# Functional Pearl: Perfect trees and bit−reversal permutations

*Ralf Hinze*

### Abstract

A famous algorithm is the Fast Fourier Transform, or FFT. An efficient iterative version of the FFT algorithm performs as a first step a bit-reversal permutation of the input list. The bit-reversal permutation swaps elements whose indices have binary representations that are the reverse of each other. Using an amortized approach this operation can be made to run in linear time on a random-access machine. An intriguing question is whether a linear-time implementation is also feasible on a pointer machine, that is in a purely functional setting. We show that the answer to this question is in the affirmative. In deriving a solution we employ several advanced programming language concepts such as nested datatypes, associated fold and unfold operators, rank-2 types, and polymorphic recursion.