Introduction

The Burrows–Wheeler transform (BWT) is a method for permuting a list with the aim of bringing repeated elements together. Its main use is as a preprocessing step in data compression. Lists with many repeated adjacent elements can be encoded compactly using simple schemes such as run length or move-to-front encoding. The result can then be fed into more advanced compressors, such as Huffman or arithmetic coding, to compress the input even more.

Clearly, the best way of bringing repeated elements together is just to sort the list. But the idea has a major flaw as a preliminary to compression: there is no way to recover the original list unless the complete sorting permutation is also produced as part of the output. Without the ability to recover the original input, data compression is pointless; and if a permutation has to be produced as well, then compression is ineffective. Instead, the BWT achieves a more modest permutation, one that brings some but not all repeated elements into adjacent positions. The main advantage of the BWT is that the transform can be inverted using a single additional piece of information, namely an integer k in the range 0 ≤ k < n, where n is the length of the (nonempty) input list. In this pearl we describe the BWT, identify the fundamental reason why inversion is possible, and use it to derive the inverse transform from its specification.