Program optimisation , naturally

One of the first things that any functional programmer learns is how to turn the naive quadratic algorithm for reversing a list into a linear-time algorithm by using an accumulating parameter. In this paper we derive another, quite different, linear-time algorithm for reversing a list. The derivation relies on a higherorder naturality [5] property of the function unzip, the function that turns a list of pairs into a pair of lists in the obvious way. The final program uses polymorphic recursion [10], and can be expressed rather neatly using nested datatypes [2]. Although neither feature is an essential component of the fast algorithm, they do simplify it, providing more evidence of the practical utility of these two concepts. Hinze [4] presents another application of these techniques, namely to the problem of computing the bit-reversal permutation of a list. We describe this problem and its solution briefly in Section 6. Indeed, it was Hinze’s application that inspired us to write this paper in the first place. We outline another application in Section 7. Whether or not the technique has wider application remains to be seen.