That obviously works and is obviously O(n^{2}).

If you like a puzzle, try improving upon this naïve algorithm before reading on.

The next approach I came up with is to sort the arguments first, and then do comparisons of all the adjacent pairs. Recall that the input numbers are complex and there’s no natural ordering. Doesn’t matter, we can just pick any old ordering, for example lexicographic ordering, where #C(a b) is less then #C(c d) if `a` < `c`, or `a` = `c` and `b` < `d`. This algorithm is O(n log(n)) for the sort (consult any algorithms book) and O(n) for the final comparison of adjacent pairs, so O(n log(n)) overall.

That seems a lot better. But there is another approach that’s even better, suggested by a good friend:

Create a hash-table and hash each argument into it, checking to see if an entry with that value already exists. If an entry already exists then that’s an equal pair, so return False; if we get to the end of the arguments then there’s no equal pair, so return True. This algorithm is O(n) unless you’re unlucky to run into some pathological hashing collision. So O(n) in all likelihood.

Of course the asymptotic behaviour of /= for large numbers of arguments is unlikely to form part of your decision to use one Lisp system or another, but it makes a nice puzzle. Unsurprisingly, the practice of using more than 2 arguments is very rare. Lisp test suites, that sort of thing. The best example I found is:

which is true when `samples` is 3 or 4. This seems obscure to me, but it is more compact than alternatives; maybe it is idiomatic in some circles.